| ofs | hex dump | ascii |
|---|---|---|
| 0000 | cb 0d 0d 0a 00 00 00 00 0d fd a7 68 b9 c7 00 00 e3 00 00 00 00 00 00 00 00 00 00 00 00 05 00 00 | ...........h.................... |
| 0020 | 00 00 00 00 00 f3 bc 01 00 00 97 00 64 00 5a 00 64 01 64 02 6c 01 5a 02 64 01 64 02 6c 03 6d 04 | ............d.Z.d.d.l.Z.d.d.l.m. |
| 0040 | 5a 05 01 00 64 01 64 03 6c 06 6d 07 5a 07 01 00 64 04 64 05 6c 08 6d 09 5a 0a 01 00 64 04 64 06 | Z...d.d.l.m.Z...d.d.l.m.Z...d.d. |
| 0060 | 6c 0b 6d 0c 5a 0c 01 00 67 00 64 07 a2 01 5a 0d 65 0a 6a 1c 00 00 00 00 00 00 00 00 00 00 00 00 | l.m.Z...g.d...Z.e.j............. |
| 0080 | 00 00 00 00 00 00 5a 0f 64 08 84 00 5a 10 64 09 84 00 5a 11 02 00 65 02 6a 24 00 00 00 00 00 00 | ......Z.d...Z.d...Z...e.j$...... |
| 00a0 | 00 00 00 00 00 00 00 00 00 00 00 00 64 0a 64 0b 67 02 ab 01 00 00 00 00 00 00 5a 13 02 00 65 02 | ............d.d.g.........Z...e. |
| 00c0 | 6a 24 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 64 01 67 01 ab 01 00 00 00 00 00 00 | j$..................d.g......... |
| 00e0 | 5a 14 02 00 65 02 6a 24 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 64 04 67 01 ab 01 | Z...e.j$..................d.g... |
| 0100 | 00 00 00 00 00 00 5a 15 02 00 65 02 6a 24 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 | ......Z...e.j$.................. |
| 0120 | 64 01 64 04 67 02 ab 01 00 00 00 00 00 00 5a 16 64 0c 84 00 5a 17 64 0d 84 00 5a 18 64 0e 84 00 | d.d.g.........Z.d...Z.d...Z.d... |
| 0140 | 5a 19 64 0f 84 00 5a 1a 64 10 84 00 5a 1b 64 11 84 00 5a 1c 64 12 84 00 5a 1d 64 25 64 13 84 01 | Z.d...Z.d...Z.d...Z.d...Z.d%d... |
| 0160 | 5a 1e 64 26 64 14 84 01 5a 1f 64 04 67 00 64 01 64 04 64 01 66 05 64 15 84 01 5a 20 64 27 64 16 | Z.d&d...Z.d.g.d.d.d.f.d...Z.d'd. |
| 0180 | 84 01 5a 21 64 17 84 00 5a 22 64 18 84 00 5a 23 64 19 84 00 5a 24 64 1a 84 00 5a 25 64 1b 84 00 | ..Z!d...Z"d...Z#d...Z$d...Z%d... |
| 01a0 | 5a 26 64 1c 84 00 5a 27 64 1d 84 00 5a 28 64 28 64 1e 84 01 5a 29 64 1f 84 00 5a 2a 64 20 84 00 | Z&d...Z'd...Z(d(d...Z)d...Z*d... |
| 01c0 | 5a 2b 64 21 84 00 5a 2c 64 22 84 00 5a 2d 02 00 47 00 64 23 84 00 64 24 65 0c ab 03 00 00 00 00 | Z+d!..Z,d"..Z-..G.d#..d$e....... |
| 01e0 | 00 00 5a 2e 79 02 29 29 61 c1 04 00 00 0a 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d | ..Z.y.))a.....================== |
| 0200 | 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d | ================================ |
| 0220 | 0a 4c 65 67 65 6e 64 72 65 20 53 65 72 69 65 73 20 28 3a 6d 6f 64 3a 60 6e 75 6d 70 79 2e 70 6f | .Legendre.Series.(:mod:`numpy.po |
| 0240 | 6c 79 6e 6f 6d 69 61 6c 2e 6c 65 67 65 6e 64 72 65 60 29 0a 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d | lynomial.legendre`).============ |
| 0260 | 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d 3d | ================================ |
| 0280 | 3d 3d 3d 3d 3d 3d 0a 0a 54 68 69 73 20 6d 6f 64 75 6c 65 20 70 72 6f 76 69 64 65 73 20 61 20 6e | ======..This.module.provides.a.n |
| 02a0 | 75 6d 62 65 72 20 6f 66 20 6f 62 6a 65 63 74 73 20 28 6d 6f 73 74 6c 79 20 66 75 6e 63 74 69 6f | umber.of.objects.(mostly.functio |
| 02c0 | 6e 73 29 20 75 73 65 66 75 6c 20 66 6f 72 0a 64 65 61 6c 69 6e 67 20 77 69 74 68 20 4c 65 67 65 | ns).useful.for.dealing.with.Lege |
| 02e0 | 6e 64 72 65 20 73 65 72 69 65 73 2c 20 69 6e 63 6c 75 64 69 6e 67 20 61 20 60 4c 65 67 65 6e 64 | ndre.series,.including.a.`Legend |
| 0300 | 72 65 60 20 63 6c 61 73 73 20 74 68 61 74 0a 65 6e 63 61 70 73 75 6c 61 74 65 73 20 74 68 65 20 | re`.class.that.encapsulates.the. |
| 0320 | 75 73 75 61 6c 20 61 72 69 74 68 6d 65 74 69 63 20 6f 70 65 72 61 74 69 6f 6e 73 2e 20 20 28 47 | usual.arithmetic.operations...(G |
| 0340 | 65 6e 65 72 61 6c 20 69 6e 66 6f 72 6d 61 74 69 6f 6e 0a 6f 6e 20 68 6f 77 20 74 68 69 73 20 6d | eneral.information.on.how.this.m |
| 0360 | 6f 64 75 6c 65 20 72 65 70 72 65 73 65 6e 74 73 20 61 6e 64 20 77 6f 72 6b 73 20 77 69 74 68 20 | odule.represents.and.works.with. |
| 0380 | 73 75 63 68 20 70 6f 6c 79 6e 6f 6d 69 61 6c 73 20 69 73 20 69 6e 20 74 68 65 0a 64 6f 63 73 74 | such.polynomials.is.in.the.docst |
| 03a0 | 72 69 6e 67 20 66 6f 72 20 69 74 73 20 22 70 61 72 65 6e 74 22 20 73 75 62 2d 70 61 63 6b 61 67 | ring.for.its."parent".sub-packag |
| 03c0 | 65 2c 20 60 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 60 29 2e 0a 0a 43 6c 61 73 73 65 73 | e,.`numpy.polynomial`)...Classes |
| 03e0 | 0a 2d 2d 2d 2d 2d 2d 2d 0a 2e 2e 20 61 75 74 6f 73 75 6d 6d 61 72 79 3a 3a 0a 20 20 20 3a 74 6f | .-------....autosummary::....:to |
| 0400 | 63 74 72 65 65 3a 20 67 65 6e 65 72 61 74 65 64 2f 0a 0a 20 20 20 20 4c 65 67 65 6e 64 72 65 0a | ctree:.generated/......Legendre. |
| 0420 | 0a 43 6f 6e 73 74 61 6e 74 73 0a 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 0a 2e 2e 20 61 75 74 6f 73 75 6d | .Constants.---------.....autosum |
| 0440 | 6d 61 72 79 3a 3a 0a 20 20 20 3a 74 6f 63 74 72 65 65 3a 20 67 65 6e 65 72 61 74 65 64 2f 0a 0a | mary::....:toctree:.generated/.. |
| 0460 | 20 20 20 6c 65 67 64 6f 6d 61 69 6e 0a 20 20 20 6c 65 67 7a 65 72 6f 0a 20 20 20 6c 65 67 6f 6e | ...legdomain....legzero....legon |
| 0480 | 65 0a 20 20 20 6c 65 67 78 0a 0a 41 72 69 74 68 6d 65 74 69 63 0a 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d | e....legx..Arithmetic.---------- |
| 04a0 | 0a 0a 2e 2e 20 61 75 74 6f 73 75 6d 6d 61 72 79 3a 3a 0a 20 20 20 3a 74 6f 63 74 72 65 65 3a 20 | .....autosummary::....:toctree:. |
| 04c0 | 67 65 6e 65 72 61 74 65 64 2f 0a 0a 20 20 20 6c 65 67 61 64 64 0a 20 20 20 6c 65 67 73 75 62 0a | generated/.....legadd....legsub. |
| 04e0 | 20 20 20 6c 65 67 6d 75 6c 78 0a 20 20 20 6c 65 67 6d 75 6c 0a 20 20 20 6c 65 67 64 69 76 0a 20 | ...legmulx....legmul....legdiv.. |
| 0500 | 20 20 6c 65 67 70 6f 77 0a 20 20 20 6c 65 67 76 61 6c 0a 20 20 20 6c 65 67 76 61 6c 32 64 0a 20 | ..legpow....legval....legval2d.. |
| 0520 | 20 20 6c 65 67 76 61 6c 33 64 0a 20 20 20 6c 65 67 67 72 69 64 32 64 0a 20 20 20 6c 65 67 67 72 | ..legval3d....leggrid2d....leggr |
| 0540 | 69 64 33 64 0a 0a 43 61 6c 63 75 6c 75 73 0a 2d 2d 2d 2d 2d 2d 2d 2d 0a 0a 2e 2e 20 61 75 74 6f | id3d..Calculus.--------.....auto |
| 0560 | 73 75 6d 6d 61 72 79 3a 3a 0a 20 20 20 3a 74 6f 63 74 72 65 65 3a 20 67 65 6e 65 72 61 74 65 64 | summary::....:toctree:.generated |
| 0580 | 2f 0a 0a 20 20 20 6c 65 67 64 65 72 0a 20 20 20 6c 65 67 69 6e 74 0a 0a 4d 69 73 63 20 46 75 6e | /.....legder....legint..Misc.Fun |
| 05a0 | 63 74 69 6f 6e 73 0a 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 0a 2e 2e 20 61 75 74 6f 73 75 | ctions.--------------.....autosu |
| 05c0 | 6d 6d 61 72 79 3a 3a 0a 20 20 20 3a 74 6f 63 74 72 65 65 3a 20 67 65 6e 65 72 61 74 65 64 2f 0a | mmary::....:toctree:.generated/. |
| 05e0 | 0a 20 20 20 6c 65 67 66 72 6f 6d 72 6f 6f 74 73 0a 20 20 20 6c 65 67 72 6f 6f 74 73 0a 20 20 20 | ....legfromroots....legroots.... |
| 0600 | 6c 65 67 76 61 6e 64 65 72 0a 20 20 20 6c 65 67 76 61 6e 64 65 72 32 64 0a 20 20 20 6c 65 67 76 | legvander....legvander2d....legv |
| 0620 | 61 6e 64 65 72 33 64 0a 20 20 20 6c 65 67 67 61 75 73 73 0a 20 20 20 6c 65 67 77 65 69 67 68 74 | ander3d....leggauss....legweight |
| 0640 | 0a 20 20 20 6c 65 67 63 6f 6d 70 61 6e 69 6f 6e 0a 20 20 20 6c 65 67 66 69 74 0a 20 20 20 6c 65 | ....legcompanion....legfit....le |
| 0660 | 67 74 72 69 6d 0a 20 20 20 6c 65 67 6c 69 6e 65 0a 20 20 20 6c 65 67 32 70 6f 6c 79 0a 20 20 20 | gtrim....legline....leg2poly.... |
| 0680 | 70 6f 6c 79 32 6c 65 67 0a 0a 53 65 65 20 61 6c 73 6f 0a 2d 2d 2d 2d 2d 2d 2d 2d 0a 6e 75 6d 70 | poly2leg..See.also.--------.nump |
| 06a0 | 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 0a 0a e9 00 00 00 00 4e 29 01 da 14 6e 6f 72 6d 61 6c 69 7a | y.polynomial.......N)...normaliz |
| 06c0 | 65 5f 61 78 69 73 5f 69 6e 64 65 78 e9 01 00 00 00 29 01 da 09 70 6f 6c 79 75 74 69 6c 73 29 01 | e_axis_index.....)...polyutils). |
| 06e0 | da 0b 41 42 43 50 6f 6c 79 42 61 73 65 29 1f da 07 6c 65 67 7a 65 72 6f da 06 6c 65 67 6f 6e 65 | ..ABCPolyBase)...legzero..legone |
| 0700 | da 04 6c 65 67 78 da 09 6c 65 67 64 6f 6d 61 69 6e da 07 6c 65 67 6c 69 6e 65 da 06 6c 65 67 61 | ..legx..legdomain..legline..lega |
| 0720 | 64 64 da 06 6c 65 67 73 75 62 da 07 6c 65 67 6d 75 6c 78 da 06 6c 65 67 6d 75 6c da 06 6c 65 67 | dd..legsub..legmulx..legmul..leg |
| 0740 | 64 69 76 da 06 6c 65 67 70 6f 77 da 06 6c 65 67 76 61 6c da 06 6c 65 67 64 65 72 da 06 6c 65 67 | div..legpow..legval..legder..leg |
| 0760 | 69 6e 74 da 08 6c 65 67 32 70 6f 6c 79 da 08 70 6f 6c 79 32 6c 65 67 da 0c 6c 65 67 66 72 6f 6d | int..leg2poly..poly2leg..legfrom |
| 0780 | 72 6f 6f 74 73 da 09 6c 65 67 76 61 6e 64 65 72 da 06 6c 65 67 66 69 74 da 07 6c 65 67 74 72 69 | roots..legvander..legfit..legtri |
| 07a0 | 6d da 08 6c 65 67 72 6f 6f 74 73 da 08 4c 65 67 65 6e 64 72 65 da 08 6c 65 67 76 61 6c 32 64 da | m..legroots..Legendre..legval2d. |
| 07c0 | 08 6c 65 67 76 61 6c 33 64 da 09 6c 65 67 67 72 69 64 32 64 da 09 6c 65 67 67 72 69 64 33 64 da | .legval3d..leggrid2d..leggrid3d. |
| 07e0 | 0b 6c 65 67 76 61 6e 64 65 72 32 64 da 0b 6c 65 67 76 61 6e 64 65 72 33 64 da 0c 6c 65 67 63 6f | .legvander2d..legvander3d..legco |
| 0800 | 6d 70 61 6e 69 6f 6e da 08 6c 65 67 67 61 75 73 73 da 09 6c 65 67 77 65 69 67 68 74 63 01 00 00 | mpanion..leggauss..legweightc... |
| 0820 | 00 00 00 00 00 00 00 00 00 06 00 00 00 03 00 00 00 f3 aa 00 00 00 97 00 74 01 00 00 00 00 00 00 | ........................t....... |
| 0840 | 00 00 6a 02 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 00 67 01 ab 01 00 00 00 00 | ..j...................|.g....... |
| 0860 | 00 00 5c 01 00 00 7d 00 74 05 00 00 00 00 00 00 00 00 7c 00 ab 01 00 00 00 00 00 00 64 01 7a 0a | ..\...}.t.........|.........d.z. |
| 0880 | 00 00 7d 01 64 02 7d 02 74 07 00 00 00 00 00 00 00 00 7c 01 64 03 64 03 ab 03 00 00 00 00 00 00 | ..}.d.}.t.........|.d.d......... |
| 08a0 | 44 00 5d 1a 00 00 7d 03 74 09 00 00 00 00 00 00 00 00 74 0b 00 00 00 00 00 00 00 00 7c 02 ab 01 | D.]...}.t.........t.........|... |
| 08c0 | 00 00 00 00 00 00 7c 00 7c 03 19 00 00 00 ab 02 00 00 00 00 00 00 7d 02 8c 1c 04 00 7c 02 53 00 | ......|.|.............}.....|.S. |
| 08e0 | 29 04 61 51 04 00 00 0a 20 20 20 20 43 6f 6e 76 65 72 74 20 61 20 70 6f 6c 79 6e 6f 6d 69 61 6c | ).aQ........Convert.a.polynomial |
| 0900 | 20 74 6f 20 61 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 2e 0a 0a 20 20 20 20 43 6f 6e 76 | .to.a.Legendre.series.......Conv |
| 0920 | 65 72 74 20 61 6e 20 61 72 72 61 79 20 72 65 70 72 65 73 65 6e 74 69 6e 67 20 74 68 65 20 63 6f | ert.an.array.representing.the.co |
| 0940 | 65 66 66 69 63 69 65 6e 74 73 20 6f 66 20 61 20 70 6f 6c 79 6e 6f 6d 69 61 6c 20 28 72 65 6c 61 | efficients.of.a.polynomial.(rela |
| 0960 | 74 69 76 65 0a 20 20 20 20 74 6f 20 74 68 65 20 22 73 74 61 6e 64 61 72 64 22 20 62 61 73 69 73 | tive.....to.the."standard".basis |
| 0980 | 29 20 6f 72 64 65 72 65 64 20 66 72 6f 6d 20 6c 6f 77 65 73 74 20 64 65 67 72 65 65 20 74 6f 20 | ).ordered.from.lowest.degree.to. |
| 09a0 | 68 69 67 68 65 73 74 2c 20 74 6f 20 61 6e 0a 20 20 20 20 61 72 72 61 79 20 6f 66 20 74 68 65 20 | highest,.to.an.....array.of.the. |
| 09c0 | 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6f 66 20 74 68 65 20 65 71 75 69 76 61 6c 65 6e 74 20 4c | coefficients.of.the.equivalent.L |
| 09e0 | 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 2c 20 6f 72 64 65 72 65 64 0a 20 20 20 20 66 72 6f 6d | egendre.series,.ordered.....from |
| 0a00 | 20 6c 6f 77 65 73 74 20 74 6f 20 68 69 67 68 65 73 74 20 64 65 67 72 65 65 2e 0a 0a 20 20 20 20 | .lowest.to.highest.degree....... |
| 0a20 | 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 70 6f | Parameters.....----------.....po |
| 0a40 | 6c 20 3a 20 61 72 72 61 79 5f 6c 69 6b 65 0a 20 20 20 20 20 20 20 20 31 2d 44 20 61 72 72 61 79 | l.:.array_like.........1-D.array |
| 0a60 | 20 63 6f 6e 74 61 69 6e 69 6e 67 20 74 68 65 20 70 6f 6c 79 6e 6f 6d 69 61 6c 20 63 6f 65 66 66 | .containing.the.polynomial.coeff |
| 0a80 | 69 63 69 65 6e 74 73 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d | icients......Returns.....------- |
| 0aa0 | 0a 20 20 20 20 63 20 3a 20 6e 64 61 72 72 61 79 0a 20 20 20 20 20 20 20 20 31 2d 44 20 61 72 72 | .....c.:.ndarray.........1-D.arr |
| 0ac0 | 61 79 20 63 6f 6e 74 61 69 6e 69 6e 67 20 74 68 65 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6f | ay.containing.the.coefficients.o |
| 0ae0 | 66 20 74 68 65 20 65 71 75 69 76 61 6c 65 6e 74 20 4c 65 67 65 6e 64 72 65 0a 20 20 20 20 20 20 | f.the.equivalent.Legendre....... |
| 0b00 | 20 20 73 65 72 69 65 73 2e 0a 0a 20 20 20 20 53 65 65 20 41 6c 73 6f 0a 20 20 20 20 2d 2d 2d 2d | ..series.......See.Also.....---- |
| 0b20 | 2d 2d 2d 2d 0a 20 20 20 20 6c 65 67 32 70 6f 6c 79 0a 0a 20 20 20 20 4e 6f 74 65 73 0a 20 20 20 | ----.....leg2poly......Notes.... |
| 0b40 | 20 2d 2d 2d 2d 2d 0a 20 20 20 20 54 68 65 20 65 61 73 79 20 77 61 79 20 74 6f 20 64 6f 20 63 6f | .-----.....The.easy.way.to.do.co |
| 0b60 | 6e 76 65 72 73 69 6f 6e 73 20 62 65 74 77 65 65 6e 20 70 6f 6c 79 6e 6f 6d 69 61 6c 20 62 61 73 | nversions.between.polynomial.bas |
| 0b80 | 69 73 20 73 65 74 73 0a 20 20 20 20 69 73 20 74 6f 20 75 73 65 20 74 68 65 20 63 6f 6e 76 65 72 | is.sets.....is.to.use.the.conver |
| 0ba0 | 74 20 6d 65 74 68 6f 64 20 6f 66 20 61 20 63 6c 61 73 73 20 69 6e 73 74 61 6e 63 65 2e 0a 0a 20 | t.method.of.a.class.instance.... |
| 0bc0 | 20 20 20 45 78 61 6d 70 6c 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 3e 3e 3e | ...Examples.....--------.....>>> |
| 0be0 | 20 69 6d 70 6f 72 74 20 6e 75 6d 70 79 20 61 73 20 6e 70 0a 20 20 20 20 3e 3e 3e 20 66 72 6f 6d | .import.numpy.as.np.....>>>.from |
| 0c00 | 20 6e 75 6d 70 79 20 69 6d 70 6f 72 74 20 70 6f 6c 79 6e 6f 6d 69 61 6c 20 61 73 20 50 0a 20 20 | .numpy.import.polynomial.as.P... |
| 0c20 | 20 20 3e 3e 3e 20 70 20 3d 20 50 2e 50 6f 6c 79 6e 6f 6d 69 61 6c 28 6e 70 2e 61 72 61 6e 67 65 | ..>>>.p.=.P.Polynomial(np.arange |
| 0c40 | 28 34 29 29 0a 20 20 20 20 3e 3e 3e 20 70 0a 20 20 20 20 50 6f 6c 79 6e 6f 6d 69 61 6c 28 5b 30 | (4)).....>>>.p.....Polynomial([0 |
| 0c60 | 2e 2c 20 20 31 2e 2c 20 20 32 2e 2c 20 20 33 2e 5d 2c 20 64 6f 6d 61 69 6e 3d 5b 2d 31 2e 2c 20 | .,..1.,..2.,..3.],.domain=[-1.,. |
| 0c80 | 20 31 2e 5d 2c 20 77 69 6e 64 6f 77 3d 5b 2d 31 2e 2c 20 20 31 2e 5d 2c 20 2e 2e 2e 0a 20 20 20 | .1.],.window=[-1.,..1.],........ |
| 0ca0 | 20 3e 3e 3e 20 63 20 3d 20 50 2e 4c 65 67 65 6e 64 72 65 28 50 2e 6c 65 67 65 6e 64 72 65 2e 70 | .>>>.c.=.P.Legendre(P.legendre.p |
| 0cc0 | 6f 6c 79 32 6c 65 67 28 70 2e 63 6f 65 66 29 29 0a 20 20 20 20 3e 3e 3e 20 63 0a 20 20 20 20 4c | oly2leg(p.coef)).....>>>.c.....L |
| 0ce0 | 65 67 65 6e 64 72 65 28 5b 20 31 2e 20 20 2c 20 20 33 2e 32 35 2c 20 20 31 2e 20 20 2c 20 20 30 | egendre([.1...,..3.25,..1...,..0 |
| 0d00 | 2e 37 35 5d 2c 20 64 6f 6d 61 69 6e 3d 5b 2d 31 2c 20 20 31 5d 2c 20 77 69 6e 64 6f 77 3d 5b 2d | .75],.domain=[-1,..1],.window=[- |
| 0d20 | 31 2c 20 20 31 5d 29 20 23 20 6d 61 79 20 76 61 72 79 0a 0a 20 20 20 20 72 04 00 00 00 72 02 00 | 1,..1]).#.may.vary......r....r.. |
| 0d40 | 00 00 e9 ff ff ff ff 29 06 da 02 70 75 da 09 61 73 5f 73 65 72 69 65 73 da 03 6c 65 6e da 05 72 | .......)...pu..as_series..len..r |
| 0d60 | 61 6e 67 65 72 0c 00 00 00 72 0e 00 00 00 29 04 da 03 70 6f 6c da 03 64 65 67 da 03 72 65 73 da | anger....r....)...pol..deg..res. |
| 0d80 | 01 69 73 04 00 00 00 20 20 20 20 fa 60 2f 68 6f 6d 65 2f 62 6c 61 63 6b 68 61 6f 2f 75 69 75 63 | .is.........`/home/blackhao/uiuc |
| 0da0 | 2d 63 6f 75 72 73 65 2d 67 72 61 70 68 2f 2e 76 65 6e 76 2f 6c 69 62 2f 70 79 74 68 6f 6e 33 2e | -course-graph/.venv/lib/python3. |
| 0dc0 | 31 32 2f 73 69 74 65 2d 70 61 63 6b 61 67 65 73 2f 6e 75 6d 70 79 2f 70 6f 6c 79 6e 6f 6d 69 61 | 12/site-packages/numpy/polynomia |
| 0de0 | 6c 2f 6c 65 67 65 6e 64 72 65 2e 70 79 72 16 00 00 00 72 16 00 00 00 64 00 00 00 73 5a 00 00 00 | l/legendre.pyr....r....d...sZ... |
| 0e00 | 80 00 f4 52 01 00 0d 0f 8f 4c 89 4c 98 23 98 15 d3 0c 1f 81 45 80 53 dc 0a 0d 88 63 8b 28 90 51 | ...R.....L.L.#......E.S....c.(.Q |
| 0e20 | 89 2c 80 43 d8 0a 0b 80 43 dc 0d 12 90 33 98 02 98 42 d3 0d 1f f2 00 01 05 2b 88 01 dc 0e 14 94 | .,.C....C....3...B.......+...... |
| 0e40 | 57 98 53 93 5c a0 33 a0 71 a1 36 d3 0e 2a 89 03 f0 03 01 05 2b e0 0b 0e 80 4a f3 00 00 00 00 63 | W.S.\.3.q.6..*......+....J.....c |
| 0e60 | 01 00 00 00 00 00 00 00 00 00 00 00 07 00 00 00 03 00 00 00 f3 30 01 00 00 97 00 64 01 64 02 6c | .....................0.....d.d.l |
| 0e80 | 00 6d 01 7d 01 6d 02 7d 02 6d 03 7d 03 01 00 74 09 00 00 00 00 00 00 00 00 6a 0a 00 00 00 00 00 | .m.}.m.}.m.}...t.........j...... |
| 0ea0 | 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 00 67 01 ab 01 00 00 00 00 00 00 5c 01 00 00 7d 00 74 | .............|.g.........\...}.t |
| 0ec0 | 0d 00 00 00 00 00 00 00 00 7c 00 ab 01 00 00 00 00 00 00 7d 04 7c 04 64 03 6b 02 00 00 72 02 7c | .........|.........}.|.d.k...r.| |
| 0ee0 | 00 53 00 7c 00 64 04 19 00 00 00 7d 05 7c 00 64 05 19 00 00 00 7d 06 74 0f 00 00 00 00 00 00 00 | .S.|.d.....}.|.d.....}.t........ |
| 0f00 | 00 7c 04 64 01 7a 0a 00 00 64 01 64 05 ab 03 00 00 00 00 00 00 44 00 5d 37 00 00 7d 07 7c 05 7d | .|.d.z...d.d.........D.]7..}.|.} |
| 0f20 | 08 02 00 7c 03 7c 00 7c 07 64 06 7a 0a 00 00 19 00 00 00 7c 06 7c 07 64 01 7a 0a 00 00 7a 05 00 | ...|.|.|.d.z.......|.|.d.z...z.. |
| 0f40 | 00 7c 07 7a 0b 00 00 ab 02 00 00 00 00 00 00 7d 05 02 00 7c 01 7c 08 02 00 7c 02 7c 06 ab 01 00 | .|.z...........}...|.|...|.|.... |
| 0f60 | 00 00 00 00 00 64 06 7c 07 7a 05 00 00 64 01 7a 0a 00 00 7a 05 00 00 7c 07 7a 0b 00 00 ab 02 00 | .....d.|.z...d.z...z...|.z...... |
| 0f80 | 00 00 00 00 00 7d 06 8c 39 04 00 02 00 7c 01 7c 05 02 00 7c 02 7c 06 ab 01 00 00 00 00 00 00 ab | .....}..9....|.|...|.|.......... |
| 0fa0 | 02 00 00 00 00 00 00 53 00 29 07 61 f1 04 00 00 0a 20 20 20 20 43 6f 6e 76 65 72 74 20 61 20 4c | .......S.).a.........Convert.a.L |
| 0fc0 | 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 74 6f 20 61 20 70 6f 6c 79 6e 6f 6d 69 61 6c 2e 0a | egendre.series.to.a.polynomial.. |
| 0fe0 | 0a 20 20 20 20 43 6f 6e 76 65 72 74 20 61 6e 20 61 72 72 61 79 20 72 65 70 72 65 73 65 6e 74 69 | .....Convert.an.array.representi |
| 1000 | 6e 67 20 74 68 65 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6f 66 20 61 20 4c 65 67 65 6e 64 72 | ng.the.coefficients.of.a.Legendr |
| 1020 | 65 20 73 65 72 69 65 73 2c 0a 20 20 20 20 6f 72 64 65 72 65 64 20 66 72 6f 6d 20 6c 6f 77 65 73 | e.series,.....ordered.from.lowes |
| 1040 | 74 20 64 65 67 72 65 65 20 74 6f 20 68 69 67 68 65 73 74 2c 20 74 6f 20 61 6e 20 61 72 72 61 79 | t.degree.to.highest,.to.an.array |
| 1060 | 20 6f 66 20 74 68 65 20 63 6f 65 66 66 69 63 69 65 6e 74 73 0a 20 20 20 20 6f 66 20 74 68 65 20 | .of.the.coefficients.....of.the. |
| 1080 | 65 71 75 69 76 61 6c 65 6e 74 20 70 6f 6c 79 6e 6f 6d 69 61 6c 20 28 72 65 6c 61 74 69 76 65 20 | equivalent.polynomial.(relative. |
| 10a0 | 74 6f 20 74 68 65 20 22 73 74 61 6e 64 61 72 64 22 20 62 61 73 69 73 29 20 6f 72 64 65 72 65 64 | to.the."standard".basis).ordered |
| 10c0 | 0a 20 20 20 20 66 72 6f 6d 20 6c 6f 77 65 73 74 20 74 6f 20 68 69 67 68 65 73 74 20 64 65 67 72 | .....from.lowest.to.highest.degr |
| 10e0 | 65 65 2e 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d | ee.......Parameters.....-------- |
| 1100 | 2d 2d 0a 20 20 20 20 63 20 3a 20 61 72 72 61 79 5f 6c 69 6b 65 0a 20 20 20 20 20 20 20 20 31 2d | --.....c.:.array_like.........1- |
| 1120 | 44 20 61 72 72 61 79 20 63 6f 6e 74 61 69 6e 69 6e 67 20 74 68 65 20 4c 65 67 65 6e 64 72 65 20 | D.array.containing.the.Legendre. |
| 1140 | 73 65 72 69 65 73 20 63 6f 65 66 66 69 63 69 65 6e 74 73 2c 20 6f 72 64 65 72 65 64 0a 20 20 20 | series.coefficients,.ordered.... |
| 1160 | 20 20 20 20 20 66 72 6f 6d 20 6c 6f 77 65 73 74 20 6f 72 64 65 72 20 74 65 72 6d 20 74 6f 20 68 | .....from.lowest.order.term.to.h |
| 1180 | 69 67 68 65 73 74 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d | ighest.......Returns.....------- |
| 11a0 | 0a 20 20 20 20 70 6f 6c 20 3a 20 6e 64 61 72 72 61 79 0a 20 20 20 20 20 20 20 20 31 2d 44 20 61 | .....pol.:.ndarray.........1-D.a |
| 11c0 | 72 72 61 79 20 63 6f 6e 74 61 69 6e 69 6e 67 20 74 68 65 20 63 6f 65 66 66 69 63 69 65 6e 74 73 | rray.containing.the.coefficients |
| 11e0 | 20 6f 66 20 74 68 65 20 65 71 75 69 76 61 6c 65 6e 74 20 70 6f 6c 79 6e 6f 6d 69 61 6c 0a 20 20 | .of.the.equivalent.polynomial... |
| 1200 | 20 20 20 20 20 20 28 72 65 6c 61 74 69 76 65 20 74 6f 20 74 68 65 20 22 73 74 61 6e 64 61 72 64 | ......(relative.to.the."standard |
| 1220 | 22 20 62 61 73 69 73 29 20 6f 72 64 65 72 65 64 20 66 72 6f 6d 20 6c 6f 77 65 73 74 20 6f 72 64 | ".basis).ordered.from.lowest.ord |
| 1240 | 65 72 20 74 65 72 6d 0a 20 20 20 20 20 20 20 20 74 6f 20 68 69 67 68 65 73 74 2e 0a 0a 20 20 20 | er.term.........to.highest...... |
| 1260 | 20 53 65 65 20 41 6c 73 6f 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 70 6f 6c 79 32 | .See.Also.....--------.....poly2 |
| 1280 | 6c 65 67 0a 0a 20 20 20 20 4e 6f 74 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 0a 20 20 20 20 54 68 65 | leg......Notes.....-----.....The |
| 12a0 | 20 65 61 73 79 20 77 61 79 20 74 6f 20 64 6f 20 63 6f 6e 76 65 72 73 69 6f 6e 73 20 62 65 74 77 | .easy.way.to.do.conversions.betw |
| 12c0 | 65 65 6e 20 70 6f 6c 79 6e 6f 6d 69 61 6c 20 62 61 73 69 73 20 73 65 74 73 0a 20 20 20 20 69 73 | een.polynomial.basis.sets.....is |
| 12e0 | 20 74 6f 20 75 73 65 20 74 68 65 20 63 6f 6e 76 65 72 74 20 6d 65 74 68 6f 64 20 6f 66 20 61 20 | .to.use.the.convert.method.of.a. |
| 1300 | 63 6c 61 73 73 20 69 6e 73 74 61 6e 63 65 2e 0a 0a 20 20 20 20 45 78 61 6d 70 6c 65 73 0a 20 20 | class.instance.......Examples... |
| 1320 | 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 3e 3e 3e 20 66 72 6f 6d 20 6e 75 6d 70 79 20 69 6d | ..--------.....>>>.from.numpy.im |
| 1340 | 70 6f 72 74 20 70 6f 6c 79 6e 6f 6d 69 61 6c 20 61 73 20 50 0a 20 20 20 20 3e 3e 3e 20 63 20 3d | port.polynomial.as.P.....>>>.c.= |
| 1360 | 20 50 2e 4c 65 67 65 6e 64 72 65 28 72 61 6e 67 65 28 34 29 29 0a 20 20 20 20 3e 3e 3e 20 63 0a | .P.Legendre(range(4)).....>>>.c. |
| 1380 | 20 20 20 20 4c 65 67 65 6e 64 72 65 28 5b 30 2e 2c 20 31 2e 2c 20 32 2e 2c 20 33 2e 5d 2c 20 64 | ....Legendre([0.,.1.,.2.,.3.],.d |
| 13a0 | 6f 6d 61 69 6e 3d 5b 2d 31 2e 2c 20 20 31 2e 5d 2c 20 77 69 6e 64 6f 77 3d 5b 2d 31 2e 2c 20 20 | omain=[-1.,..1.],.window=[-1.,.. |
| 13c0 | 31 2e 5d 2c 20 73 79 6d 62 6f 6c 3d 27 78 27 29 0a 20 20 20 20 3e 3e 3e 20 70 20 3d 20 63 2e 63 | 1.],.symbol='x').....>>>.p.=.c.c |
| 13e0 | 6f 6e 76 65 72 74 28 6b 69 6e 64 3d 50 2e 50 6f 6c 79 6e 6f 6d 69 61 6c 29 0a 20 20 20 20 3e 3e | onvert(kind=P.Polynomial).....>> |
| 1400 | 3e 20 70 0a 20 20 20 20 50 6f 6c 79 6e 6f 6d 69 61 6c 28 5b 2d 31 2e 20 2c 20 2d 33 2e 35 2c 20 | >.p.....Polynomial([-1..,.-3.5,. |
| 1420 | 20 33 2e 20 2c 20 20 37 2e 35 5d 2c 20 64 6f 6d 61 69 6e 3d 5b 2d 31 2e 2c 20 20 31 2e 5d 2c 20 | .3..,..7.5],.domain=[-1.,..1.],. |
| 1440 | 77 69 6e 64 6f 77 3d 5b 2d 31 2e 2c 20 2e 2e 2e 0a 20 20 20 20 3e 3e 3e 20 50 2e 6c 65 67 65 6e | window=[-1.,.........>>>.P.legen |
| 1460 | 64 72 65 2e 6c 65 67 32 70 6f 6c 79 28 72 61 6e 67 65 28 34 29 29 0a 20 20 20 20 61 72 72 61 79 | dre.leg2poly(range(4)).....array |
| 1480 | 28 5b 2d 31 2e 20 2c 20 2d 33 2e 35 2c 20 20 33 2e 20 2c 20 20 37 2e 35 5d 29 0a 0a 0a 20 20 20 | ([-1..,.-3.5,..3..,..7.5])...... |
| 14a0 | 20 72 04 00 00 00 29 03 da 07 70 6f 6c 79 61 64 64 da 08 70 6f 6c 79 6d 75 6c 78 da 07 70 6f 6c | .r....)...polyadd..polymulx..pol |
| 14c0 | 79 73 75 62 e9 03 00 00 00 e9 fe ff ff ff 72 27 00 00 00 e9 02 00 00 00 29 08 da 0a 70 6f 6c 79 | ysub..........r'........)...poly |
| 14e0 | 6e 6f 6d 69 61 6c 72 33 00 00 00 72 34 00 00 00 72 35 00 00 00 72 28 00 00 00 72 29 00 00 00 72 | nomialr3...r4...r5...r(...r)...r |
| 1500 | 2a 00 00 00 72 2b 00 00 00 29 09 da 01 63 72 33 00 00 00 72 34 00 00 00 72 35 00 00 00 da 01 6e | *...r+...)...cr3...r4...r5.....n |
| 1520 | da 02 63 30 da 02 63 31 72 2f 00 00 00 da 03 74 6d 70 73 09 00 00 00 20 20 20 20 20 20 20 20 20 | ..c0..c1r/.....tmps............. |
| 1540 | 72 30 00 00 00 72 15 00 00 00 72 15 00 00 00 95 00 00 00 73 bd 00 00 00 80 00 f7 5a 01 00 05 37 | r0...r....r........s.......Z...7 |
| 1560 | d1 04 36 e4 0a 0c 8f 2c 89 2c 98 01 90 73 d3 0a 1b 81 43 80 51 dc 08 0b 88 41 8b 06 80 41 d8 07 | ..6....,.,...s....C.Q....A...A.. |
| 1580 | 08 88 31 82 75 d8 0f 10 88 08 e0 0d 0e 88 72 89 55 88 02 d8 0d 0e 88 72 89 55 88 02 e4 11 16 90 | ..1.u.........r.U......r.U...... |
| 15a0 | 71 98 31 91 75 98 61 a0 12 d3 11 24 f2 00 03 09 40 01 88 41 d8 12 14 88 43 d9 11 18 98 11 98 31 | q.1.u.a....$....@..A....C......1 |
| 15c0 | 98 71 99 35 99 18 a0 42 a8 21 a8 61 a9 25 a1 4c b0 41 d1 23 35 d3 11 36 88 42 d9 11 18 98 13 99 | .q.5...B.!.a.%.L.A.#5..6.B...... |
| 15e0 | 78 a8 02 9b 7c a8 71 b0 31 a9 75 b0 71 a9 79 d1 1f 39 b8 51 d1 1e 3e d3 11 3f 89 42 f0 07 03 09 | x...|.q.1.u.q.y..9.Q..>..?.B.... |
| 1600 | 40 01 f1 08 00 10 17 90 72 99 38 a0 42 9b 3c d3 0f 28 d0 08 28 72 31 00 00 00 67 00 00 00 00 00 | @.......r.8.B.<..(..(r1...g..... |
| 1620 | 00 f0 bf e7 00 00 00 00 00 00 f0 3f 63 02 00 00 00 00 00 00 00 00 00 00 00 04 00 00 00 03 00 00 | ...........?c................... |
| 1640 | 00 f3 66 00 00 00 97 00 7c 01 64 01 6b 37 00 00 72 17 74 01 00 00 00 00 00 00 00 00 6a 02 00 00 | ..f.....|.d.k7..r.t.........j... |
| 1660 | 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 00 7c 01 67 02 ab 01 00 00 00 00 00 00 53 00 | ................|.|.g.........S. |
| 1680 | 74 01 00 00 00 00 00 00 00 00 6a 02 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 00 | t.........j...................|. |
| 16a0 | 67 01 ab 01 00 00 00 00 00 00 53 00 29 02 61 bc 02 00 00 0a 20 20 20 20 4c 65 67 65 6e 64 72 65 | g.........S.).a.........Legendre |
| 16c0 | 20 73 65 72 69 65 73 20 77 68 6f 73 65 20 67 72 61 70 68 20 69 73 20 61 20 73 74 72 61 69 67 68 | .series.whose.graph.is.a.straigh |
| 16e0 | 74 20 6c 69 6e 65 2e 0a 0a 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 20 2d 2d | t.line.........Parameters.....-- |
| 1700 | 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 6f 66 66 2c 20 73 63 6c 20 3a 20 73 63 61 6c 61 72 73 0a | --------.....off,.scl.:.scalars. |
| 1720 | 20 20 20 20 20 20 20 20 54 68 65 20 73 70 65 63 69 66 69 65 64 20 6c 69 6e 65 20 69 73 20 67 69 | ........The.specified.line.is.gi |
| 1740 | 76 65 6e 20 62 79 20 60 60 6f 66 66 20 2b 20 73 63 6c 2a 78 60 60 2e 0a 0a 20 20 20 20 52 65 74 | ven.by.``off.+.scl*x``.......Ret |
| 1760 | 75 72 6e 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 79 20 3a 20 6e 64 61 72 72 61 79 | urns.....-------.....y.:.ndarray |
| 1780 | 0a 20 20 20 20 20 20 20 20 54 68 69 73 20 6d 6f 64 75 6c 65 27 73 20 72 65 70 72 65 73 65 6e 74 | .........This.module's.represent |
| 17a0 | 61 74 69 6f 6e 20 6f 66 20 74 68 65 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 66 6f 72 | ation.of.the.Legendre.series.for |
| 17c0 | 0a 20 20 20 20 20 20 20 20 60 60 6f 66 66 20 2b 20 73 63 6c 2a 78 60 60 2e 0a 0a 20 20 20 20 53 | .........``off.+.scl*x``.......S |
| 17e0 | 65 65 20 41 6c 73 6f 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 6e 75 6d 70 79 2e 70 | ee.Also.....--------.....numpy.p |
| 1800 | 6f 6c 79 6e 6f 6d 69 61 6c 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 2e 70 6f 6c 79 6c 69 6e 65 0a 20 20 | olynomial.polynomial.polyline... |
| 1820 | 20 20 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 2e 63 68 65 62 79 73 68 65 76 2e 63 68 65 | ..numpy.polynomial.chebyshev.che |
| 1840 | 62 6c 69 6e 65 0a 20 20 20 20 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 2e 6c 61 67 75 65 | bline.....numpy.polynomial.lague |
| 1860 | 72 72 65 2e 6c 61 67 6c 69 6e 65 0a 20 20 20 20 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c | rre.lagline.....numpy.polynomial |
| 1880 | 2e 68 65 72 6d 69 74 65 2e 68 65 72 6d 6c 69 6e 65 0a 20 20 20 20 6e 75 6d 70 79 2e 70 6f 6c 79 | .hermite.hermline.....numpy.poly |
| 18a0 | 6e 6f 6d 69 61 6c 2e 68 65 72 6d 69 74 65 5f 65 2e 68 65 72 6d 65 6c 69 6e 65 0a 0a 20 20 20 20 | nomial.hermite_e.hermeline...... |
| 18c0 | 45 78 61 6d 70 6c 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 3e 3e 3e 20 69 6d | Examples.....--------.....>>>.im |
| 18e0 | 70 6f 72 74 20 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 2e 6c 65 67 65 6e 64 72 65 20 61 | port.numpy.polynomial.legendre.a |
| 1900 | 73 20 4c 0a 20 20 20 20 3e 3e 3e 20 4c 2e 6c 65 67 6c 69 6e 65 28 33 2c 32 29 0a 20 20 20 20 61 | s.L.....>>>.L.legline(3,2).....a |
| 1920 | 72 72 61 79 28 5b 33 2c 20 32 5d 29 0a 20 20 20 20 3e 3e 3e 20 4c 2e 6c 65 67 76 61 6c 28 2d 33 | rray([3,.2]).....>>>.L.legval(-3 |
| 1940 | 2c 20 4c 2e 6c 65 67 6c 69 6e 65 28 33 2c 32 29 29 20 23 20 73 68 6f 75 6c 64 20 62 65 20 2d 33 | ,.L.legline(3,2)).#.should.be.-3 |
| 1960 | 0a 20 20 20 20 2d 33 2e 30 0a 0a 20 20 20 20 72 02 00 00 00 29 02 da 02 6e 70 da 05 61 72 72 61 | .....-3.0......r....)...np..arra |
| 1980 | 79 29 02 da 03 6f 66 66 da 03 73 63 6c 73 02 00 00 00 20 20 72 30 00 00 00 72 0b 00 00 00 72 0b | y)...off..scls......r0...r....r. |
| 19a0 | 00 00 00 e5 00 00 00 73 2f 00 00 00 80 00 f0 44 01 00 08 0b 88 61 82 78 dc 0f 11 8f 78 89 78 98 | .......s/......D.....a.x....x.x. |
| 19c0 | 13 98 63 98 0a d3 0f 23 d0 08 23 e4 0f 11 8f 78 89 78 98 13 98 05 8b 7f d0 08 1e 72 31 00 00 00 | ..c....#..#....x.x.........r1... |
| 19e0 | 63 01 00 00 00 00 00 00 00 00 00 00 00 05 00 00 00 03 00 00 00 f3 40 00 00 00 97 00 74 01 00 00 | c.....................@.....t... |
| 1a00 | 00 00 00 00 00 00 6a 02 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 74 04 00 00 00 00 | ......j...................t..... |
| 1a20 | 00 00 00 00 74 06 00 00 00 00 00 00 00 00 7c 00 ab 03 00 00 00 00 00 00 53 00 29 01 61 ae 06 00 | ....t.........|.........S.).a... |
| 1a40 | 00 0a 20 20 20 20 47 65 6e 65 72 61 74 65 20 61 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 | ......Generate.a.Legendre.series |
| 1a60 | 20 77 69 74 68 20 67 69 76 65 6e 20 72 6f 6f 74 73 2e 0a 0a 20 20 20 20 54 68 65 20 66 75 6e 63 | .with.given.roots.......The.func |
| 1a80 | 74 69 6f 6e 20 72 65 74 75 72 6e 73 20 74 68 65 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6f 66 | tion.returns.the.coefficients.of |
| 1aa0 | 20 74 68 65 20 70 6f 6c 79 6e 6f 6d 69 61 6c 0a 0a 20 20 20 20 2e 2e 20 6d 61 74 68 3a 3a 20 70 | .the.polynomial.........math::.p |
| 1ac0 | 28 78 29 20 3d 20 28 78 20 2d 20 72 5f 30 29 20 2a 20 28 78 20 2d 20 72 5f 31 29 20 2a 20 2e 2e | (x).=.(x.-.r_0).*.(x.-.r_1).*... |
| 1ae0 | 2e 20 2a 20 28 78 20 2d 20 72 5f 6e 29 2c 0a 0a 20 20 20 20 69 6e 20 4c 65 67 65 6e 64 72 65 20 | ..*.(x.-.r_n),......in.Legendre. |
| 1b00 | 66 6f 72 6d 2c 20 77 68 65 72 65 20 74 68 65 20 3a 6d 61 74 68 3a 60 72 5f 6e 60 20 61 72 65 20 | form,.where.the.:math:`r_n`.are. |
| 1b20 | 74 68 65 20 72 6f 6f 74 73 20 73 70 65 63 69 66 69 65 64 20 69 6e 20 60 72 6f 6f 74 73 60 2e 0a | the.roots.specified.in.`roots`.. |
| 1b40 | 20 20 20 20 49 66 20 61 20 7a 65 72 6f 20 68 61 73 20 6d 75 6c 74 69 70 6c 69 63 69 74 79 20 6e | ....If.a.zero.has.multiplicity.n |
| 1b60 | 2c 20 74 68 65 6e 20 69 74 20 6d 75 73 74 20 61 70 70 65 61 72 20 69 6e 20 60 72 6f 6f 74 73 60 | ,.then.it.must.appear.in.`roots` |
| 1b80 | 20 6e 20 74 69 6d 65 73 2e 0a 20 20 20 20 46 6f 72 20 69 6e 73 74 61 6e 63 65 2c 20 69 66 20 32 | .n.times......For.instance,.if.2 |
| 1ba0 | 20 69 73 20 61 20 72 6f 6f 74 20 6f 66 20 6d 75 6c 74 69 70 6c 69 63 69 74 79 20 74 68 72 65 65 | .is.a.root.of.multiplicity.three |
| 1bc0 | 20 61 6e 64 20 33 20 69 73 20 61 20 72 6f 6f 74 20 6f 66 0a 20 20 20 20 6d 75 6c 74 69 70 6c 69 | .and.3.is.a.root.of.....multipli |
| 1be0 | 63 69 74 79 20 32 2c 20 74 68 65 6e 20 60 72 6f 6f 74 73 60 20 6c 6f 6f 6b 73 20 73 6f 6d 65 74 | city.2,.then.`roots`.looks.somet |
| 1c00 | 68 69 6e 67 20 6c 69 6b 65 20 5b 32 2c 20 32 2c 20 32 2c 20 33 2c 20 33 5d 2e 20 54 68 65 0a 20 | hing.like.[2,.2,.2,.3,.3]..The.. |
| 1c20 | 20 20 20 72 6f 6f 74 73 20 63 61 6e 20 61 70 70 65 61 72 20 69 6e 20 61 6e 79 20 6f 72 64 65 72 | ...roots.can.appear.in.any.order |
| 1c40 | 2e 0a 0a 20 20 20 20 49 66 20 74 68 65 20 72 65 74 75 72 6e 65 64 20 63 6f 65 66 66 69 63 69 65 | .......If.the.returned.coefficie |
| 1c60 | 6e 74 73 20 61 72 65 20 60 63 60 2c 20 74 68 65 6e 0a 0a 20 20 20 20 2e 2e 20 6d 61 74 68 3a 3a | nts.are.`c`,.then.........math:: |
| 1c80 | 20 70 28 78 29 20 3d 20 63 5f 30 20 2b 20 63 5f 31 20 2a 20 4c 5f 31 28 78 29 20 2b 20 2e 2e 2e | .p(x).=.c_0.+.c_1.*.L_1(x).+.... |
| 1ca0 | 20 2b 20 20 63 5f 6e 20 2a 20 4c 5f 6e 28 78 29 0a 0a 20 20 20 20 54 68 65 20 63 6f 65 66 66 69 | .+..c_n.*.L_n(x)......The.coeffi |
| 1cc0 | 63 69 65 6e 74 20 6f 66 20 74 68 65 20 6c 61 73 74 20 74 65 72 6d 20 69 73 20 6e 6f 74 20 67 65 | cient.of.the.last.term.is.not.ge |
| 1ce0 | 6e 65 72 61 6c 6c 79 20 31 20 66 6f 72 20 6d 6f 6e 69 63 0a 20 20 20 20 70 6f 6c 79 6e 6f 6d 69 | nerally.1.for.monic.....polynomi |
| 1d00 | 61 6c 73 20 69 6e 20 4c 65 67 65 6e 64 72 65 20 66 6f 72 6d 2e 0a 0a 20 20 20 20 50 61 72 61 6d | als.in.Legendre.form.......Param |
| 1d20 | 65 74 65 72 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 72 6f 6f 74 73 20 3a | eters.....----------.....roots.: |
| 1d40 | 20 61 72 72 61 79 5f 6c 69 6b 65 0a 20 20 20 20 20 20 20 20 53 65 71 75 65 6e 63 65 20 63 6f 6e | .array_like.........Sequence.con |
| 1d60 | 74 61 69 6e 69 6e 67 20 74 68 65 20 72 6f 6f 74 73 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 0a | taining.the.roots.......Returns. |
| 1d80 | 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 6f 75 74 20 3a 20 6e 64 61 72 72 61 79 0a 20 20 | ....-------.....out.:.ndarray... |
| 1da0 | 20 20 20 20 20 20 31 2d 44 20 61 72 72 61 79 20 6f 66 20 63 6f 65 66 66 69 63 69 65 6e 74 73 2e | ......1-D.array.of.coefficients. |
| 1dc0 | 20 20 49 66 20 61 6c 6c 20 72 6f 6f 74 73 20 61 72 65 20 72 65 61 6c 20 74 68 65 6e 20 60 6f 75 | ..If.all.roots.are.real.then.`ou |
| 1de0 | 74 60 20 69 73 20 61 0a 20 20 20 20 20 20 20 20 72 65 61 6c 20 61 72 72 61 79 2c 20 69 66 20 73 | t`.is.a.........real.array,.if.s |
| 1e00 | 6f 6d 65 20 6f 66 20 74 68 65 20 72 6f 6f 74 73 20 61 72 65 20 63 6f 6d 70 6c 65 78 2c 20 74 68 | ome.of.the.roots.are.complex,.th |
| 1e20 | 65 6e 20 60 6f 75 74 60 20 69 73 20 63 6f 6d 70 6c 65 78 0a 20 20 20 20 20 20 20 20 65 76 65 6e | en.`out`.is.complex.........even |
| 1e40 | 20 69 66 20 61 6c 6c 20 74 68 65 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 69 6e 20 74 68 65 20 | .if.all.the.coefficients.in.the. |
| 1e60 | 72 65 73 75 6c 74 20 61 72 65 20 72 65 61 6c 20 28 73 65 65 20 45 78 61 6d 70 6c 65 73 0a 20 20 | result.are.real.(see.Examples... |
| 1e80 | 20 20 20 20 20 20 62 65 6c 6f 77 29 2e 0a 0a 20 20 20 20 53 65 65 20 41 6c 73 6f 0a 20 20 20 20 | ......below).......See.Also..... |
| 1ea0 | 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 2e 70 6f | --------.....numpy.polynomial.po |
| 1ec0 | 6c 79 6e 6f 6d 69 61 6c 2e 70 6f 6c 79 66 72 6f 6d 72 6f 6f 74 73 0a 20 20 20 20 6e 75 6d 70 79 | lynomial.polyfromroots.....numpy |
| 1ee0 | 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 2e 63 68 65 62 79 73 68 65 76 2e 63 68 65 62 66 72 6f 6d 72 6f | .polynomial.chebyshev.chebfromro |
| 1f00 | 6f 74 73 0a 20 20 20 20 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 2e 6c 61 67 75 65 72 72 | ots.....numpy.polynomial.laguerr |
| 1f20 | 65 2e 6c 61 67 66 72 6f 6d 72 6f 6f 74 73 0a 20 20 20 20 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d | e.lagfromroots.....numpy.polynom |
| 1f40 | 69 61 6c 2e 68 65 72 6d 69 74 65 2e 68 65 72 6d 66 72 6f 6d 72 6f 6f 74 73 0a 20 20 20 20 6e 75 | ial.hermite.hermfromroots.....nu |
| 1f60 | 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 2e 68 65 72 6d 69 74 65 5f 65 2e 68 65 72 6d 65 66 72 | mpy.polynomial.hermite_e.hermefr |
| 1f80 | 6f 6d 72 6f 6f 74 73 0a 0a 20 20 20 20 45 78 61 6d 70 6c 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d | omroots......Examples.....------ |
| 1fa0 | 2d 2d 0a 20 20 20 20 3e 3e 3e 20 69 6d 70 6f 72 74 20 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 | --.....>>>.import.numpy.polynomi |
| 1fc0 | 61 6c 2e 6c 65 67 65 6e 64 72 65 20 61 73 20 4c 0a 20 20 20 20 3e 3e 3e 20 4c 2e 6c 65 67 66 72 | al.legendre.as.L.....>>>.L.legfr |
| 1fe0 | 6f 6d 72 6f 6f 74 73 28 28 2d 31 2c 30 2c 31 29 29 20 23 20 78 5e 33 20 2d 20 78 20 72 65 6c 61 | omroots((-1,0,1)).#.x^3.-.x.rela |
| 2000 | 74 69 76 65 20 74 6f 20 74 68 65 20 73 74 61 6e 64 61 72 64 20 62 61 73 69 73 0a 20 20 20 20 61 | tive.to.the.standard.basis.....a |
| 2020 | 72 72 61 79 28 5b 20 30 2e 20 2c 20 2d 30 2e 34 2c 20 20 30 2e 20 2c 20 20 30 2e 34 5d 29 0a 20 | rray([.0..,.-0.4,..0..,..0.4]).. |
| 2040 | 20 20 20 3e 3e 3e 20 6a 20 3d 20 63 6f 6d 70 6c 65 78 28 30 2c 31 29 0a 20 20 20 20 3e 3e 3e 20 | ...>>>.j.=.complex(0,1).....>>>. |
| 2060 | 4c 2e 6c 65 67 66 72 6f 6d 72 6f 6f 74 73 28 28 2d 6a 2c 6a 29 29 20 23 20 78 5e 32 20 2b 20 31 | L.legfromroots((-j,j)).#.x^2.+.1 |
| 2080 | 20 72 65 6c 61 74 69 76 65 20 74 6f 20 74 68 65 20 73 74 61 6e 64 61 72 64 20 62 61 73 69 73 0a | .relative.to.the.standard.basis. |
| 20a0 | 20 20 20 20 61 72 72 61 79 28 5b 20 31 2e 33 33 33 33 33 33 33 33 2b 30 2e 6a 2c 20 20 30 2e 30 | ....array([.1.33333333+0.j,..0.0 |
| 20c0 | 30 30 30 30 30 30 30 2b 30 2e 6a 2c 20 20 30 2e 36 36 36 36 36 36 36 37 2b 30 2e 6a 5d 29 20 23 | 0000000+0.j,..0.66666667+0.j]).# |
| 20e0 | 20 6d 61 79 20 76 61 72 79 0a 0a 20 20 20 20 29 04 72 28 00 00 00 da 0a 5f 66 72 6f 6d 72 6f 6f | .may.vary......).r(....._fromroo |
| 2100 | 74 73 72 0b 00 00 00 72 0f 00 00 00 29 01 da 05 72 6f 6f 74 73 73 01 00 00 00 20 72 30 00 00 00 | tsr....r....)...rootss.....r0... |
| 2120 | 72 17 00 00 00 72 17 00 00 00 0d 01 00 00 73 18 00 00 00 80 00 f4 68 01 00 0c 0e 8f 3d 89 3d 9c | r....r........s.......h.....=.=. |
| 2140 | 17 a4 26 a8 25 d3 0b 30 d0 04 30 72 31 00 00 00 63 02 00 00 00 00 00 00 00 00 00 00 00 04 00 00 | ..&.%..0..0r1...c............... |
| 2160 | 00 03 00 00 00 f3 2e 00 00 00 97 00 74 01 00 00 00 00 00 00 00 00 6a 02 00 00 00 00 00 00 00 00 | ............t.........j......... |
| 2180 | 00 00 00 00 00 00 00 00 00 00 7c 00 7c 01 ab 02 00 00 00 00 00 00 53 00 29 01 61 fa 03 00 00 0a | ..........|.|.........S.).a..... |
| 21a0 | 20 20 20 20 41 64 64 20 6f 6e 65 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 74 6f 20 61 | ....Add.one.Legendre.series.to.a |
| 21c0 | 6e 6f 74 68 65 72 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 20 74 68 65 20 73 75 6d 20 6f 66 20 | nother.......Returns.the.sum.of. |
| 21e0 | 74 77 6f 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 60 63 31 60 20 2b 20 60 63 32 60 2e | two.Legendre.series.`c1`.+.`c2`. |
| 2200 | 20 20 54 68 65 20 61 72 67 75 6d 65 6e 74 73 0a 20 20 20 20 61 72 65 20 73 65 71 75 65 6e 63 65 | ..The.arguments.....are.sequence |
| 2220 | 73 20 6f 66 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6f 72 64 65 72 65 64 20 66 72 6f 6d 20 6c | s.of.coefficients.ordered.from.l |
| 2240 | 6f 77 65 73 74 20 6f 72 64 65 72 20 74 65 72 6d 20 74 6f 0a 20 20 20 20 68 69 67 68 65 73 74 2c | owest.order.term.to.....highest, |
| 2260 | 20 69 2e 65 2e 2c 20 5b 31 2c 32 2c 33 5d 20 72 65 70 72 65 73 65 6e 74 73 20 74 68 65 20 73 65 | .i.e.,.[1,2,3].represents.the.se |
| 2280 | 72 69 65 73 20 60 60 50 5f 30 20 2b 20 32 2a 50 5f 31 20 2b 20 33 2a 50 5f 32 60 60 2e 0a 0a 20 | ries.``P_0.+.2*P_1.+.3*P_2``.... |
| 22a0 | 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 | ...Parameters.....----------.... |
| 22c0 | 20 63 31 2c 20 63 32 20 3a 20 61 72 72 61 79 5f 6c 69 6b 65 0a 20 20 20 20 20 20 20 20 31 2d 44 | .c1,.c2.:.array_like.........1-D |
| 22e0 | 20 61 72 72 61 79 73 20 6f 66 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 63 6f 65 66 66 | .arrays.of.Legendre.series.coeff |
| 2300 | 69 63 69 65 6e 74 73 20 6f 72 64 65 72 65 64 20 66 72 6f 6d 20 6c 6f 77 20 74 6f 0a 20 20 20 20 | icients.ordered.from.low.to..... |
| 2320 | 20 20 20 20 68 69 67 68 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 0a 20 20 20 20 2d 2d 2d 2d 2d | ....high.......Returns.....----- |
| 2340 | 2d 2d 0a 20 20 20 20 6f 75 74 20 3a 20 6e 64 61 72 72 61 79 0a 20 20 20 20 20 20 20 20 41 72 72 | --.....out.:.ndarray.........Arr |
| 2360 | 61 79 20 72 65 70 72 65 73 65 6e 74 69 6e 67 20 74 68 65 20 4c 65 67 65 6e 64 72 65 20 73 65 72 | ay.representing.the.Legendre.ser |
| 2380 | 69 65 73 20 6f 66 20 74 68 65 69 72 20 73 75 6d 2e 0a 0a 20 20 20 20 53 65 65 20 41 6c 73 6f 0a | ies.of.their.sum.......See.Also. |
| 23a0 | 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 6c 65 67 73 75 62 2c 20 6c 65 67 6d 75 6c 78 | ....--------.....legsub,.legmulx |
| 23c0 | 2c 20 6c 65 67 6d 75 6c 2c 20 6c 65 67 64 69 76 2c 20 6c 65 67 70 6f 77 0a 0a 20 20 20 20 4e 6f | ,.legmul,.legdiv,.legpow......No |
| 23e0 | 74 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 0a 20 20 20 20 55 6e 6c 69 6b 65 20 6d 75 6c 74 69 70 6c | tes.....-----.....Unlike.multipl |
| 2400 | 69 63 61 74 69 6f 6e 2c 20 64 69 76 69 73 69 6f 6e 2c 20 65 74 63 2e 2c 20 74 68 65 20 73 75 6d | ication,.division,.etc.,.the.sum |
| 2420 | 20 6f 66 20 74 77 6f 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 0a 20 20 20 20 69 73 20 61 | .of.two.Legendre.series.....is.a |
| 2440 | 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 28 77 69 74 68 6f 75 74 20 68 61 76 69 6e 67 | .Legendre.series.(without.having |
| 2460 | 20 74 6f 20 22 72 65 70 72 6f 6a 65 63 74 22 20 74 68 65 20 72 65 73 75 6c 74 20 6f 6e 74 6f 0a | .to."reproject".the.result.onto. |
| 2480 | 20 20 20 20 74 68 65 20 62 61 73 69 73 20 73 65 74 29 20 73 6f 20 61 64 64 69 74 69 6f 6e 2c 20 | ....the.basis.set).so.addition,. |
| 24a0 | 6a 75 73 74 20 6c 69 6b 65 20 74 68 61 74 20 6f 66 20 22 73 74 61 6e 64 61 72 64 22 20 70 6f 6c | just.like.that.of."standard".pol |
| 24c0 | 79 6e 6f 6d 69 61 6c 73 2c 0a 20 20 20 20 69 73 20 73 69 6d 70 6c 79 20 22 63 6f 6d 70 6f 6e 65 | ynomials,.....is.simply."compone |
| 24e0 | 6e 74 2d 77 69 73 65 2e 22 0a 0a 20 20 20 20 45 78 61 6d 70 6c 65 73 0a 20 20 20 20 2d 2d 2d 2d | nt-wise."......Examples.....---- |
| 2500 | 2d 2d 2d 2d 0a 20 20 20 20 3e 3e 3e 20 66 72 6f 6d 20 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 | ----.....>>>.from.numpy.polynomi |
| 2520 | 61 6c 20 69 6d 70 6f 72 74 20 6c 65 67 65 6e 64 72 65 20 61 73 20 4c 0a 20 20 20 20 3e 3e 3e 20 | al.import.legendre.as.L.....>>>. |
| 2540 | 63 31 20 3d 20 28 31 2c 32 2c 33 29 0a 20 20 20 20 3e 3e 3e 20 63 32 20 3d 20 28 33 2c 32 2c 31 | c1.=.(1,2,3).....>>>.c2.=.(3,2,1 |
| 2560 | 29 0a 20 20 20 20 3e 3e 3e 20 4c 2e 6c 65 67 61 64 64 28 63 31 2c 63 32 29 0a 20 20 20 20 61 72 | ).....>>>.L.legadd(c1,c2).....ar |
| 2580 | 72 61 79 28 5b 34 2e 2c 20 20 34 2e 2c 20 20 34 2e 5d 29 0a 0a 20 20 20 20 29 02 72 28 00 00 00 | ray([4.,..4.,..4.])......).r(... |
| 25a0 | da 04 5f 61 64 64 a9 02 72 3d 00 00 00 da 02 63 32 73 02 00 00 00 20 20 72 30 00 00 00 72 0c 00 | .._add..r=.....c2s......r0...r.. |
| 25c0 | 00 00 72 0c 00 00 00 44 01 00 00 73 15 00 00 00 80 00 f4 4e 01 00 0c 0e 8f 37 89 37 90 32 90 72 | ..r....D...s.......N.....7.7.2.r |
| 25e0 | 8b 3f d0 04 1a 72 31 00 00 00 63 02 00 00 00 00 00 00 00 00 00 00 00 04 00 00 00 03 00 00 00 f3 | .?...r1...c..................... |
| 2600 | 2e 00 00 00 97 00 74 01 00 00 00 00 00 00 00 00 6a 02 00 00 00 00 00 00 00 00 00 00 00 00 00 00 | ......t.........j............... |
| 2620 | 00 00 00 00 7c 00 7c 01 ab 02 00 00 00 00 00 00 53 00 29 01 61 51 04 00 00 0a 20 20 20 20 53 75 | ....|.|.........S.).aQ........Su |
| 2640 | 62 74 72 61 63 74 20 6f 6e 65 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 66 72 6f 6d 20 | btract.one.Legendre.series.from. |
| 2660 | 61 6e 6f 74 68 65 72 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 20 74 68 65 20 64 69 66 66 65 72 | another.......Returns.the.differ |
| 2680 | 65 6e 63 65 20 6f 66 20 74 77 6f 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 60 63 31 60 | ence.of.two.Legendre.series.`c1` |
| 26a0 | 20 2d 20 60 63 32 60 2e 20 20 54 68 65 0a 20 20 20 20 73 65 71 75 65 6e 63 65 73 20 6f 66 20 63 | .-.`c2`...The.....sequences.of.c |
| 26c0 | 6f 65 66 66 69 63 69 65 6e 74 73 20 61 72 65 20 66 72 6f 6d 20 6c 6f 77 65 73 74 20 6f 72 64 65 | oefficients.are.from.lowest.orde |
| 26e0 | 72 20 74 65 72 6d 20 74 6f 20 68 69 67 68 65 73 74 2c 20 69 2e 65 2e 2c 0a 20 20 20 20 5b 31 2c | r.term.to.highest,.i.e.,.....[1, |
| 2700 | 32 2c 33 5d 20 72 65 70 72 65 73 65 6e 74 73 20 74 68 65 20 73 65 72 69 65 73 20 60 60 50 5f 30 | 2,3].represents.the.series.``P_0 |
| 2720 | 20 2b 20 32 2a 50 5f 31 20 2b 20 33 2a 50 5f 32 60 60 2e 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 | .+.2*P_1.+.3*P_2``.......Paramet |
| 2740 | 65 72 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 63 31 2c 20 63 32 20 3a 20 | ers.....----------.....c1,.c2.:. |
| 2760 | 61 72 72 61 79 5f 6c 69 6b 65 0a 20 20 20 20 20 20 20 20 31 2d 44 20 61 72 72 61 79 73 20 6f 66 | array_like.........1-D.arrays.of |
| 2780 | 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6f 72 | .Legendre.series.coefficients.or |
| 27a0 | 64 65 72 65 64 20 66 72 6f 6d 20 6c 6f 77 20 74 6f 0a 20 20 20 20 20 20 20 20 68 69 67 68 2e 0a | dered.from.low.to.........high.. |
| 27c0 | 0a 20 20 20 20 52 65 74 75 72 6e 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 6f 75 74 | .....Returns.....-------.....out |
| 27e0 | 20 3a 20 6e 64 61 72 72 61 79 0a 20 20 20 20 20 20 20 20 4f 66 20 4c 65 67 65 6e 64 72 65 20 73 | .:.ndarray.........Of.Legendre.s |
| 2800 | 65 72 69 65 73 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 72 65 70 72 65 73 65 6e 74 69 6e 67 20 | eries.coefficients.representing. |
| 2820 | 74 68 65 69 72 20 64 69 66 66 65 72 65 6e 63 65 2e 0a 0a 20 20 20 20 53 65 65 20 41 6c 73 6f 0a | their.difference.......See.Also. |
| 2840 | 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 6c 65 67 61 64 64 2c 20 6c 65 67 6d 75 6c 78 | ....--------.....legadd,.legmulx |
| 2860 | 2c 20 6c 65 67 6d 75 6c 2c 20 6c 65 67 64 69 76 2c 20 6c 65 67 70 6f 77 0a 0a 20 20 20 20 4e 6f | ,.legmul,.legdiv,.legpow......No |
| 2880 | 74 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 0a 20 20 20 20 55 6e 6c 69 6b 65 20 6d 75 6c 74 69 70 6c | tes.....-----.....Unlike.multipl |
| 28a0 | 69 63 61 74 69 6f 6e 2c 20 64 69 76 69 73 69 6f 6e 2c 20 65 74 63 2e 2c 20 74 68 65 20 64 69 66 | ication,.division,.etc.,.the.dif |
| 28c0 | 66 65 72 65 6e 63 65 20 6f 66 20 74 77 6f 20 4c 65 67 65 6e 64 72 65 0a 20 20 20 20 73 65 72 69 | ference.of.two.Legendre.....seri |
| 28e0 | 65 73 20 69 73 20 61 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 28 77 69 74 68 6f 75 74 | es.is.a.Legendre.series.(without |
| 2900 | 20 68 61 76 69 6e 67 20 74 6f 20 22 72 65 70 72 6f 6a 65 63 74 22 20 74 68 65 20 72 65 73 75 6c | .having.to."reproject".the.resul |
| 2920 | 74 0a 20 20 20 20 6f 6e 74 6f 20 74 68 65 20 62 61 73 69 73 20 73 65 74 29 20 73 6f 20 73 75 62 | t.....onto.the.basis.set).so.sub |
| 2940 | 74 72 61 63 74 69 6f 6e 2c 20 6a 75 73 74 20 6c 69 6b 65 20 74 68 61 74 20 6f 66 20 22 73 74 61 | traction,.just.like.that.of."sta |
| 2960 | 6e 64 61 72 64 22 0a 20 20 20 20 70 6f 6c 79 6e 6f 6d 69 61 6c 73 2c 20 69 73 20 73 69 6d 70 6c | ndard".....polynomials,.is.simpl |
| 2980 | 79 20 22 63 6f 6d 70 6f 6e 65 6e 74 2d 77 69 73 65 2e 22 0a 0a 20 20 20 20 45 78 61 6d 70 6c 65 | y."component-wise."......Example |
| 29a0 | 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 3e 3e 3e 20 66 72 6f 6d 20 6e 75 6d 70 | s.....--------.....>>>.from.nump |
| 29c0 | 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 20 69 6d 70 6f 72 74 20 6c 65 67 65 6e 64 72 65 20 61 73 20 | y.polynomial.import.legendre.as. |
| 29e0 | 4c 0a 20 20 20 20 3e 3e 3e 20 63 31 20 3d 20 28 31 2c 32 2c 33 29 0a 20 20 20 20 3e 3e 3e 20 63 | L.....>>>.c1.=.(1,2,3).....>>>.c |
| 2a00 | 32 20 3d 20 28 33 2c 32 2c 31 29 0a 20 20 20 20 3e 3e 3e 20 4c 2e 6c 65 67 73 75 62 28 63 31 2c | 2.=.(3,2,1).....>>>.L.legsub(c1, |
| 2a20 | 63 32 29 0a 20 20 20 20 61 72 72 61 79 28 5b 2d 32 2e 2c 20 20 30 2e 2c 20 20 32 2e 5d 29 0a 20 | c2).....array([-2.,..0.,..2.]).. |
| 2a40 | 20 20 20 3e 3e 3e 20 4c 2e 6c 65 67 73 75 62 28 63 32 2c 63 31 29 20 23 20 2d 43 2e 6c 65 67 73 | ...>>>.L.legsub(c2,c1).#.-C.legs |
| 2a60 | 75 62 28 63 31 2c 63 32 29 0a 20 20 20 20 61 72 72 61 79 28 5b 20 32 2e 2c 20 20 30 2e 2c 20 2d | ub(c1,c2).....array([.2.,..0.,.- |
| 2a80 | 32 2e 5d 29 0a 0a 20 20 20 20 29 02 72 28 00 00 00 da 04 5f 73 75 62 72 4a 00 00 00 73 02 00 00 | 2.])......).r(....._subrJ...s... |
| 2aa0 | 00 20 20 72 30 00 00 00 72 0d 00 00 00 72 0d 00 00 00 6e 01 00 00 73 15 00 00 00 80 00 f4 52 01 | ...r0...r....r....n...s.......R. |
| 2ac0 | 00 0c 0e 8f 37 89 37 90 32 90 72 8b 3f d0 04 1a 72 31 00 00 00 63 01 00 00 00 00 00 00 00 00 00 | ....7.7.2.r.?...r1...c.......... |
| 2ae0 | 00 00 06 00 00 00 03 00 00 00 f3 80 01 00 00 97 00 74 01 00 00 00 00 00 00 00 00 6a 02 00 00 00 | .................t.........j.... |
| 2b00 | 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 00 67 01 ab 01 00 00 00 00 00 00 5c 01 00 00 7d | ...............|.g.........\...} |
| 2b20 | 00 74 05 00 00 00 00 00 00 00 00 7c 00 ab 01 00 00 00 00 00 00 64 01 6b 28 00 00 72 0a 7c 00 64 | .t.........|.........d.k(..r.|.d |
| 2b40 | 02 19 00 00 00 64 02 6b 28 00 00 72 02 7c 00 53 00 74 07 00 00 00 00 00 00 00 00 6a 08 00 00 00 | .....d.k(..r.|.S.t.........j.... |
| 2b60 | 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 74 05 00 00 00 00 00 00 00 00 7c 00 ab 01 00 00 00 | ...............t.........|...... |
| 2b80 | 00 00 00 64 01 7a 00 00 00 7c 00 6a 0a 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ac | ...d.z...|.j.................... |
| 2ba0 | 03 ab 02 00 00 00 00 00 00 7d 01 7c 00 64 02 19 00 00 00 64 02 7a 05 00 00 7c 01 64 02 3c 00 00 | .........}.|.d.....d.z...|.d.<.. |
| 2bc0 | 00 7c 00 64 02 19 00 00 00 7c 01 64 01 3c 00 00 00 74 0d 00 00 00 00 00 00 00 00 64 01 74 05 00 | .|.d.....|.d.<...t.........d.t.. |
| 2be0 | 00 00 00 00 00 00 00 7c 00 ab 01 00 00 00 00 00 00 ab 02 00 00 00 00 00 00 44 00 5d 35 00 00 7d | .......|.................D.]5..} |
| 2c00 | 02 7c 02 64 01 7a 00 00 00 7d 03 7c 02 64 01 7a 0a 00 00 7d 04 7c 02 7c 03 7a 00 00 00 7d 05 7c | .|.d.z...}.|.d.z...}.|.|.z...}.| |
| 2c20 | 00 7c 02 19 00 00 00 7c 03 7a 05 00 00 7c 05 7a 0b 00 00 7c 01 7c 03 3c 00 00 00 7c 01 7c 04 78 | .|.....|.z...|.z...|.|.<...|.|.x |
| 2c40 | 02 78 02 19 00 00 00 7c 00 7c 02 19 00 00 00 7c 02 7a 05 00 00 7c 05 7a 0b 00 00 7a 0d 00 00 63 | .x.....|.|.....|.z...|.z...z...c |
| 2c60 | 03 63 02 3c 00 00 00 8c 37 04 00 7c 01 53 00 29 04 61 19 03 00 00 4d 75 6c 74 69 70 6c 79 20 61 | .c.<....7..|.S.).a....Multiply.a |
| 2c80 | 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 62 79 20 78 2e 0a 0a 20 20 20 20 4d 75 6c 74 | .Legendre.series.by.x.......Mult |
| 2ca0 | 69 70 6c 79 20 74 68 65 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 60 63 60 20 62 79 20 | iply.the.Legendre.series.`c`.by. |
| 2cc0 | 78 2c 20 77 68 65 72 65 20 78 20 69 73 20 74 68 65 20 69 6e 64 65 70 65 6e 64 65 6e 74 0a 20 20 | x,.where.x.is.the.independent... |
| 2ce0 | 20 20 76 61 72 69 61 62 6c 65 2e 0a 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 | ..variable........Parameters.... |
| 2d00 | 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 63 20 3a 20 61 72 72 61 79 5f 6c 69 6b 65 0a 20 | .----------.....c.:.array_like.. |
| 2d20 | 20 20 20 20 20 20 20 31 2d 44 20 61 72 72 61 79 20 6f 66 20 4c 65 67 65 6e 64 72 65 20 73 65 72 | .......1-D.array.of.Legendre.ser |
| 2d40 | 69 65 73 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6f 72 64 65 72 65 64 20 66 72 6f 6d 20 6c 6f | ies.coefficients.ordered.from.lo |
| 2d60 | 77 20 74 6f 0a 20 20 20 20 20 20 20 20 68 69 67 68 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 0a | w.to.........high.......Returns. |
| 2d80 | 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 6f 75 74 20 3a 20 6e 64 61 72 72 61 79 0a 20 20 | ....-------.....out.:.ndarray... |
| 2da0 | 20 20 20 20 20 20 41 72 72 61 79 20 72 65 70 72 65 73 65 6e 74 69 6e 67 20 74 68 65 20 72 65 73 | ......Array.representing.the.res |
| 2dc0 | 75 6c 74 20 6f 66 20 74 68 65 20 6d 75 6c 74 69 70 6c 69 63 61 74 69 6f 6e 2e 0a 0a 20 20 20 20 | ult.of.the.multiplication....... |
| 2de0 | 53 65 65 20 41 6c 73 6f 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 6c 65 67 61 64 64 | See.Also.....--------.....legadd |
| 2e00 | 2c 20 6c 65 67 73 75 62 2c 20 6c 65 67 6d 75 6c 2c 20 6c 65 67 64 69 76 2c 20 6c 65 67 70 6f 77 | ,.legsub,.legmul,.legdiv,.legpow |
| 2e20 | 0a 0a 20 20 20 20 4e 6f 74 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 0a 20 20 20 20 54 68 65 20 6d 75 | ......Notes.....-----.....The.mu |
| 2e40 | 6c 74 69 70 6c 69 63 61 74 69 6f 6e 20 75 73 65 73 20 74 68 65 20 72 65 63 75 72 73 69 6f 6e 20 | ltiplication.uses.the.recursion. |
| 2e60 | 72 65 6c 61 74 69 6f 6e 73 68 69 70 20 66 6f 72 20 4c 65 67 65 6e 64 72 65 0a 20 20 20 20 70 6f | relationship.for.Legendre.....po |
| 2e80 | 6c 79 6e 6f 6d 69 61 6c 73 20 69 6e 20 74 68 65 20 66 6f 72 6d 0a 0a 20 20 20 20 2e 2e 20 6d 61 | lynomials.in.the.form.........ma |
| 2ea0 | 74 68 3a 3a 0a 0a 20 20 20 20 20 20 78 50 5f 69 28 78 29 20 3d 20 28 28 69 20 2b 20 31 29 2a 50 | th::........xP_i(x).=.((i.+.1)*P |
| 2ec0 | 5f 7b 69 20 2b 20 31 7d 28 78 29 20 2b 20 69 2a 50 5f 7b 69 20 2d 20 31 7d 28 78 29 29 2f 28 32 | _{i.+.1}(x).+.i*P_{i.-.1}(x))/(2 |
| 2ee0 | 69 20 2b 20 31 29 0a 0a 20 20 20 20 45 78 61 6d 70 6c 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d | i.+.1)......Examples.....------- |
| 2f00 | 2d 0a 20 20 20 20 3e 3e 3e 20 66 72 6f 6d 20 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 20 | -.....>>>.from.numpy.polynomial. |
| 2f20 | 69 6d 70 6f 72 74 20 6c 65 67 65 6e 64 72 65 20 61 73 20 4c 0a 20 20 20 20 3e 3e 3e 20 4c 2e 6c | import.legendre.as.L.....>>>.L.l |
| 2f40 | 65 67 6d 75 6c 78 28 5b 31 2c 32 2c 33 5d 29 0a 20 20 20 20 61 72 72 61 79 28 5b 20 30 2e 36 36 | egmulx([1,2,3]).....array([.0.66 |
| 2f60 | 36 36 36 36 36 37 2c 20 32 2e 32 2c 20 31 2e 33 33 33 33 33 33 33 33 2c 20 31 2e 38 5d 29 20 23 | 666667,.2.2,.1.33333333,.1.8]).# |
| 2f80 | 20 6d 61 79 20 76 61 72 79 0a 0a 20 20 20 20 72 04 00 00 00 72 02 00 00 00 a9 01 da 05 64 74 79 | .may.vary......r....r........dty |
| 2fa0 | 70 65 29 07 72 28 00 00 00 72 29 00 00 00 72 2a 00 00 00 72 41 00 00 00 da 05 65 6d 70 74 79 72 | pe).r(...r)...r*...rA.....emptyr |
| 2fc0 | 50 00 00 00 72 2b 00 00 00 29 06 72 3a 00 00 00 da 03 70 72 64 72 2f 00 00 00 da 01 6a da 01 6b | P...r+...).r:.....prdr/.....j..k |
| 2fe0 | da 01 73 73 06 00 00 00 20 20 20 20 20 20 72 30 00 00 00 72 0e 00 00 00 72 0e 00 00 00 9a 01 00 | ..ss..........r0...r....r....... |
| 3000 | 00 73 d6 00 00 00 80 00 f4 4e 01 00 0b 0d 8f 2c 89 2c 98 01 90 73 d3 0a 1b 81 43 80 51 e4 07 0a | .s.......N.....,.,...s....C.Q... |
| 3020 | 88 31 83 76 90 11 82 7b 90 71 98 11 91 74 98 71 92 79 d8 0f 10 88 08 e4 0a 0c 8f 28 89 28 94 33 | .1.v...{.q...t.q.y.........(.(.3 |
| 3040 | 90 71 93 36 98 41 91 3a a0 51 a7 57 a1 57 d4 0a 2d 80 43 d8 0d 0e 88 71 89 54 90 41 89 58 80 43 | .q.6.A.:.Q.W.W..-.C....q.T.A.X.C |
| 3060 | 88 01 81 46 d8 0d 0e 88 71 89 54 80 43 88 01 81 46 dc 0d 12 90 31 94 63 98 21 93 66 d3 0d 1d f2 | ...F....q.T.C...F....1.c.!.f.... |
| 3080 | 00 05 05 21 88 01 d8 0c 0d 90 01 89 45 88 01 d8 0c 0d 90 01 89 45 88 01 d8 0c 0d 90 01 89 45 88 | ...!........E........E........E. |
| 30a0 | 01 d8 12 13 90 41 91 24 98 11 91 28 98 61 91 1e 88 03 88 41 89 06 d8 08 0b 88 41 8b 06 90 31 90 | .....A.$...(.a.....A......A...1. |
| 30c0 | 51 91 34 98 21 91 38 98 71 91 2e d1 08 20 8c 06 f0 0b 05 05 21 f0 0c 00 0c 0f 80 4a 72 31 00 00 | Q.4.!.8.q...........!......Jr1.. |
| 30e0 | 00 63 02 00 00 00 00 00 00 00 00 00 00 00 07 00 00 00 03 00 00 00 f3 08 02 00 00 97 00 74 01 00 | .c...........................t.. |
| 3100 | 00 00 00 00 00 00 00 6a 02 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 00 7c 01 67 | .......j...................|.|.g |
| 3120 | 02 ab 01 00 00 00 00 00 00 5c 02 00 00 7d 00 7d 01 74 05 00 00 00 00 00 00 00 00 7c 00 ab 01 00 | .........\...}.}.t.........|.... |
| 3140 | 00 00 00 00 00 74 05 00 00 00 00 00 00 00 00 7c 01 ab 01 00 00 00 00 00 00 6b 44 00 00 72 05 7c | .....t.........|.........kD..r.| |
| 3160 | 01 7d 02 7c 00 7d 03 6e 04 7c 00 7d 02 7c 01 7d 03 74 05 00 00 00 00 00 00 00 00 7c 02 ab 01 00 | .}.|.}.n.|.}.|.}.t.........|.... |
| 3180 | 00 00 00 00 00 64 01 6b 28 00 00 72 0b 7c 02 64 02 19 00 00 00 7c 03 7a 05 00 00 7d 04 64 02 7d | .....d.k(..r.|.d.....|.z...}.d.} |
| 31a0 | 00 6e 9b 74 05 00 00 00 00 00 00 00 00 7c 02 ab 01 00 00 00 00 00 00 64 03 6b 28 00 00 72 11 7c | .n.t.........|.........d.k(..r.| |
| 31c0 | 02 64 02 19 00 00 00 7c 03 7a 05 00 00 7d 04 7c 02 64 01 19 00 00 00 7c 03 7a 05 00 00 7d 00 6e | .d.....|.z...}.|.d.....|.z...}.n |
| 31e0 | 7c 74 05 00 00 00 00 00 00 00 00 7c 02 ab 01 00 00 00 00 00 00 7d 05 7c 02 64 04 19 00 00 00 7c | |t.........|.........}.|.d.....| |
| 3200 | 03 7a 05 00 00 7d 04 7c 02 64 05 19 00 00 00 7c 03 7a 05 00 00 7d 00 74 07 00 00 00 00 00 00 00 | .z...}.|.d.....|.z...}.t........ |
| 3220 | 00 64 06 74 05 00 00 00 00 00 00 00 00 7c 02 ab 01 00 00 00 00 00 00 64 01 7a 00 00 00 ab 02 00 | .d.t.........|.........d.z...... |
| 3240 | 00 00 00 00 00 44 00 5d 46 00 00 7d 06 7c 04 7d 07 7c 05 64 01 7a 0a 00 00 7d 05 74 09 00 00 00 | .....D.]F..}.|.}.|.d.z...}.t.... |
| 3260 | 00 00 00 00 00 7c 02 7c 06 0b 00 19 00 00 00 7c 03 7a 05 00 00 7c 00 7c 05 64 01 7a 0a 00 00 7a | .....|.|.......|.z...|.|.d.z...z |
| 3280 | 05 00 00 7c 05 7a 0b 00 00 ab 02 00 00 00 00 00 00 7d 04 74 0b 00 00 00 00 00 00 00 00 7c 07 74 | ...|.z...........}.t.........|.t |
| 32a0 | 0d 00 00 00 00 00 00 00 00 7c 00 ab 01 00 00 00 00 00 00 64 03 7c 05 7a 05 00 00 64 01 7a 0a 00 | .........|.........d.|.z...d.z.. |
| 32c0 | 00 7a 05 00 00 7c 05 7a 0b 00 00 ab 02 00 00 00 00 00 00 7d 00 8c 48 04 00 74 0b 00 00 00 00 00 | .z...|.z...........}..H..t...... |
| 32e0 | 00 00 00 7c 04 74 0d 00 00 00 00 00 00 00 00 7c 00 ab 01 00 00 00 00 00 00 ab 02 00 00 00 00 00 | ...|.t.........|................ |
| 3300 | 00 53 00 29 07 61 b9 04 00 00 0a 20 20 20 20 4d 75 6c 74 69 70 6c 79 20 6f 6e 65 20 4c 65 67 65 | .S.).a.........Multiply.one.Lege |
| 3320 | 6e 64 72 65 20 73 65 72 69 65 73 20 62 79 20 61 6e 6f 74 68 65 72 2e 0a 0a 20 20 20 20 52 65 74 | ndre.series.by.another.......Ret |
| 3340 | 75 72 6e 73 20 74 68 65 20 70 72 6f 64 75 63 74 20 6f 66 20 74 77 6f 20 4c 65 67 65 6e 64 72 65 | urns.the.product.of.two.Legendre |
| 3360 | 20 73 65 72 69 65 73 20 60 63 31 60 20 2a 20 60 63 32 60 2e 20 20 54 68 65 20 61 72 67 75 6d 65 | .series.`c1`.*.`c2`...The.argume |
| 3380 | 6e 74 73 0a 20 20 20 20 61 72 65 20 73 65 71 75 65 6e 63 65 73 20 6f 66 20 63 6f 65 66 66 69 63 | nts.....are.sequences.of.coeffic |
| 33a0 | 69 65 6e 74 73 2c 20 66 72 6f 6d 20 6c 6f 77 65 73 74 20 6f 72 64 65 72 20 22 74 65 72 6d 22 20 | ients,.from.lowest.order."term". |
| 33c0 | 74 6f 20 68 69 67 68 65 73 74 2c 0a 20 20 20 20 65 2e 67 2e 2c 20 5b 31 2c 32 2c 33 5d 20 72 65 | to.highest,.....e.g.,.[1,2,3].re |
| 33e0 | 70 72 65 73 65 6e 74 73 20 74 68 65 20 73 65 72 69 65 73 20 60 60 50 5f 30 20 2b 20 32 2a 50 5f | presents.the.series.``P_0.+.2*P_ |
| 3400 | 31 20 2b 20 33 2a 50 5f 32 60 60 2e 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 | 1.+.3*P_2``.......Parameters.... |
| 3420 | 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 63 31 2c 20 63 32 20 3a 20 61 72 72 61 79 5f 6c | .----------.....c1,.c2.:.array_l |
| 3440 | 69 6b 65 0a 20 20 20 20 20 20 20 20 31 2d 44 20 61 72 72 61 79 73 20 6f 66 20 4c 65 67 65 6e 64 | ike.........1-D.arrays.of.Legend |
| 3460 | 72 65 20 73 65 72 69 65 73 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6f 72 64 65 72 65 64 20 66 | re.series.coefficients.ordered.f |
| 3480 | 72 6f 6d 20 6c 6f 77 20 74 6f 0a 20 20 20 20 20 20 20 20 68 69 67 68 2e 0a 0a 20 20 20 20 52 65 | rom.low.to.........high.......Re |
| 34a0 | 74 75 72 6e 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 6f 75 74 20 3a 20 6e 64 61 72 | turns.....-------.....out.:.ndar |
| 34c0 | 72 61 79 0a 20 20 20 20 20 20 20 20 4f 66 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 63 | ray.........Of.Legendre.series.c |
| 34e0 | 6f 65 66 66 69 63 69 65 6e 74 73 20 72 65 70 72 65 73 65 6e 74 69 6e 67 20 74 68 65 69 72 20 70 | oefficients.representing.their.p |
| 3500 | 72 6f 64 75 63 74 2e 0a 0a 20 20 20 20 53 65 65 20 41 6c 73 6f 0a 20 20 20 20 2d 2d 2d 2d 2d 2d | roduct.......See.Also.....------ |
| 3520 | 2d 2d 0a 20 20 20 20 6c 65 67 61 64 64 2c 20 6c 65 67 73 75 62 2c 20 6c 65 67 6d 75 6c 78 2c 20 | --.....legadd,.legsub,.legmulx,. |
| 3540 | 6c 65 67 64 69 76 2c 20 6c 65 67 70 6f 77 0a 0a 20 20 20 20 4e 6f 74 65 73 0a 20 20 20 20 2d 2d | legdiv,.legpow......Notes.....-- |
| 3560 | 2d 2d 2d 0a 20 20 20 20 49 6e 20 67 65 6e 65 72 61 6c 2c 20 74 68 65 20 28 70 6f 6c 79 6e 6f 6d | ---.....In.general,.the.(polynom |
| 3580 | 69 61 6c 29 20 70 72 6f 64 75 63 74 20 6f 66 20 74 77 6f 20 43 2d 73 65 72 69 65 73 20 72 65 73 | ial).product.of.two.C-series.res |
| 35a0 | 75 6c 74 73 20 69 6e 20 74 65 72 6d 73 0a 20 20 20 20 74 68 61 74 20 61 72 65 20 6e 6f 74 20 69 | ults.in.terms.....that.are.not.i |
| 35c0 | 6e 20 74 68 65 20 4c 65 67 65 6e 64 72 65 20 70 6f 6c 79 6e 6f 6d 69 61 6c 20 62 61 73 69 73 20 | n.the.Legendre.polynomial.basis. |
| 35e0 | 73 65 74 2e 20 20 54 68 75 73 2c 20 74 6f 20 65 78 70 72 65 73 73 0a 20 20 20 20 74 68 65 20 70 | set...Thus,.to.express.....the.p |
| 3600 | 72 6f 64 75 63 74 20 61 73 20 61 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 2c 20 69 74 20 | roduct.as.a.Legendre.series,.it. |
| 3620 | 69 73 20 6e 65 63 65 73 73 61 72 79 20 74 6f 20 22 72 65 70 72 6f 6a 65 63 74 22 20 74 68 65 0a | is.necessary.to."reproject".the. |
| 3640 | 20 20 20 20 70 72 6f 64 75 63 74 20 6f 6e 74 6f 20 73 61 69 64 20 62 61 73 69 73 20 73 65 74 2c | ....product.onto.said.basis.set, |
| 3660 | 20 77 68 69 63 68 20 6d 61 79 20 70 72 6f 64 75 63 65 20 22 75 6e 69 6e 74 75 69 74 69 76 65 22 | .which.may.produce."unintuitive" |
| 3680 | 20 28 62 75 74 0a 20 20 20 20 63 6f 72 72 65 63 74 29 20 72 65 73 75 6c 74 73 3b 20 73 65 65 20 | .(but.....correct).results;.see. |
| 36a0 | 45 78 61 6d 70 6c 65 73 20 73 65 63 74 69 6f 6e 20 62 65 6c 6f 77 2e 0a 0a 20 20 20 20 45 78 61 | Examples.section.below.......Exa |
| 36c0 | 6d 70 6c 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 3e 3e 3e 20 66 72 6f 6d 20 | mples.....--------.....>>>.from. |
| 36e0 | 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 20 69 6d 70 6f 72 74 20 6c 65 67 65 6e 64 72 65 | numpy.polynomial.import.legendre |
| 3700 | 20 61 73 20 4c 0a 20 20 20 20 3e 3e 3e 20 63 31 20 3d 20 28 31 2c 32 2c 33 29 0a 20 20 20 20 3e | .as.L.....>>>.c1.=.(1,2,3).....> |
| 3720 | 3e 3e 20 63 32 20 3d 20 28 33 2c 32 29 0a 20 20 20 20 3e 3e 3e 20 4c 2e 6c 65 67 6d 75 6c 28 63 | >>.c2.=.(3,2).....>>>.L.legmul(c |
| 3740 | 31 2c 63 32 29 20 23 20 6d 75 6c 74 69 70 6c 69 63 61 74 69 6f 6e 20 72 65 71 75 69 72 65 73 20 | 1,c2).#.multiplication.requires. |
| 3760 | 22 72 65 70 72 6f 6a 65 63 74 69 6f 6e 22 0a 20 20 20 20 61 72 72 61 79 28 5b 20 20 34 2e 33 33 | "reprojection".....array([..4.33 |
| 3780 | 33 33 33 33 33 33 2c 20 20 31 30 2e 34 20 20 20 20 20 20 20 2c 20 20 31 31 2e 36 36 36 36 36 36 | 333333,..10.4.......,..11.666666 |
| 37a0 | 36 37 2c 20 20 20 33 2e 36 20 20 20 20 20 20 20 5d 29 20 23 20 6d 61 79 20 76 61 72 79 0a 0a 20 | 67,...3.6.......]).#.may.vary... |
| 37c0 | 20 20 20 72 04 00 00 00 72 02 00 00 00 72 38 00 00 00 72 37 00 00 00 72 27 00 00 00 72 36 00 00 | ...r....r....r8...r7...r'...r6.. |
| 37e0 | 00 29 07 72 28 00 00 00 72 29 00 00 00 72 2a 00 00 00 72 2b 00 00 00 72 0d 00 00 00 72 0c 00 00 | .).r(...r)...r*...r+...r....r... |
| 3800 | 00 72 0e 00 00 00 29 08 72 3d 00 00 00 72 4b 00 00 00 72 3a 00 00 00 da 02 78 73 72 3c 00 00 00 | .r....).r=...rK...r:.....xsr<... |
| 3820 | da 02 6e 64 72 2f 00 00 00 72 3e 00 00 00 73 08 00 00 00 20 20 20 20 20 20 20 20 72 30 00 00 00 | ..ndr/...r>...s............r0... |
| 3840 | 72 0f 00 00 00 72 0f 00 00 00 d2 01 00 00 73 2a 01 00 00 80 00 f4 52 01 00 10 12 8f 7c 89 7c 98 | r....r........s*......R.....|.|. |
| 3860 | 52 a0 12 98 48 d3 0f 25 81 48 80 52 88 12 e4 07 0a 88 32 83 77 94 13 90 52 93 17 d2 07 18 d8 0c | R...H..%.H.R......2.w...R....... |
| 3880 | 0e 88 01 d8 0d 0f 89 02 e0 0c 0e 88 01 d8 0d 0f 88 02 e4 07 0a 88 31 83 76 90 11 82 7b d8 0d 0e | ......................1.v...{... |
| 38a0 | 88 71 89 54 90 42 89 59 88 02 d8 0d 0e 89 02 dc 09 0c 88 51 8b 16 90 31 8a 1b d8 0d 0e 88 71 89 | .q.T.B.Y...........Q...1......q. |
| 38c0 | 54 90 42 89 59 88 02 d8 0d 0e 88 71 89 54 90 42 89 59 89 02 e4 0d 10 90 11 8b 56 88 02 d8 0d 0e | T.B.Y......q.T.B.Y........V..... |
| 38e0 | 88 72 89 55 90 52 89 5a 88 02 d8 0d 0e 88 72 89 55 90 52 89 5a 88 02 dc 11 16 90 71 9c 23 98 61 | .r.U.R.Z......r.U.R.Z......q.#.a |
| 3900 | 9b 26 a0 31 99 2a d3 11 25 f2 00 04 09 40 01 88 41 d8 12 14 88 43 d8 11 13 90 61 91 16 88 42 dc | .&.1.*..%....@..A....C....a...B. |
| 3920 | 11 17 98 01 98 31 98 22 99 05 a0 02 99 0a a0 52 a8 32 b0 01 a9 36 a1 5d b0 62 d1 24 38 d3 11 39 | .....1.".......R.2...6.].b.$8..9 |
| 3940 | 88 42 dc 11 17 98 03 9c 67 a0 62 9b 6b a8 51 b0 12 a9 56 b0 61 a9 5a d1 1e 38 b8 42 d1 1d 3e d3 | .B......g.b.k.Q...V.a.Z..8.B..>. |
| 3960 | 11 3f 89 42 f0 09 04 09 40 01 f4 0a 00 0c 12 90 22 94 67 98 62 93 6b d3 0b 22 d0 04 22 72 31 00 | .?.B....@.......".g.b.k..".."r1. |
| 3980 | 00 00 63 02 00 00 00 00 00 00 00 00 00 00 00 05 00 00 00 03 00 00 00 f3 38 00 00 00 97 00 74 01 | ..c.....................8.....t. |
| 39a0 | 00 00 00 00 00 00 00 00 6a 02 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 74 04 00 00 | ........j...................t... |
| 39c0 | 00 00 00 00 00 00 7c 00 7c 01 ab 03 00 00 00 00 00 00 53 00 29 01 61 8d 05 00 00 0a 20 20 20 20 | ......|.|.........S.).a......... |
| 39e0 | 44 69 76 69 64 65 20 6f 6e 65 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 62 79 20 61 6e | Divide.one.Legendre.series.by.an |
| 3a00 | 6f 74 68 65 72 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 20 74 68 65 20 71 75 6f 74 69 65 6e 74 | other.......Returns.the.quotient |
| 3a20 | 2d 77 69 74 68 2d 72 65 6d 61 69 6e 64 65 72 20 6f 66 20 74 77 6f 20 4c 65 67 65 6e 64 72 65 20 | -with-remainder.of.two.Legendre. |
| 3a40 | 73 65 72 69 65 73 0a 20 20 20 20 60 63 31 60 20 2f 20 60 63 32 60 2e 20 20 54 68 65 20 61 72 67 | series.....`c1`./.`c2`...The.arg |
| 3a60 | 75 6d 65 6e 74 73 20 61 72 65 20 73 65 71 75 65 6e 63 65 73 20 6f 66 20 63 6f 65 66 66 69 63 69 | uments.are.sequences.of.coeffici |
| 3a80 | 65 6e 74 73 20 66 72 6f 6d 20 6c 6f 77 65 73 74 0a 20 20 20 20 6f 72 64 65 72 20 22 74 65 72 6d | ents.from.lowest.....order."term |
| 3aa0 | 22 20 74 6f 20 68 69 67 68 65 73 74 2c 20 65 2e 67 2e 2c 20 5b 31 2c 32 2c 33 5d 20 72 65 70 72 | ".to.highest,.e.g.,.[1,2,3].repr |
| 3ac0 | 65 73 65 6e 74 73 20 74 68 65 20 73 65 72 69 65 73 0a 20 20 20 20 60 60 50 5f 30 20 2b 20 32 2a | esents.the.series.....``P_0.+.2* |
| 3ae0 | 50 5f 31 20 2b 20 33 2a 50 5f 32 60 60 2e 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 | P_1.+.3*P_2``.......Parameters.. |
| 3b00 | 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 63 31 2c 20 63 32 20 3a 20 61 72 72 61 79 | ...----------.....c1,.c2.:.array |
| 3b20 | 5f 6c 69 6b 65 0a 20 20 20 20 20 20 20 20 31 2d 44 20 61 72 72 61 79 73 20 6f 66 20 4c 65 67 65 | _like.........1-D.arrays.of.Lege |
| 3b40 | 6e 64 72 65 20 73 65 72 69 65 73 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6f 72 64 65 72 65 64 | ndre.series.coefficients.ordered |
| 3b60 | 20 66 72 6f 6d 20 6c 6f 77 20 74 6f 0a 20 20 20 20 20 20 20 20 68 69 67 68 2e 0a 0a 20 20 20 20 | .from.low.to.........high....... |
| 3b80 | 52 65 74 75 72 6e 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 71 75 6f 2c 20 72 65 6d | Returns.....-------.....quo,.rem |
| 3ba0 | 20 3a 20 6e 64 61 72 72 61 79 73 0a 20 20 20 20 20 20 20 20 4f 66 20 4c 65 67 65 6e 64 72 65 20 | .:.ndarrays.........Of.Legendre. |
| 3bc0 | 73 65 72 69 65 73 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 72 65 70 72 65 73 65 6e 74 69 6e 67 | series.coefficients.representing |
| 3be0 | 20 74 68 65 20 71 75 6f 74 69 65 6e 74 20 61 6e 64 0a 20 20 20 20 20 20 20 20 72 65 6d 61 69 6e | .the.quotient.and.........remain |
| 3c00 | 64 65 72 2e 0a 0a 20 20 20 20 53 65 65 20 41 6c 73 6f 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a | der.......See.Also.....--------. |
| 3c20 | 20 20 20 20 6c 65 67 61 64 64 2c 20 6c 65 67 73 75 62 2c 20 6c 65 67 6d 75 6c 78 2c 20 6c 65 67 | ....legadd,.legsub,.legmulx,.leg |
| 3c40 | 6d 75 6c 2c 20 6c 65 67 70 6f 77 0a 0a 20 20 20 20 4e 6f 74 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d | mul,.legpow......Notes.....----- |
| 3c60 | 0a 20 20 20 20 49 6e 20 67 65 6e 65 72 61 6c 2c 20 74 68 65 20 28 70 6f 6c 79 6e 6f 6d 69 61 6c | .....In.general,.the.(polynomial |
| 3c80 | 29 20 64 69 76 69 73 69 6f 6e 20 6f 66 20 6f 6e 65 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 | ).division.of.one.Legendre.serie |
| 3ca0 | 73 20 62 79 20 61 6e 6f 74 68 65 72 0a 20 20 20 20 72 65 73 75 6c 74 73 20 69 6e 20 71 75 6f 74 | s.by.another.....results.in.quot |
| 3cc0 | 69 65 6e 74 20 61 6e 64 20 72 65 6d 61 69 6e 64 65 72 20 74 65 72 6d 73 20 74 68 61 74 20 61 72 | ient.and.remainder.terms.that.ar |
| 3ce0 | 65 20 6e 6f 74 20 69 6e 20 74 68 65 20 4c 65 67 65 6e 64 72 65 0a 20 20 20 20 70 6f 6c 79 6e 6f | e.not.in.the.Legendre.....polyno |
| 3d00 | 6d 69 61 6c 20 62 61 73 69 73 20 73 65 74 2e 20 20 54 68 75 73 2c 20 74 6f 20 65 78 70 72 65 73 | mial.basis.set...Thus,.to.expres |
| 3d20 | 73 20 74 68 65 73 65 20 72 65 73 75 6c 74 73 20 61 73 20 61 20 4c 65 67 65 6e 64 72 65 0a 20 20 | s.these.results.as.a.Legendre... |
| 3d40 | 20 20 73 65 72 69 65 73 2c 20 69 74 20 69 73 20 6e 65 63 65 73 73 61 72 79 20 74 6f 20 22 72 65 | ..series,.it.is.necessary.to."re |
| 3d60 | 70 72 6f 6a 65 63 74 22 20 74 68 65 20 72 65 73 75 6c 74 73 20 6f 6e 74 6f 20 74 68 65 20 4c 65 | project".the.results.onto.the.Le |
| 3d80 | 67 65 6e 64 72 65 0a 20 20 20 20 62 61 73 69 73 20 73 65 74 2c 20 77 68 69 63 68 20 6d 61 79 20 | gendre.....basis.set,.which.may. |
| 3da0 | 70 72 6f 64 75 63 65 20 22 75 6e 69 6e 74 75 69 74 69 76 65 22 20 28 62 75 74 20 63 6f 72 72 65 | produce."unintuitive".(but.corre |
| 3dc0 | 63 74 29 20 72 65 73 75 6c 74 73 3b 20 73 65 65 0a 20 20 20 20 45 78 61 6d 70 6c 65 73 20 73 65 | ct).results;.see.....Examples.se |
| 3de0 | 63 74 69 6f 6e 20 62 65 6c 6f 77 2e 0a 0a 20 20 20 20 45 78 61 6d 70 6c 65 73 0a 20 20 20 20 2d | ction.below.......Examples.....- |
| 3e00 | 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 3e 3e 3e 20 66 72 6f 6d 20 6e 75 6d 70 79 2e 70 6f 6c 79 6e | -------.....>>>.from.numpy.polyn |
| 3e20 | 6f 6d 69 61 6c 20 69 6d 70 6f 72 74 20 6c 65 67 65 6e 64 72 65 20 61 73 20 4c 0a 20 20 20 20 3e | omial.import.legendre.as.L.....> |
| 3e40 | 3e 3e 20 63 31 20 3d 20 28 31 2c 32 2c 33 29 0a 20 20 20 20 3e 3e 3e 20 63 32 20 3d 20 28 33 2c | >>.c1.=.(1,2,3).....>>>.c2.=.(3, |
| 3e60 | 32 2c 31 29 0a 20 20 20 20 3e 3e 3e 20 4c 2e 6c 65 67 64 69 76 28 63 31 2c 63 32 29 20 23 20 71 | 2,1).....>>>.L.legdiv(c1,c2).#.q |
| 3e80 | 75 6f 74 69 65 6e 74 20 22 69 6e 74 75 69 74 69 76 65 2c 22 20 72 65 6d 61 69 6e 64 65 72 20 6e | uotient."intuitive,".remainder.n |
| 3ea0 | 6f 74 0a 20 20 20 20 28 61 72 72 61 79 28 5b 33 2e 5d 29 2c 20 61 72 72 61 79 28 5b 2d 38 2e 2c | ot.....(array([3.]),.array([-8., |
| 3ec0 | 20 2d 34 2e 5d 29 29 0a 20 20 20 20 3e 3e 3e 20 63 32 20 3d 20 28 30 2c 31 2c 32 2c 33 29 0a 20 | .-4.])).....>>>.c2.=.(0,1,2,3).. |
| 3ee0 | 20 20 20 3e 3e 3e 20 4c 2e 6c 65 67 64 69 76 28 63 32 2c 63 31 29 20 23 20 6e 65 69 74 68 65 72 | ...>>>.L.legdiv(c2,c1).#.neither |
| 3f00 | 20 22 69 6e 74 75 69 74 69 76 65 22 0a 20 20 20 20 28 61 72 72 61 79 28 5b 2d 30 2e 30 37 34 30 | ."intuitive".....(array([-0.0740 |
| 3f20 | 37 34 30 37 2c 20 20 31 2e 36 36 36 36 36 36 36 37 5d 29 2c 20 61 72 72 61 79 28 5b 2d 31 2e 30 | 7407,..1.66666667]),.array([-1.0 |
| 3f40 | 33 37 30 33 37 30 34 2c 20 2d 32 2e 35 31 38 35 31 38 35 32 5d 29 29 20 23 20 6d 61 79 20 76 61 | 3703704,.-2.51851852])).#.may.va |
| 3f60 | 72 79 0a 0a 20 20 20 20 29 03 72 28 00 00 00 da 04 5f 64 69 76 72 0f 00 00 00 72 4a 00 00 00 73 | ry......).r(....._divr....rJ...s |
| 3f80 | 02 00 00 00 20 20 72 30 00 00 00 72 10 00 00 00 72 10 00 00 00 16 02 00 00 73 18 00 00 00 80 00 | ......r0...r....r........s...... |
| 3fa0 | f4 5c 01 00 0c 0e 8f 37 89 37 94 36 98 32 98 72 d3 0b 22 d0 04 22 72 31 00 00 00 63 03 00 00 00 | .\.....7.7.6.2.r..".."r1...c.... |
| 3fc0 | 00 00 00 00 00 00 00 00 06 00 00 00 03 00 00 00 f3 3a 00 00 00 97 00 74 01 00 00 00 00 00 00 00 | .................:.....t........ |
| 3fe0 | 00 6a 02 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 74 04 00 00 00 00 00 00 00 00 7c | .j...................t.........| |
| 4000 | 00 7c 01 7c 02 ab 04 00 00 00 00 00 00 53 00 29 01 61 e7 02 00 00 52 61 69 73 65 20 61 20 4c 65 | .|.|.........S.).a....Raise.a.Le |
| 4020 | 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 74 6f 20 61 20 70 6f 77 65 72 2e 0a 0a 20 20 20 20 52 | gendre.series.to.a.power.......R |
| 4040 | 65 74 75 72 6e 73 20 74 68 65 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 60 63 60 20 72 | eturns.the.Legendre.series.`c`.r |
| 4060 | 61 69 73 65 64 20 74 6f 20 74 68 65 20 70 6f 77 65 72 20 60 70 6f 77 60 2e 20 54 68 65 0a 20 20 | aised.to.the.power.`pow`..The... |
| 4080 | 20 20 61 72 67 75 6d 65 6e 74 20 60 63 60 20 69 73 20 61 20 73 65 71 75 65 6e 63 65 20 6f 66 20 | ..argument.`c`.is.a.sequence.of. |
| 40a0 | 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6f 72 64 65 72 65 64 20 66 72 6f 6d 20 6c 6f 77 20 74 6f | coefficients.ordered.from.low.to |
| 40c0 | 20 68 69 67 68 2e 0a 20 20 20 20 69 2e 65 2e 2c 20 5b 31 2c 32 2c 33 5d 20 69 73 20 74 68 65 20 | .high......i.e.,.[1,2,3].is.the. |
| 40e0 | 73 65 72 69 65 73 20 20 60 60 50 5f 30 20 2b 20 32 2a 50 5f 31 20 2b 20 33 2a 50 5f 32 2e 60 60 | series..``P_0.+.2*P_1.+.3*P_2.`` |
| 4100 | 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a | ......Parameters.....----------. |
| 4120 | 20 20 20 20 63 20 3a 20 61 72 72 61 79 5f 6c 69 6b 65 0a 20 20 20 20 20 20 20 20 31 2d 44 20 61 | ....c.:.array_like.........1-D.a |
| 4140 | 72 72 61 79 20 6f 66 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 63 6f 65 66 66 69 63 69 | rray.of.Legendre.series.coeffici |
| 4160 | 65 6e 74 73 20 6f 72 64 65 72 65 64 20 66 72 6f 6d 20 6c 6f 77 20 74 6f 0a 20 20 20 20 20 20 20 | ents.ordered.from.low.to........ |
| 4180 | 20 68 69 67 68 2e 0a 20 20 20 20 70 6f 77 20 3a 20 69 6e 74 65 67 65 72 0a 20 20 20 20 20 20 20 | .high......pow.:.integer........ |
| 41a0 | 20 50 6f 77 65 72 20 74 6f 20 77 68 69 63 68 20 74 68 65 20 73 65 72 69 65 73 20 77 69 6c 6c 20 | .Power.to.which.the.series.will. |
| 41c0 | 62 65 20 72 61 69 73 65 64 0a 20 20 20 20 6d 61 78 70 6f 77 65 72 20 3a 20 69 6e 74 65 67 65 72 | be.raised.....maxpower.:.integer |
| 41e0 | 2c 20 6f 70 74 69 6f 6e 61 6c 0a 20 20 20 20 20 20 20 20 4d 61 78 69 6d 75 6d 20 70 6f 77 65 72 | ,.optional.........Maximum.power |
| 4200 | 20 61 6c 6c 6f 77 65 64 2e 20 54 68 69 73 20 69 73 20 6d 61 69 6e 6c 79 20 74 6f 20 6c 69 6d 69 | .allowed..This.is.mainly.to.limi |
| 4220 | 74 20 67 72 6f 77 74 68 20 6f 66 20 74 68 65 20 73 65 72 69 65 73 0a 20 20 20 20 20 20 20 20 74 | t.growth.of.the.series.........t |
| 4240 | 6f 20 75 6e 6d 61 6e 61 67 65 61 62 6c 65 20 73 69 7a 65 2e 20 44 65 66 61 75 6c 74 20 69 73 20 | o.unmanageable.size..Default.is. |
| 4260 | 31 36 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 | 16......Returns.....-------..... |
| 4280 | 63 6f 65 66 20 3a 20 6e 64 61 72 72 61 79 0a 20 20 20 20 20 20 20 20 4c 65 67 65 6e 64 72 65 20 | coef.:.ndarray.........Legendre. |
| 42a0 | 73 65 72 69 65 73 20 6f 66 20 70 6f 77 65 72 2e 0a 0a 20 20 20 20 53 65 65 20 41 6c 73 6f 0a 20 | series.of.power.......See.Also.. |
| 42c0 | 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 6c 65 67 61 64 64 2c 20 6c 65 67 73 75 62 2c 20 | ...--------.....legadd,.legsub,. |
| 42e0 | 6c 65 67 6d 75 6c 78 2c 20 6c 65 67 6d 75 6c 2c 20 6c 65 67 64 69 76 0a 0a 20 20 20 20 29 03 72 | legmulx,.legmul,.legdiv......).r |
| 4300 | 28 00 00 00 da 04 5f 70 6f 77 72 0f 00 00 00 29 03 72 3a 00 00 00 da 03 70 6f 77 da 08 6d 61 78 | (....._powr....).r:.....pow..max |
| 4320 | 70 6f 77 65 72 73 03 00 00 00 20 20 20 72 30 00 00 00 72 11 00 00 00 72 11 00 00 00 47 02 00 00 | powers.......r0...r....r....G... |
| 4340 | 73 19 00 00 00 80 00 f4 38 00 0c 0e 8f 37 89 37 94 36 98 31 98 63 a0 38 d3 0b 2c d0 04 2c 72 31 | s.......8....7.7.6.1.c.8..,..,r1 |
| 4360 | 00 00 00 63 04 00 00 00 00 00 00 00 00 00 00 00 07 00 00 00 03 00 00 00 f3 22 03 00 00 97 00 74 | ...c.....................".....t |
| 4380 | 01 00 00 00 00 00 00 00 00 6a 02 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 00 64 | .........j...................|.d |
| 43a0 | 01 64 02 ac 03 ab 03 00 00 00 00 00 00 7d 00 7c 00 6a 04 00 00 00 00 00 00 00 00 00 00 00 00 00 | .d...........}.|.j.............. |
| 43c0 | 00 00 00 00 00 6a 06 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 64 04 76 00 72 1f 7c | .....j...................d.v.r.| |
| 43e0 | 00 6a 09 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 74 00 00 00 00 00 00 00 00 00 6a | .j...................t.........j |
| 4400 | 0a 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ab 01 00 00 00 00 00 00 7d 00 74 0d 00 | ...........................}.t.. |
| 4420 | 00 00 00 00 00 00 00 6a 0e 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 01 64 05 ab | .......j...................|.d.. |
| 4440 | 02 00 00 00 00 00 00 7d 04 74 0d 00 00 00 00 00 00 00 00 6a 0e 00 00 00 00 00 00 00 00 00 00 00 | .......}.t.........j............ |
| 4460 | 00 00 00 00 00 00 00 7c 03 64 06 ab 02 00 00 00 00 00 00 7d 05 7c 04 64 07 6b 02 00 00 72 0b 74 | .......|.d.........}.|.d.k...r.t |
| 4480 | 11 00 00 00 00 00 00 00 00 64 08 ab 01 00 00 00 00 00 00 82 01 74 13 00 00 00 00 00 00 00 00 7c | .........d...........t.........| |
| 44a0 | 05 7c 00 6a 14 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ab 02 00 00 00 00 00 00 7d | .|.j...........................} |
| 44c0 | 05 7c 04 64 07 6b 28 00 00 72 02 7c 00 53 00 74 01 00 00 00 00 00 00 00 00 6a 16 00 00 00 00 00 | .|.d.k(..r.|.S.t.........j...... |
| 44e0 | 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 00 7c 05 64 07 ab 03 00 00 00 00 00 00 7d 00 74 19 00 | .............|.|.d.........}.t.. |
| 4500 | 00 00 00 00 00 00 00 7c 00 ab 01 00 00 00 00 00 00 7d 06 7c 04 7c 06 6b 5c 00 00 72 09 7c 00 64 | .......|.........}.|.|.k\..r.|.d |
| 4520 | 09 64 01 1a 00 64 07 7a 05 00 00 7d 00 6e 9f 74 1b 00 00 00 00 00 00 00 00 7c 04 ab 01 00 00 00 | .d...d.z...}.n.t.........|...... |
| 4540 | 00 00 00 44 00 5d 91 00 00 7d 07 7c 06 64 01 7a 0a 00 00 7d 06 7c 00 7c 02 7a 12 00 00 7d 00 74 | ...D.]...}.|.d.z...}.|.|.z...}.t |
| 4560 | 01 00 00 00 00 00 00 00 00 6a 1c 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 06 66 | .........j...................|.f |
| 4580 | 01 7c 00 6a 1e 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 64 01 64 09 1a 00 7a 00 00 | .|.j...................d.d...z.. |
| 45a0 | 00 7c 00 6a 04 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ac 0a ab 02 00 00 00 00 00 | .|.j............................ |
| 45c0 | 00 7d 08 74 1b 00 00 00 00 00 00 00 00 7c 06 64 0b 64 0c ab 03 00 00 00 00 00 00 44 00 5d 29 00 | .}.t.........|.d.d.........D.]). |
| 45e0 | 00 7d 09 64 0b 7c 09 7a 05 00 00 64 01 7a 0a 00 00 7c 00 7c 09 19 00 00 00 7a 05 00 00 7c 08 7c | .}.d.|.z...d.z...|.|.....z...|.| |
| 4600 | 09 64 01 7a 0a 00 00 3c 00 00 00 7c 00 7c 09 64 0b 7a 0a 00 00 78 02 78 02 19 00 00 00 7c 00 7c | .d.z...<...|.|.d.z...x.x.....|.| |
| 4620 | 09 19 00 00 00 7a 0d 00 00 63 03 63 02 3c 00 00 00 8c 2b 04 00 7c 06 64 01 6b 44 00 00 72 0b 64 | .....z...c.c.<....+..|.d.kD..r.d |
| 4640 | 0d 7c 00 64 0b 19 00 00 00 7a 05 00 00 7c 08 64 01 3c 00 00 00 7c 00 64 01 19 00 00 00 7c 08 64 | .|.d.....z...|.d.<...|.d.....|.d |
| 4660 | 07 3c 00 00 00 7c 08 7d 00 8c 93 04 00 74 01 00 00 00 00 00 00 00 00 6a 16 00 00 00 00 00 00 00 | .<...|.}.....t.........j........ |
| 4680 | 00 00 00 00 00 00 00 00 00 00 00 7c 00 64 07 7c 05 ab 03 00 00 00 00 00 00 7d 00 7c 00 53 00 29 | ...........|.d.|.........}.|.S.) |
| 46a0 | 0e 61 5f 07 00 00 0a 20 20 20 20 44 69 66 66 65 72 65 6e 74 69 61 74 65 20 61 20 4c 65 67 65 6e | .a_........Differentiate.a.Legen |
| 46c0 | 64 72 65 20 73 65 72 69 65 73 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 20 74 68 65 20 4c 65 67 | dre.series.......Returns.the.Leg |
| 46e0 | 65 6e 64 72 65 20 73 65 72 69 65 73 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 60 63 60 20 64 69 | endre.series.coefficients.`c`.di |
| 4700 | 66 66 65 72 65 6e 74 69 61 74 65 64 20 60 6d 60 20 74 69 6d 65 73 0a 20 20 20 20 61 6c 6f 6e 67 | fferentiated.`m`.times.....along |
| 4720 | 20 60 61 78 69 73 60 2e 20 20 41 74 20 65 61 63 68 20 69 74 65 72 61 74 69 6f 6e 20 74 68 65 20 | .`axis`...At.each.iteration.the. |
| 4740 | 72 65 73 75 6c 74 20 69 73 20 6d 75 6c 74 69 70 6c 69 65 64 20 62 79 20 60 73 63 6c 60 20 28 74 | result.is.multiplied.by.`scl`.(t |
| 4760 | 68 65 0a 20 20 20 20 73 63 61 6c 69 6e 67 20 66 61 63 74 6f 72 20 69 73 20 66 6f 72 20 75 73 65 | he.....scaling.factor.is.for.use |
| 4780 | 20 69 6e 20 61 20 6c 69 6e 65 61 72 20 63 68 61 6e 67 65 20 6f 66 20 76 61 72 69 61 62 6c 65 29 | .in.a.linear.change.of.variable) |
| 47a0 | 2e 20 54 68 65 20 61 72 67 75 6d 65 6e 74 0a 20 20 20 20 60 63 60 20 69 73 20 61 6e 20 61 72 72 | ..The.argument.....`c`.is.an.arr |
| 47c0 | 61 79 20 6f 66 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 66 72 6f 6d 20 6c 6f 77 20 74 6f 20 68 | ay.of.coefficients.from.low.to.h |
| 47e0 | 69 67 68 20 64 65 67 72 65 65 20 61 6c 6f 6e 67 20 65 61 63 68 0a 20 20 20 20 61 78 69 73 2c 20 | igh.degree.along.each.....axis,. |
| 4800 | 65 2e 67 2e 2c 20 5b 31 2c 32 2c 33 5d 20 72 65 70 72 65 73 65 6e 74 73 20 74 68 65 20 73 65 72 | e.g.,.[1,2,3].represents.the.ser |
| 4820 | 69 65 73 20 60 60 31 2a 4c 5f 30 20 2b 20 32 2a 4c 5f 31 20 2b 20 33 2a 4c 5f 32 60 60 0a 20 20 | ies.``1*L_0.+.2*L_1.+.3*L_2``... |
| 4840 | 20 20 77 68 69 6c 65 20 5b 5b 31 2c 32 5d 2c 5b 31 2c 32 5d 5d 20 72 65 70 72 65 73 65 6e 74 73 | ..while.[[1,2],[1,2]].represents |
| 4860 | 20 60 60 31 2a 4c 5f 30 28 78 29 2a 4c 5f 30 28 79 29 20 2b 20 31 2a 4c 5f 31 28 78 29 2a 4c 5f | .``1*L_0(x)*L_0(y).+.1*L_1(x)*L_ |
| 4880 | 30 28 79 29 20 2b 0a 20 20 20 20 32 2a 4c 5f 30 28 78 29 2a 4c 5f 31 28 79 29 20 2b 20 32 2a 4c | 0(y).+.....2*L_0(x)*L_1(y).+.2*L |
| 48a0 | 5f 31 28 78 29 2a 4c 5f 31 28 79 29 60 60 20 69 66 20 61 78 69 73 3d 30 20 69 73 20 60 60 78 60 | _1(x)*L_1(y)``.if.axis=0.is.``x` |
| 48c0 | 60 20 61 6e 64 20 61 78 69 73 3d 31 20 69 73 0a 20 20 20 20 60 60 79 60 60 2e 0a 0a 20 20 20 20 | `.and.axis=1.is.....``y``....... |
| 48e0 | 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 63 20 | Parameters.....----------.....c. |
| 4900 | 3a 20 61 72 72 61 79 5f 6c 69 6b 65 0a 20 20 20 20 20 20 20 20 41 72 72 61 79 20 6f 66 20 4c 65 | :.array_like.........Array.of.Le |
| 4920 | 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 63 6f 65 66 66 69 63 69 65 6e 74 73 2e 20 49 66 20 63 | gendre.series.coefficients..If.c |
| 4940 | 20 69 73 20 6d 75 6c 74 69 64 69 6d 65 6e 73 69 6f 6e 61 6c 20 74 68 65 0a 20 20 20 20 20 20 20 | .is.multidimensional.the........ |
| 4960 | 20 64 69 66 66 65 72 65 6e 74 20 61 78 69 73 20 63 6f 72 72 65 73 70 6f 6e 64 20 74 6f 20 64 69 | .different.axis.correspond.to.di |
| 4980 | 66 66 65 72 65 6e 74 20 76 61 72 69 61 62 6c 65 73 20 77 69 74 68 20 74 68 65 20 64 65 67 72 65 | fferent.variables.with.the.degre |
| 49a0 | 65 20 69 6e 0a 20 20 20 20 20 20 20 20 65 61 63 68 20 61 78 69 73 20 67 69 76 65 6e 20 62 79 20 | e.in.........each.axis.given.by. |
| 49c0 | 74 68 65 20 63 6f 72 72 65 73 70 6f 6e 64 69 6e 67 20 69 6e 64 65 78 2e 0a 20 20 20 20 6d 20 3a | the.corresponding.index......m.: |
| 49e0 | 20 69 6e 74 2c 20 6f 70 74 69 6f 6e 61 6c 0a 20 20 20 20 20 20 20 20 4e 75 6d 62 65 72 20 6f 66 | .int,.optional.........Number.of |
| 4a00 | 20 64 65 72 69 76 61 74 69 76 65 73 20 74 61 6b 65 6e 2c 20 6d 75 73 74 20 62 65 20 6e 6f 6e 2d | .derivatives.taken,.must.be.non- |
| 4a20 | 6e 65 67 61 74 69 76 65 2e 20 28 44 65 66 61 75 6c 74 3a 20 31 29 0a 20 20 20 20 73 63 6c 20 3a | negative..(Default:.1).....scl.: |
| 4a40 | 20 73 63 61 6c 61 72 2c 20 6f 70 74 69 6f 6e 61 6c 0a 20 20 20 20 20 20 20 20 45 61 63 68 20 64 | .scalar,.optional.........Each.d |
| 4a60 | 69 66 66 65 72 65 6e 74 69 61 74 69 6f 6e 20 69 73 20 6d 75 6c 74 69 70 6c 69 65 64 20 62 79 20 | ifferentiation.is.multiplied.by. |
| 4a80 | 60 73 63 6c 60 2e 20 20 54 68 65 20 65 6e 64 20 72 65 73 75 6c 74 20 69 73 0a 20 20 20 20 20 20 | `scl`...The.end.result.is....... |
| 4aa0 | 20 20 6d 75 6c 74 69 70 6c 69 63 61 74 69 6f 6e 20 62 79 20 60 60 73 63 6c 2a 2a 6d 60 60 2e 20 | ..multiplication.by.``scl**m``.. |
| 4ac0 | 20 54 68 69 73 20 69 73 20 66 6f 72 20 75 73 65 20 69 6e 20 61 20 6c 69 6e 65 61 72 20 63 68 61 | .This.is.for.use.in.a.linear.cha |
| 4ae0 | 6e 67 65 20 6f 66 0a 20 20 20 20 20 20 20 20 76 61 72 69 61 62 6c 65 2e 20 28 44 65 66 61 75 6c | nge.of.........variable..(Defaul |
| 4b00 | 74 3a 20 31 29 0a 20 20 20 20 61 78 69 73 20 3a 20 69 6e 74 2c 20 6f 70 74 69 6f 6e 61 6c 0a 20 | t:.1).....axis.:.int,.optional.. |
| 4b20 | 20 20 20 20 20 20 20 41 78 69 73 20 6f 76 65 72 20 77 68 69 63 68 20 74 68 65 20 64 65 72 69 76 | .......Axis.over.which.the.deriv |
| 4b40 | 61 74 69 76 65 20 69 73 20 74 61 6b 65 6e 2e 20 28 44 65 66 61 75 6c 74 3a 20 30 29 2e 0a 0a 20 | ative.is.taken..(Default:.0).... |
| 4b60 | 20 20 20 52 65 74 75 72 6e 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 64 65 72 20 3a | ...Returns.....-------.....der.: |
| 4b80 | 20 6e 64 61 72 72 61 79 0a 20 20 20 20 20 20 20 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 | .ndarray.........Legendre.series |
| 4ba0 | 20 6f 66 20 74 68 65 20 64 65 72 69 76 61 74 69 76 65 2e 0a 0a 20 20 20 20 53 65 65 20 41 6c 73 | .of.the.derivative.......See.Als |
| 4bc0 | 6f 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 6c 65 67 69 6e 74 0a 0a 20 20 20 20 4e | o.....--------.....legint......N |
| 4be0 | 6f 74 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 0a 20 20 20 20 49 6e 20 67 65 6e 65 72 61 6c 2c 20 74 | otes.....-----.....In.general,.t |
| 4c00 | 68 65 20 72 65 73 75 6c 74 20 6f 66 20 64 69 66 66 65 72 65 6e 74 69 61 74 69 6e 67 20 61 20 4c | he.result.of.differentiating.a.L |
| 4c20 | 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 64 6f 65 73 20 6e 6f 74 0a 20 20 20 20 72 65 73 65 | egendre.series.does.not.....rese |
| 4c40 | 6d 62 6c 65 20 74 68 65 20 73 61 6d 65 20 6f 70 65 72 61 74 69 6f 6e 20 6f 6e 20 61 20 70 6f 77 | mble.the.same.operation.on.a.pow |
| 4c60 | 65 72 20 73 65 72 69 65 73 2e 20 54 68 75 73 20 74 68 65 20 72 65 73 75 6c 74 20 6f 66 20 74 68 | er.series..Thus.the.result.of.th |
| 4c80 | 69 73 0a 20 20 20 20 66 75 6e 63 74 69 6f 6e 20 6d 61 79 20 62 65 20 22 75 6e 69 6e 74 75 69 74 | is.....function.may.be."unintuit |
| 4ca0 | 69 76 65 2c 22 20 61 6c 62 65 69 74 20 63 6f 72 72 65 63 74 3b 20 73 65 65 20 45 78 61 6d 70 6c | ive,".albeit.correct;.see.Exampl |
| 4cc0 | 65 73 20 73 65 63 74 69 6f 6e 0a 20 20 20 20 62 65 6c 6f 77 2e 0a 0a 20 20 20 20 45 78 61 6d 70 | es.section.....below.......Examp |
| 4ce0 | 6c 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 3e 3e 3e 20 66 72 6f 6d 20 6e 75 | les.....--------.....>>>.from.nu |
| 4d00 | 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 20 69 6d 70 6f 72 74 20 6c 65 67 65 6e 64 72 65 20 61 | mpy.polynomial.import.legendre.a |
| 4d20 | 73 20 4c 0a 20 20 20 20 3e 3e 3e 20 63 20 3d 20 28 31 2c 32 2c 33 2c 34 29 0a 20 20 20 20 3e 3e | s.L.....>>>.c.=.(1,2,3,4).....>> |
| 4d40 | 3e 20 4c 2e 6c 65 67 64 65 72 28 63 29 0a 20 20 20 20 61 72 72 61 79 28 5b 20 20 36 2e 2c 20 20 | >.L.legder(c).....array([..6.,.. |
| 4d60 | 20 39 2e 2c 20 20 32 30 2e 5d 29 0a 20 20 20 20 3e 3e 3e 20 4c 2e 6c 65 67 64 65 72 28 63 2c 20 | .9.,..20.]).....>>>.L.legder(c,. |
| 4d80 | 33 29 0a 20 20 20 20 61 72 72 61 79 28 5b 36 30 2e 5d 29 0a 20 20 20 20 3e 3e 3e 20 4c 2e 6c 65 | 3).....array([60.]).....>>>.L.le |
| 4da0 | 67 64 65 72 28 63 2c 20 73 63 6c 3d 2d 31 29 0a 20 20 20 20 61 72 72 61 79 28 5b 20 2d 36 2e 2c | gder(c,.scl=-1).....array([.-6., |
| 4dc0 | 20 20 2d 39 2e 2c 20 2d 32 30 2e 5d 29 0a 20 20 20 20 3e 3e 3e 20 4c 2e 6c 65 67 64 65 72 28 63 | ..-9.,.-20.]).....>>>.L.legder(c |
| 4de0 | 2c 20 32 2c 2d 31 29 0a 20 20 20 20 61 72 72 61 79 28 5b 20 20 39 2e 2c 20 20 36 30 2e 5d 29 0a | ,.2,-1).....array([..9.,..60.]). |
| 4e00 | 0a 20 20 20 20 72 04 00 00 00 54 a9 02 da 05 6e 64 6d 69 6e da 04 63 6f 70 79 fa 0d 3f 62 42 68 | .....r....T....ndmin..copy..?bBh |
| 4e20 | 48 69 49 6c 4c 71 51 70 50 7a 17 74 68 65 20 6f 72 64 65 72 20 6f 66 20 64 65 72 69 76 61 74 69 | HiIlLqQpPz.the.order.of.derivati |
| 4e40 | 6f 6e fa 08 74 68 65 20 61 78 69 73 72 02 00 00 00 7a 2c 54 68 65 20 6f 72 64 65 72 20 6f 66 20 | on..the.axisr....z,The.order.of. |
| 4e60 | 64 65 72 69 76 61 74 69 6f 6e 20 6d 75 73 74 20 62 65 20 6e 6f 6e 2d 6e 65 67 61 74 69 76 65 4e | derivation.must.be.non-negativeN |
| 4e80 | 72 4f 00 00 00 72 38 00 00 00 72 27 00 00 00 72 36 00 00 00 29 10 72 41 00 00 00 72 42 00 00 00 | rO...r8...r'...r6...).rA...rB... |
| 4ea0 | 72 50 00 00 00 da 04 63 68 61 72 da 06 61 73 74 79 70 65 da 06 64 6f 75 62 6c 65 72 28 00 00 00 | rP.....char..astype..doubler(... |
| 4ec0 | da 07 5f 61 73 5f 69 6e 74 da 0a 56 61 6c 75 65 45 72 72 6f 72 72 03 00 00 00 da 04 6e 64 69 6d | .._as_int..ValueErrorr......ndim |
| 4ee0 | da 08 6d 6f 76 65 61 78 69 73 72 2a 00 00 00 72 2b 00 00 00 72 51 00 00 00 da 05 73 68 61 70 65 | ..moveaxisr*...r+...rQ.....shape |
| 4f00 | 29 0a 72 3a 00 00 00 da 01 6d 72 44 00 00 00 da 04 61 78 69 73 da 03 63 6e 74 da 05 69 61 78 69 | ).r:.....mrD.....axis..cnt..iaxi |
| 4f20 | 73 72 3b 00 00 00 72 2f 00 00 00 da 03 64 65 72 72 53 00 00 00 73 0a 00 00 00 20 20 20 20 20 20 | sr;...r/.....derrS...s.......... |
| 4f40 | 20 20 20 20 72 30 00 00 00 72 13 00 00 00 72 13 00 00 00 66 02 00 00 73 94 01 00 00 80 00 f4 74 | ....r0...r....r....f...s.......t |
| 4f60 | 01 00 09 0b 8f 08 89 08 90 11 98 21 a0 24 d4 08 27 80 41 d8 07 08 87 77 81 77 87 7c 81 7c 90 7f | ...........!.$..'.A....w.w.|.|.. |
| 4f80 | d1 07 26 d8 0c 0d 8f 48 89 48 94 52 97 59 91 59 d3 0c 1f 88 01 dc 0a 0c 8f 2a 89 2a 90 51 d0 18 | ..&....H.H.R.Y.Y.........*.*.Q.. |
| 4fa0 | 31 d3 0a 32 80 43 dc 0c 0e 8f 4a 89 4a 90 74 98 5a d3 0c 28 80 45 d8 07 0a 88 51 82 77 dc 0e 18 | 1..2.C....J.J.t.Z..(.E....Q.w... |
| 4fc0 | d0 19 47 d3 0e 48 d0 08 48 dc 0c 20 a0 15 a8 01 af 06 a9 06 d3 0c 2f 80 45 e0 07 0a 88 61 82 78 | ..G..H..H............./.E....a.x |
| 4fe0 | d8 0f 10 88 08 e4 08 0a 8f 0b 89 0b 90 41 90 75 98 61 d3 08 20 80 41 dc 08 0b 88 41 8b 06 80 41 | .............A.u.a....A....A...A |
| 5000 | d8 07 0a 88 61 82 78 d8 0c 0d 88 62 88 71 88 45 90 41 89 49 89 01 e4 11 16 90 73 93 1a f2 00 0a | ....a.x....b.q.E.A.I......s..... |
| 5020 | 09 14 88 41 d8 10 11 90 41 91 05 88 41 d8 0c 0d 90 13 89 48 88 41 dc 12 14 97 28 91 28 98 41 98 | ...A....A...A......H.A....(.(.A. |
| 5040 | 34 a0 21 a7 27 a1 27 a8 21 a8 22 a0 2b d1 1b 2d b0 51 b7 57 b1 57 d4 12 3d 88 43 dc 15 1a 98 31 | 4.!.'.'.!.".+..-.Q.W.W..=.C....1 |
| 5060 | 98 61 a0 12 93 5f f2 00 02 0d 21 90 01 d8 1e 1f a0 21 99 65 a0 61 99 69 a8 31 a8 51 a9 34 d1 1d | .a..._....!......!.e.a.i.1.Q.4.. |
| 5080 | 2f 90 03 90 41 98 01 91 45 91 0a d8 10 11 90 21 90 61 91 25 93 08 98 41 98 61 99 44 d1 10 20 94 | /...A...E......!.a.%...A.a.D.... |
| 50a0 | 08 f0 05 02 0d 21 f0 06 00 10 11 90 31 8a 75 d8 19 1a 98 51 98 71 99 54 99 18 90 03 90 41 91 06 | .....!......1.u....Q.q.T.....A.. |
| 50c0 | d8 15 16 90 71 91 54 88 43 90 01 89 46 d8 10 13 89 41 f0 15 0a 09 14 f4 16 00 09 0b 8f 0b 89 0b | ....q.T.C...F....A.............. |
| 50e0 | 90 41 90 71 98 25 d3 08 20 80 41 d8 0b 0c 80 48 72 31 00 00 00 63 06 00 00 00 00 00 00 00 00 00 | .A.q.%....A....Hr1...c.......... |
| 5100 | 00 00 09 00 00 00 03 00 00 00 f3 da 04 00 00 97 00 74 01 00 00 00 00 00 00 00 00 6a 02 00 00 00 | .................t.........j.... |
| 5120 | 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 00 64 01 64 02 ac 03 ab 03 00 00 00 00 00 00 7d | ...............|.d.d...........} |
| 5140 | 00 7c 00 6a 04 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 6a 06 00 00 00 00 00 00 00 | .|.j...................j........ |
| 5160 | 00 00 00 00 00 00 00 00 00 00 00 64 04 76 00 72 1f 7c 00 6a 09 00 00 00 00 00 00 00 00 00 00 00 | ...........d.v.r.|.j............ |
| 5180 | 00 00 00 00 00 00 00 74 00 00 00 00 00 00 00 00 00 6a 0a 00 00 00 00 00 00 00 00 00 00 00 00 00 | .......t.........j.............. |
| 51a0 | 00 00 00 00 00 ab 01 00 00 00 00 00 00 7d 00 74 01 00 00 00 00 00 00 00 00 6a 0c 00 00 00 00 00 | .............}.t.........j...... |
| 51c0 | 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 02 ab 01 00 00 00 00 00 00 73 03 7c 02 67 01 7d 02 74 | .............|.........s.|.g.}.t |
| 51e0 | 0f 00 00 00 00 00 00 00 00 6a 10 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 01 64 | .........j...................|.d |
| 5200 | 05 ab 02 00 00 00 00 00 00 7d 06 74 0f 00 00 00 00 00 00 00 00 6a 10 00 00 00 00 00 00 00 00 00 | .........}.t.........j.......... |
| 5220 | 00 00 00 00 00 00 00 00 00 7c 05 64 06 ab 02 00 00 00 00 00 00 7d 07 7c 06 64 07 6b 02 00 00 72 | .........|.d.........}.|.d.k...r |
| 5240 | 0b 74 13 00 00 00 00 00 00 00 00 64 08 ab 01 00 00 00 00 00 00 82 01 74 15 00 00 00 00 00 00 00 | .t.........d...........t........ |
| 5260 | 00 7c 02 ab 01 00 00 00 00 00 00 7c 06 6b 44 00 00 72 0b 74 13 00 00 00 00 00 00 00 00 64 09 ab | .|.........|.kD..r.t.........d.. |
| 5280 | 01 00 00 00 00 00 00 82 01 74 01 00 00 00 00 00 00 00 00 6a 16 00 00 00 00 00 00 00 00 00 00 00 | .........t.........j............ |
| 52a0 | 00 00 00 00 00 00 00 7c 03 ab 01 00 00 00 00 00 00 64 07 6b 37 00 00 72 0b 74 13 00 00 00 00 00 | .......|.........d.k7..r.t...... |
| 52c0 | 00 00 00 64 0a ab 01 00 00 00 00 00 00 82 01 74 01 00 00 00 00 00 00 00 00 6a 16 00 00 00 00 00 | ...d...........t.........j...... |
| 52e0 | 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 04 ab 01 00 00 00 00 00 00 64 07 6b 37 00 00 72 0b 74 | .............|.........d.k7..r.t |
| 5300 | 13 00 00 00 00 00 00 00 00 64 0b ab 01 00 00 00 00 00 00 82 01 74 19 00 00 00 00 00 00 00 00 7c | .........d...........t.........| |
| 5320 | 07 7c 00 6a 16 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ab 02 00 00 00 00 00 00 7d | .|.j...........................} |
| 5340 | 07 7c 06 64 07 6b 28 00 00 72 02 7c 00 53 00 74 01 00 00 00 00 00 00 00 00 6a 1a 00 00 00 00 00 | .|.d.k(..r.|.S.t.........j...... |
| 5360 | 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 00 7c 07 64 07 ab 03 00 00 00 00 00 00 7d 00 74 1d 00 | .............|.|.d.........}.t.. |
| 5380 | 00 00 00 00 00 00 00 7c 02 ab 01 00 00 00 00 00 00 64 07 67 01 7c 06 74 15 00 00 00 00 00 00 00 | .......|.........d.g.|.t........ |
| 53a0 | 00 7c 02 ab 01 00 00 00 00 00 00 7a 0a 00 00 7a 05 00 00 7a 00 00 00 7d 02 74 1f 00 00 00 00 00 | .|.........z...z...z...}.t...... |
| 53c0 | 00 00 00 7c 06 ab 01 00 00 00 00 00 00 44 00 5d f1 00 00 7d 08 74 15 00 00 00 00 00 00 00 00 7c | ...|.........D.]...}.t.........| |
| 53e0 | 00 ab 01 00 00 00 00 00 00 7d 09 7c 00 7c 04 7a 12 00 00 7d 00 7c 09 64 01 6b 28 00 00 72 2c 74 | .........}.|.|.z...}.|.d.k(..r,t |
| 5400 | 01 00 00 00 00 00 00 00 00 6a 20 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 00 64 | .........j...................|.d |
| 5420 | 07 19 00 00 00 64 07 6b 28 00 00 ab 01 00 00 00 00 00 00 72 11 7c 00 64 07 78 02 78 02 19 00 00 | .....d.k(..........r.|.d.x.x.... |
| 5440 | 00 7c 02 7c 08 19 00 00 00 7a 0d 00 00 63 03 63 02 3c 00 00 00 8c 44 74 01 00 00 00 00 00 00 00 | .|.|.....z...c.c.<....Dt........ |
| 5460 | 00 6a 22 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 09 64 01 7a 00 00 00 66 01 7c | .j"..................|.d.z...f.| |
| 5480 | 00 6a 24 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 64 01 64 0c 1a 00 7a 00 00 00 7c | .j$..................d.d...z...| |
| 54a0 | 00 6a 04 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ac 0d ab 02 00 00 00 00 00 00 7d | .j.............................} |
| 54c0 | 0a 7c 00 64 07 19 00 00 00 64 07 7a 05 00 00 7c 0a 64 07 3c 00 00 00 7c 00 64 07 19 00 00 00 7c | .|.d.....d.z...|.d.<...|.d.....| |
| 54e0 | 0a 64 01 3c 00 00 00 7c 09 64 01 6b 44 00 00 72 0b 7c 00 64 01 19 00 00 00 64 0e 7a 0b 00 00 7c | .d.<...|.d.kD..r.|.d.....d.z...| |
| 5500 | 0a 64 0f 3c 00 00 00 74 1f 00 00 00 00 00 00 00 00 64 0f 7c 09 ab 02 00 00 00 00 00 00 44 00 5d | .d.<...t.........d.|.........D.] |
| 5520 | 28 00 00 7d 0b 7c 00 7c 0b 19 00 00 00 64 0f 7c 0b 7a 05 00 00 64 01 7a 00 00 00 7a 0b 00 00 7d | (..}.|.|.....d.|.z...d.z...z...} |
| 5540 | 0c 7c 0c 7c 0a 7c 0b 64 01 7a 00 00 00 3c 00 00 00 7c 0a 7c 0b 64 01 7a 0a 00 00 78 02 78 02 19 | .|.|.|.d.z...<...|.|.d.z...x.x.. |
| 5560 | 00 00 00 7c 0c 7a 17 00 00 63 03 63 02 3c 00 00 00 8c 2a 04 00 7c 0a 64 07 78 02 78 02 19 00 00 | ...|.z...c.c.<....*..|.d.x.x.... |
| 5580 | 00 7c 02 7c 08 19 00 00 00 74 27 00 00 00 00 00 00 00 00 7c 03 7c 0a ab 02 00 00 00 00 00 00 7a | .|.|.....t'........|.|.........z |
| 55a0 | 0a 00 00 7a 0d 00 00 63 03 63 02 3c 00 00 00 7c 0a 7d 00 8c f3 04 00 74 01 00 00 00 00 00 00 00 | ...z...c.c.<...|.}.....t........ |
| 55c0 | 00 6a 1a 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 00 64 07 7c 07 ab 03 00 00 00 | .j...................|.d.|...... |
| 55e0 | 00 00 00 7d 00 7c 00 53 00 29 10 61 4e 0d 00 00 0a 20 20 20 20 49 6e 74 65 67 72 61 74 65 20 61 | ...}.|.S.).aN........Integrate.a |
| 5600 | 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 20 74 | .Legendre.series.......Returns.t |
| 5620 | 68 65 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 | he.Legendre.series.coefficients. |
| 5640 | 60 63 60 20 69 6e 74 65 67 72 61 74 65 64 20 60 6d 60 20 74 69 6d 65 73 20 66 72 6f 6d 0a 20 20 | `c`.integrated.`m`.times.from... |
| 5660 | 20 20 60 6c 62 6e 64 60 20 61 6c 6f 6e 67 20 60 61 78 69 73 60 2e 20 41 74 20 65 61 63 68 20 69 | ..`lbnd`.along.`axis`..At.each.i |
| 5680 | 74 65 72 61 74 69 6f 6e 20 74 68 65 20 72 65 73 75 6c 74 69 6e 67 20 73 65 72 69 65 73 20 69 73 | teration.the.resulting.series.is |
| 56a0 | 0a 20 20 20 20 2a 2a 6d 75 6c 74 69 70 6c 69 65 64 2a 2a 20 62 79 20 60 73 63 6c 60 20 61 6e 64 | .....**multiplied**.by.`scl`.and |
| 56c0 | 20 61 6e 20 69 6e 74 65 67 72 61 74 69 6f 6e 20 63 6f 6e 73 74 61 6e 74 2c 20 60 6b 60 2c 20 69 | .an.integration.constant,.`k`,.i |
| 56e0 | 73 20 61 64 64 65 64 2e 0a 20 20 20 20 54 68 65 20 73 63 61 6c 69 6e 67 20 66 61 63 74 6f 72 20 | s.added......The.scaling.factor. |
| 5700 | 69 73 20 66 6f 72 20 75 73 65 20 69 6e 20 61 20 6c 69 6e 65 61 72 20 63 68 61 6e 67 65 20 6f 66 | is.for.use.in.a.linear.change.of |
| 5720 | 20 76 61 72 69 61 62 6c 65 2e 20 20 28 22 42 75 79 65 72 0a 20 20 20 20 62 65 77 61 72 65 22 3a | .variable...("Buyer.....beware": |
| 5740 | 20 6e 6f 74 65 20 74 68 61 74 2c 20 64 65 70 65 6e 64 69 6e 67 20 6f 6e 20 77 68 61 74 20 6f 6e | .note.that,.depending.on.what.on |
| 5760 | 65 20 69 73 20 64 6f 69 6e 67 2c 20 6f 6e 65 20 6d 61 79 20 77 61 6e 74 20 60 73 63 6c 60 0a 20 | e.is.doing,.one.may.want.`scl`.. |
| 5780 | 20 20 20 74 6f 20 62 65 20 74 68 65 20 72 65 63 69 70 72 6f 63 61 6c 20 6f 66 20 77 68 61 74 20 | ...to.be.the.reciprocal.of.what. |
| 57a0 | 6f 6e 65 20 6d 69 67 68 74 20 65 78 70 65 63 74 3b 20 66 6f 72 20 6d 6f 72 65 20 69 6e 66 6f 72 | one.might.expect;.for.more.infor |
| 57c0 | 6d 61 74 69 6f 6e 2c 0a 20 20 20 20 73 65 65 20 74 68 65 20 4e 6f 74 65 73 20 73 65 63 74 69 6f | mation,.....see.the.Notes.sectio |
| 57e0 | 6e 20 62 65 6c 6f 77 2e 29 20 20 54 68 65 20 61 72 67 75 6d 65 6e 74 20 60 63 60 20 69 73 20 61 | n.below.)..The.argument.`c`.is.a |
| 5800 | 6e 20 61 72 72 61 79 20 6f 66 0a 20 20 20 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 66 72 6f 6d | n.array.of.....coefficients.from |
| 5820 | 20 6c 6f 77 20 74 6f 20 68 69 67 68 20 64 65 67 72 65 65 20 61 6c 6f 6e 67 20 65 61 63 68 20 61 | .low.to.high.degree.along.each.a |
| 5840 | 78 69 73 2c 20 65 2e 67 2e 2c 20 5b 31 2c 32 2c 33 5d 0a 20 20 20 20 72 65 70 72 65 73 65 6e 74 | xis,.e.g.,.[1,2,3].....represent |
| 5860 | 73 20 74 68 65 20 73 65 72 69 65 73 20 60 60 4c 5f 30 20 2b 20 32 2a 4c 5f 31 20 2b 20 33 2a 4c | s.the.series.``L_0.+.2*L_1.+.3*L |
| 5880 | 5f 32 60 60 20 77 68 69 6c 65 20 5b 5b 31 2c 32 5d 2c 5b 31 2c 32 5d 5d 0a 20 20 20 20 72 65 70 | _2``.while.[[1,2],[1,2]].....rep |
| 58a0 | 72 65 73 65 6e 74 73 20 60 60 31 2a 4c 5f 30 28 78 29 2a 4c 5f 30 28 79 29 20 2b 20 31 2a 4c 5f | resents.``1*L_0(x)*L_0(y).+.1*L_ |
| 58c0 | 31 28 78 29 2a 4c 5f 30 28 79 29 20 2b 20 32 2a 4c 5f 30 28 78 29 2a 4c 5f 31 28 79 29 20 2b 0a | 1(x)*L_0(y).+.2*L_0(x)*L_1(y).+. |
| 58e0 | 20 20 20 20 32 2a 4c 5f 31 28 78 29 2a 4c 5f 31 28 79 29 60 60 20 69 66 20 61 78 69 73 3d 30 20 | ....2*L_1(x)*L_1(y)``.if.axis=0. |
| 5900 | 69 73 20 60 60 78 60 60 20 61 6e 64 20 61 78 69 73 3d 31 20 69 73 20 60 60 79 60 60 2e 0a 0a 20 | is.``x``.and.axis=1.is.``y``.... |
| 5920 | 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 | ...Parameters.....----------.... |
| 5940 | 20 63 20 3a 20 61 72 72 61 79 5f 6c 69 6b 65 0a 20 20 20 20 20 20 20 20 41 72 72 61 79 20 6f 66 | .c.:.array_like.........Array.of |
| 5960 | 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 63 6f 65 66 66 69 63 69 65 6e 74 73 2e 20 49 | .Legendre.series.coefficients..I |
| 5980 | 66 20 63 20 69 73 20 6d 75 6c 74 69 64 69 6d 65 6e 73 69 6f 6e 61 6c 20 74 68 65 0a 20 20 20 20 | f.c.is.multidimensional.the..... |
| 59a0 | 20 20 20 20 64 69 66 66 65 72 65 6e 74 20 61 78 69 73 20 63 6f 72 72 65 73 70 6f 6e 64 20 74 6f | ....different.axis.correspond.to |
| 59c0 | 20 64 69 66 66 65 72 65 6e 74 20 76 61 72 69 61 62 6c 65 73 20 77 69 74 68 20 74 68 65 20 64 65 | .different.variables.with.the.de |
| 59e0 | 67 72 65 65 20 69 6e 0a 20 20 20 20 20 20 20 20 65 61 63 68 20 61 78 69 73 20 67 69 76 65 6e 20 | gree.in.........each.axis.given. |
| 5a00 | 62 79 20 74 68 65 20 63 6f 72 72 65 73 70 6f 6e 64 69 6e 67 20 69 6e 64 65 78 2e 0a 20 20 20 20 | by.the.corresponding.index...... |
| 5a20 | 6d 20 3a 20 69 6e 74 2c 20 6f 70 74 69 6f 6e 61 6c 0a 20 20 20 20 20 20 20 20 4f 72 64 65 72 20 | m.:.int,.optional.........Order. |
| 5a40 | 6f 66 20 69 6e 74 65 67 72 61 74 69 6f 6e 2c 20 6d 75 73 74 20 62 65 20 70 6f 73 69 74 69 76 65 | of.integration,.must.be.positive |
| 5a60 | 2e 20 28 44 65 66 61 75 6c 74 3a 20 31 29 0a 20 20 20 20 6b 20 3a 20 7b 5b 5d 2c 20 6c 69 73 74 | ..(Default:.1).....k.:.{[],.list |
| 5a80 | 2c 20 73 63 61 6c 61 72 7d 2c 20 6f 70 74 69 6f 6e 61 6c 0a 20 20 20 20 20 20 20 20 49 6e 74 65 | ,.scalar},.optional.........Inte |
| 5aa0 | 67 72 61 74 69 6f 6e 20 63 6f 6e 73 74 61 6e 74 28 73 29 2e 20 20 54 68 65 20 76 61 6c 75 65 20 | gration.constant(s)...The.value. |
| 5ac0 | 6f 66 20 74 68 65 20 66 69 72 73 74 20 69 6e 74 65 67 72 61 6c 20 61 74 0a 20 20 20 20 20 20 20 | of.the.first.integral.at........ |
| 5ae0 | 20 60 60 6c 62 6e 64 60 60 20 69 73 20 74 68 65 20 66 69 72 73 74 20 76 61 6c 75 65 20 69 6e 20 | .``lbnd``.is.the.first.value.in. |
| 5b00 | 74 68 65 20 6c 69 73 74 2c 20 74 68 65 20 76 61 6c 75 65 20 6f 66 20 74 68 65 20 73 65 63 6f 6e | the.list,.the.value.of.the.secon |
| 5b20 | 64 0a 20 20 20 20 20 20 20 20 69 6e 74 65 67 72 61 6c 20 61 74 20 60 60 6c 62 6e 64 60 60 20 69 | d.........integral.at.``lbnd``.i |
| 5b40 | 73 20 74 68 65 20 73 65 63 6f 6e 64 20 76 61 6c 75 65 2c 20 65 74 63 2e 20 20 49 66 20 60 60 6b | s.the.second.value,.etc...If.``k |
| 5b60 | 20 3d 3d 20 5b 5d 60 60 20 28 74 68 65 0a 20 20 20 20 20 20 20 20 64 65 66 61 75 6c 74 29 2c 20 | .==.[]``.(the.........default),. |
| 5b80 | 61 6c 6c 20 63 6f 6e 73 74 61 6e 74 73 20 61 72 65 20 73 65 74 20 74 6f 20 7a 65 72 6f 2e 20 20 | all.constants.are.set.to.zero... |
| 5ba0 | 49 66 20 60 60 6d 20 3d 3d 20 31 60 60 2c 20 61 20 73 69 6e 67 6c 65 0a 20 20 20 20 20 20 20 20 | If.``m.==.1``,.a.single......... |
| 5bc0 | 73 63 61 6c 61 72 20 63 61 6e 20 62 65 20 67 69 76 65 6e 20 69 6e 73 74 65 61 64 20 6f 66 20 61 | scalar.can.be.given.instead.of.a |
| 5be0 | 20 6c 69 73 74 2e 0a 20 20 20 20 6c 62 6e 64 20 3a 20 73 63 61 6c 61 72 2c 20 6f 70 74 69 6f 6e | .list......lbnd.:.scalar,.option |
| 5c00 | 61 6c 0a 20 20 20 20 20 20 20 20 54 68 65 20 6c 6f 77 65 72 20 62 6f 75 6e 64 20 6f 66 20 74 68 | al.........The.lower.bound.of.th |
| 5c20 | 65 20 69 6e 74 65 67 72 61 6c 2e 20 28 44 65 66 61 75 6c 74 3a 20 30 29 0a 20 20 20 20 73 63 6c | e.integral..(Default:.0).....scl |
| 5c40 | 20 3a 20 73 63 61 6c 61 72 2c 20 6f 70 74 69 6f 6e 61 6c 0a 20 20 20 20 20 20 20 20 46 6f 6c 6c | .:.scalar,.optional.........Foll |
| 5c60 | 6f 77 69 6e 67 20 65 61 63 68 20 69 6e 74 65 67 72 61 74 69 6f 6e 20 74 68 65 20 72 65 73 75 6c | owing.each.integration.the.resul |
| 5c80 | 74 20 69 73 20 2a 6d 75 6c 74 69 70 6c 69 65 64 2a 20 62 79 20 60 73 63 6c 60 0a 20 20 20 20 20 | t.is.*multiplied*.by.`scl`...... |
| 5ca0 | 20 20 20 62 65 66 6f 72 65 20 74 68 65 20 69 6e 74 65 67 72 61 74 69 6f 6e 20 63 6f 6e 73 74 61 | ...before.the.integration.consta |
| 5cc0 | 6e 74 20 69 73 20 61 64 64 65 64 2e 20 28 44 65 66 61 75 6c 74 3a 20 31 29 0a 20 20 20 20 61 78 | nt.is.added..(Default:.1).....ax |
| 5ce0 | 69 73 20 3a 20 69 6e 74 2c 20 6f 70 74 69 6f 6e 61 6c 0a 20 20 20 20 20 20 20 20 41 78 69 73 20 | is.:.int,.optional.........Axis. |
| 5d00 | 6f 76 65 72 20 77 68 69 63 68 20 74 68 65 20 69 6e 74 65 67 72 61 6c 20 69 73 20 74 61 6b 65 6e | over.which.the.integral.is.taken |
| 5d20 | 2e 20 28 44 65 66 61 75 6c 74 3a 20 30 29 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 0a 20 20 20 | ..(Default:.0).......Returns.... |
| 5d40 | 20 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 53 20 3a 20 6e 64 61 72 72 61 79 0a 20 20 20 20 20 20 20 | .-------.....S.:.ndarray........ |
| 5d60 | 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 63 6f 65 66 66 69 63 69 65 6e 74 20 61 72 72 | .Legendre.series.coefficient.arr |
| 5d80 | 61 79 20 6f 66 20 74 68 65 20 69 6e 74 65 67 72 61 6c 2e 0a 0a 20 20 20 20 52 61 69 73 65 73 0a | ay.of.the.integral.......Raises. |
| 5da0 | 20 20 20 20 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 56 61 6c 75 65 45 72 72 6f 72 0a 20 20 20 20 20 20 | ....------.....ValueError....... |
| 5dc0 | 20 20 49 66 20 60 60 6d 20 3c 20 30 60 60 2c 20 60 60 6c 65 6e 28 6b 29 20 3e 20 6d 60 60 2c 20 | ..If.``m.<.0``,.``len(k).>.m``,. |
| 5de0 | 60 60 6e 70 2e 6e 64 69 6d 28 6c 62 6e 64 29 20 21 3d 20 30 60 60 2c 20 6f 72 0a 20 20 20 20 20 | ``np.ndim(lbnd).!=.0``,.or...... |
| 5e00 | 20 20 20 60 60 6e 70 2e 6e 64 69 6d 28 73 63 6c 29 20 21 3d 20 30 60 60 2e 0a 0a 20 20 20 20 53 | ...``np.ndim(scl).!=.0``.......S |
| 5e20 | 65 65 20 41 6c 73 6f 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 6c 65 67 64 65 72 0a | ee.Also.....--------.....legder. |
| 5e40 | 0a 20 20 20 20 4e 6f 74 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 0a 20 20 20 20 4e 6f 74 65 20 74 68 | .....Notes.....-----.....Note.th |
| 5e60 | 61 74 20 74 68 65 20 72 65 73 75 6c 74 20 6f 66 20 65 61 63 68 20 69 6e 74 65 67 72 61 74 69 6f | at.the.result.of.each.integratio |
| 5e80 | 6e 20 69 73 20 2a 6d 75 6c 74 69 70 6c 69 65 64 2a 20 62 79 20 60 73 63 6c 60 2e 0a 20 20 20 20 | n.is.*multiplied*.by.`scl`...... |
| 5ea0 | 57 68 79 20 69 73 20 74 68 69 73 20 69 6d 70 6f 72 74 61 6e 74 20 74 6f 20 6e 6f 74 65 3f 20 20 | Why.is.this.important.to.note?.. |
| 5ec0 | 53 61 79 20 6f 6e 65 20 69 73 20 6d 61 6b 69 6e 67 20 61 20 6c 69 6e 65 61 72 20 63 68 61 6e 67 | Say.one.is.making.a.linear.chang |
| 5ee0 | 65 20 6f 66 0a 20 20 20 20 76 61 72 69 61 62 6c 65 20 3a 6d 61 74 68 3a 60 75 20 3d 20 61 78 20 | e.of.....variable.:math:`u.=.ax. |
| 5f00 | 2b 20 62 60 20 69 6e 20 61 6e 20 69 6e 74 65 67 72 61 6c 20 72 65 6c 61 74 69 76 65 20 74 6f 20 | +.b`.in.an.integral.relative.to. |
| 5f20 | 60 78 60 2e 20 20 54 68 65 6e 0a 20 20 20 20 3a 6d 61 74 68 3a 60 64 78 20 3d 20 64 75 2f 61 60 | `x`...Then.....:math:`dx.=.du/a` |
| 5f40 | 2c 20 73 6f 20 6f 6e 65 20 77 69 6c 6c 20 6e 65 65 64 20 74 6f 20 73 65 74 20 60 73 63 6c 60 20 | ,.so.one.will.need.to.set.`scl`. |
| 5f60 | 65 71 75 61 6c 20 74 6f 0a 20 20 20 20 3a 6d 61 74 68 3a 60 31 2f 61 60 20 2d 20 70 65 72 68 61 | equal.to.....:math:`1/a`.-.perha |
| 5f80 | 70 73 20 6e 6f 74 20 77 68 61 74 20 6f 6e 65 20 77 6f 75 6c 64 20 68 61 76 65 20 66 69 72 73 74 | ps.not.what.one.would.have.first |
| 5fa0 | 20 74 68 6f 75 67 68 74 2e 0a 0a 20 20 20 20 41 6c 73 6f 20 6e 6f 74 65 20 74 68 61 74 2c 20 69 | .thought.......Also.note.that,.i |
| 5fc0 | 6e 20 67 65 6e 65 72 61 6c 2c 20 74 68 65 20 72 65 73 75 6c 74 20 6f 66 20 69 6e 74 65 67 72 61 | n.general,.the.result.of.integra |
| 5fe0 | 74 69 6e 67 20 61 20 43 2d 73 65 72 69 65 73 20 6e 65 65 64 73 0a 20 20 20 20 74 6f 20 62 65 20 | ting.a.C-series.needs.....to.be. |
| 6000 | 22 72 65 70 72 6f 6a 65 63 74 65 64 22 20 6f 6e 74 6f 20 74 68 65 20 43 2d 73 65 72 69 65 73 20 | "reprojected".onto.the.C-series. |
| 6020 | 62 61 73 69 73 20 73 65 74 2e 20 20 54 68 75 73 2c 20 74 79 70 69 63 61 6c 6c 79 2c 0a 20 20 20 | basis.set...Thus,.typically,.... |
| 6040 | 20 74 68 65 20 72 65 73 75 6c 74 20 6f 66 20 74 68 69 73 20 66 75 6e 63 74 69 6f 6e 20 69 73 20 | .the.result.of.this.function.is. |
| 6060 | 22 75 6e 69 6e 74 75 69 74 69 76 65 2c 22 20 61 6c 62 65 69 74 20 63 6f 72 72 65 63 74 3b 20 73 | "unintuitive,".albeit.correct;.s |
| 6080 | 65 65 0a 20 20 20 20 45 78 61 6d 70 6c 65 73 20 73 65 63 74 69 6f 6e 20 62 65 6c 6f 77 2e 0a 0a | ee.....Examples.section.below... |
| 60a0 | 20 20 20 20 45 78 61 6d 70 6c 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 3e 3e | ....Examples.....--------.....>> |
| 60c0 | 3e 20 66 72 6f 6d 20 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 20 69 6d 70 6f 72 74 20 6c | >.from.numpy.polynomial.import.l |
| 60e0 | 65 67 65 6e 64 72 65 20 61 73 20 4c 0a 20 20 20 20 3e 3e 3e 20 63 20 3d 20 28 31 2c 32 2c 33 29 | egendre.as.L.....>>>.c.=.(1,2,3) |
| 6100 | 0a 20 20 20 20 3e 3e 3e 20 4c 2e 6c 65 67 69 6e 74 28 63 29 0a 20 20 20 20 61 72 72 61 79 28 5b | .....>>>.L.legint(c).....array([ |
| 6120 | 20 30 2e 33 33 33 33 33 33 33 33 2c 20 20 30 2e 34 20 20 20 20 20 20 20 2c 20 20 30 2e 36 36 36 | .0.33333333,..0.4.......,..0.666 |
| 6140 | 36 36 36 36 37 2c 20 20 30 2e 36 20 20 20 20 20 20 20 5d 29 20 23 20 6d 61 79 20 76 61 72 79 0a | 66667,..0.6.......]).#.may.vary. |
| 6160 | 20 20 20 20 3e 3e 3e 20 4c 2e 6c 65 67 69 6e 74 28 63 2c 20 33 29 0a 20 20 20 20 61 72 72 61 79 | ....>>>.L.legint(c,.3).....array |
| 6180 | 28 5b 20 20 31 2e 36 36 36 36 36 36 36 37 65 2d 30 32 2c 20 20 2d 31 2e 37 38 35 37 31 34 32 39 | ([..1.66666667e-02,..-1.78571429 |
| 61a0 | 65 2d 30 32 2c 20 20 20 34 2e 37 36 31 39 30 34 37 36 65 2d 30 32 2c 20 23 20 6d 61 79 20 76 61 | e-02,...4.76190476e-02,.#.may.va |
| 61c0 | 72 79 0a 20 20 20 20 20 20 20 20 20 20 20 20 20 2d 31 2e 37 33 34 37 32 33 34 38 65 2d 31 38 2c | ry..............-1.73472348e-18, |
| 61e0 | 20 20 20 31 2e 39 30 34 37 36 31 39 30 65 2d 30 32 2c 20 20 20 39 2e 35 32 33 38 30 39 35 32 65 | ...1.90476190e-02,...9.52380952e |
| 6200 | 2d 30 33 5d 29 0a 20 20 20 20 3e 3e 3e 20 4c 2e 6c 65 67 69 6e 74 28 63 2c 20 6b 3d 33 29 0a 20 | -03]).....>>>.L.legint(c,.k=3).. |
| 6220 | 20 20 20 20 61 72 72 61 79 28 5b 20 33 2e 33 33 33 33 33 33 33 33 2c 20 20 30 2e 34 20 20 20 20 | ....array([.3.33333333,..0.4.... |
| 6240 | 20 20 20 2c 20 20 30 2e 36 36 36 36 36 36 36 37 2c 20 20 30 2e 36 20 20 20 20 20 20 20 5d 29 20 | ...,..0.66666667,..0.6.......]). |
| 6260 | 23 20 6d 61 79 20 76 61 72 79 0a 20 20 20 20 3e 3e 3e 20 4c 2e 6c 65 67 69 6e 74 28 63 2c 20 6c | #.may.vary.....>>>.L.legint(c,.l |
| 6280 | 62 6e 64 3d 2d 32 29 0a 20 20 20 20 61 72 72 61 79 28 5b 20 37 2e 33 33 33 33 33 33 33 33 2c 20 | bnd=-2).....array([.7.33333333,. |
| 62a0 | 20 30 2e 34 20 20 20 20 20 20 20 2c 20 20 30 2e 36 36 36 36 36 36 36 37 2c 20 20 30 2e 36 20 20 | .0.4.......,..0.66666667,..0.6.. |
| 62c0 | 20 20 20 20 20 5d 29 20 23 20 6d 61 79 20 76 61 72 79 0a 20 20 20 20 3e 3e 3e 20 4c 2e 6c 65 67 | .....]).#.may.vary.....>>>.L.leg |
| 62e0 | 69 6e 74 28 63 2c 20 73 63 6c 3d 32 29 0a 20 20 20 20 61 72 72 61 79 28 5b 20 30 2e 36 36 36 36 | int(c,.scl=2).....array([.0.6666 |
| 6300 | 36 36 36 37 2c 20 20 30 2e 38 20 20 20 20 20 20 20 2c 20 20 31 2e 33 33 33 33 33 33 33 33 2c 20 | 6667,..0.8.......,..1.33333333,. |
| 6320 | 20 31 2e 32 20 20 20 20 20 20 20 5d 29 20 23 20 6d 61 79 20 76 61 72 79 0a 0a 20 20 20 20 72 04 | .1.2.......]).#.may.vary......r. |
| 6340 | 00 00 00 54 72 60 00 00 00 72 63 00 00 00 7a 18 74 68 65 20 6f 72 64 65 72 20 6f 66 20 69 6e 74 | ...Tr`...rc...z.the.order.of.int |
| 6360 | 65 67 72 61 74 69 6f 6e 72 64 00 00 00 72 02 00 00 00 7a 2d 54 68 65 20 6f 72 64 65 72 20 6f 66 | egrationrd...r....z-The.order.of |
| 6380 | 20 69 6e 74 65 67 72 61 74 69 6f 6e 20 6d 75 73 74 20 62 65 20 6e 6f 6e 2d 6e 65 67 61 74 69 76 | .integration.must.be.non-negativ |
| 63a0 | 65 7a 1e 54 6f 6f 20 6d 61 6e 79 20 69 6e 74 65 67 72 61 74 69 6f 6e 20 63 6f 6e 73 74 61 6e 74 | ez.Too.many.integration.constant |
| 63c0 | 73 7a 16 6c 62 6e 64 20 6d 75 73 74 20 62 65 20 61 20 73 63 61 6c 61 72 2e 7a 15 73 63 6c 20 6d | sz.lbnd.must.be.a.scalar.z.scl.m |
| 63e0 | 75 73 74 20 62 65 20 61 20 73 63 61 6c 61 72 2e 4e 72 4f 00 00 00 72 36 00 00 00 72 38 00 00 00 | ust.be.a.scalar.NrO...r6...r8... |
| 6400 | 29 14 72 41 00 00 00 72 42 00 00 00 72 50 00 00 00 72 65 00 00 00 72 66 00 00 00 72 67 00 00 00 | ).rA...rB...rP...re...rf...rg... |
| 6420 | da 08 69 74 65 72 61 62 6c 65 72 28 00 00 00 72 68 00 00 00 72 69 00 00 00 72 2a 00 00 00 72 6a | ..iterabler(...rh...ri...r*...rj |
| 6440 | 00 00 00 72 03 00 00 00 72 6b 00 00 00 da 04 6c 69 73 74 72 2b 00 00 00 da 03 61 6c 6c 72 51 00 | ...r....rk.....listr+.....allrQ. |
| 6460 | 00 00 72 6c 00 00 00 72 12 00 00 00 29 0d 72 3a 00 00 00 72 6d 00 00 00 72 54 00 00 00 da 04 6c | ..rl...r....).r:...rm...rT.....l |
| 6480 | 62 6e 64 72 44 00 00 00 72 6e 00 00 00 72 6f 00 00 00 72 70 00 00 00 72 2f 00 00 00 72 3b 00 00 | bndrD...rn...ro...rp...r/...r;.. |
| 64a0 | 00 72 3e 00 00 00 72 53 00 00 00 da 01 74 73 0d 00 00 00 20 20 20 20 20 20 20 20 20 20 20 20 20 | .r>...rS.....ts................. |
| 64c0 | 72 30 00 00 00 72 14 00 00 00 72 14 00 00 00 c0 02 00 00 73 4f 02 00 00 80 00 f4 66 02 00 09 0b | r0...r....r........sO......f.... |
| 64e0 | 8f 08 89 08 90 11 98 21 a0 24 d4 08 27 80 41 d8 07 08 87 77 81 77 87 7c 81 7c 90 7f d1 07 26 d8 | .......!.$..'.A....w.w.|.|....&. |
| 6500 | 0c 0d 8f 48 89 48 94 52 97 59 91 59 d3 0c 1f 88 01 dc 0b 0d 8f 3b 89 3b 90 71 8c 3e d8 0d 0e 88 | ...H.H.R.Y.Y.........;.;.q.>.... |
| 6520 | 43 88 01 dc 0a 0c 8f 2a 89 2a 90 51 d0 18 32 d3 0a 33 80 43 dc 0c 0e 8f 4a 89 4a 90 74 98 5a d3 | C......*.*.Q..2..3.C....J.J.t.Z. |
| 6540 | 0c 28 80 45 d8 07 0a 88 51 82 77 dc 0e 18 d0 19 48 d3 0e 49 d0 08 49 dc 07 0a 88 31 83 76 90 03 | .(.E....Q.w.....H..I..I....1.v.. |
| 6560 | 82 7c dc 0e 18 d0 19 39 d3 0e 3a d0 08 3a dc 07 09 87 77 81 77 88 74 83 7d 98 01 d2 07 19 dc 0e | .|.....9..:..:....w.w.t.}....... |
| 6580 | 18 d0 19 31 d3 0e 32 d0 08 32 dc 07 09 87 77 81 77 88 73 83 7c 90 71 d2 07 18 dc 0e 18 d0 19 30 | ...1..2..2....w.w.s.|.q........0 |
| 65a0 | d3 0e 31 d0 08 31 dc 0c 20 a0 15 a8 01 af 06 a9 06 d3 0c 2f 80 45 e0 07 0a 88 61 82 78 d8 0f 10 | ..1..1............./.E....a.x... |
| 65c0 | 88 08 e4 08 0a 8f 0b 89 0b 90 41 90 75 98 61 d3 08 20 80 41 dc 08 0c 88 51 8b 07 90 31 90 23 98 | ..........A.u.a....A....Q...1.#. |
| 65e0 | 13 9c 73 a0 31 9b 76 99 1c d1 12 26 d1 08 26 80 41 dc 0d 12 90 33 8b 5a f2 00 10 05 14 88 01 dc | ..s.1.v....&..&.A....3.Z........ |
| 6600 | 0c 0f 90 01 8b 46 88 01 d8 08 09 88 53 89 08 88 01 d8 0b 0c 90 01 8a 36 94 62 97 66 91 66 98 51 | .....F......S..........6.b.f.f.Q |
| 6620 | 98 71 99 54 a0 51 99 59 d4 16 27 d8 0c 0d 88 61 8b 44 90 41 90 61 91 44 89 4c 8c 44 e4 12 14 97 | .q.T.Q.Y..'....a.D.A.a.D.L.D.... |
| 6640 | 28 91 28 98 41 a0 01 99 45 98 38 a0 61 a7 67 a1 67 a8 61 a8 62 a0 6b d1 1b 31 b8 11 bf 17 b9 17 | (.(.A...E.8.a.g.g.a.b.k..1...... |
| 6660 | d4 12 41 88 43 d8 15 16 90 71 91 54 98 41 91 58 88 43 90 01 89 46 d8 15 16 90 71 91 54 88 43 90 | ..A.C....q.T.A.X.C...F....q.T.C. |
| 6680 | 01 89 46 d8 0f 10 90 31 8a 75 d8 19 1a 98 31 99 14 a0 01 99 18 90 03 90 41 91 06 dc 15 1a 98 31 | ..F....1.u....1.........A......1 |
| 66a0 | 98 61 93 5b f2 00 03 0d 20 90 01 d8 14 15 90 61 91 44 98 41 a0 01 99 45 a0 41 99 49 d1 14 26 90 | .a.[...........a.D.A...E.A.I..&. |
| 66c0 | 01 d8 1d 1e 90 03 90 41 98 01 91 45 91 0a d8 10 13 90 41 98 01 91 45 93 0a 98 61 91 0f 94 0a f0 | .......A...E......A...E...a..... |
| 66e0 | 07 03 0d 20 f0 08 00 0d 10 90 01 8b 46 90 61 98 01 91 64 9c 56 a0 44 a8 23 d3 1d 2e d1 16 2e d1 | ............F.a...d.V.D.#....... |
| 6700 | 0c 2e 8b 46 d8 10 13 89 41 f0 21 10 05 14 f4 22 00 09 0b 8f 0b 89 0b 90 41 90 71 98 25 d3 08 20 | ...F....A.!...."........A.q.%... |
| 6720 | 80 41 d8 0b 0c 80 48 72 31 00 00 00 63 03 00 00 00 00 00 00 00 00 00 00 00 06 00 00 00 03 00 00 | .A....Hr1...c................... |
| 6740 | 00 f3 b0 02 00 00 97 00 74 01 00 00 00 00 00 00 00 00 6a 02 00 00 00 00 00 00 00 00 00 00 00 00 | ........t.........j............. |
| 6760 | 00 00 00 00 00 00 7c 01 64 01 64 02 ac 03 ab 03 00 00 00 00 00 00 7d 01 7c 01 6a 04 00 00 00 00 | ......|.d.d...........}.|.j..... |
| 6780 | 00 00 00 00 00 00 00 00 00 00 00 00 00 00 6a 06 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 | ..............j................. |
| 67a0 | 00 00 64 04 76 00 72 1f 7c 01 6a 09 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 74 00 | ..d.v.r.|.j...................t. |
| 67c0 | 00 00 00 00 00 00 00 00 6a 0a 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ab 01 00 00 | ........j....................... |
| 67e0 | 00 00 00 00 7d 01 74 0d 00 00 00 00 00 00 00 00 7c 00 74 0e 00 00 00 00 00 00 00 00 74 10 00 00 | ....}.t.........|.t.........t... |
| 6800 | 00 00 00 00 00 00 66 02 ab 02 00 00 00 00 00 00 72 15 74 01 00 00 00 00 00 00 00 00 6a 12 00 00 | ......f.........r.t.........j... |
| 6820 | 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 00 ab 01 00 00 00 00 00 00 7d 00 74 0d 00 00 | ................|.........}.t... |
| 6840 | 00 00 00 00 00 00 7c 00 74 00 00 00 00 00 00 00 00 00 6a 14 00 00 00 00 00 00 00 00 00 00 00 00 | ......|.t.........j............. |
| 6860 | 00 00 00 00 00 00 ab 02 00 00 00 00 00 00 72 2d 7c 02 72 2b 7c 01 6a 17 00 00 00 00 00 00 00 00 | ..............r-|.r+|.j......... |
| 6880 | 00 00 00 00 00 00 00 00 00 00 7c 01 6a 18 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 | ..........|.j................... |
| 68a0 | 64 05 7c 00 6a 1a 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7a 05 00 00 7a 00 00 00 | d.|.j...................z...z... |
| 68c0 | ab 01 00 00 00 00 00 00 7d 01 74 1d 00 00 00 00 00 00 00 00 7c 01 ab 01 00 00 00 00 00 00 64 01 | ........}.t.........|.........d. |
| 68e0 | 6b 28 00 00 72 08 7c 01 64 06 19 00 00 00 7d 03 64 06 7d 04 6e 78 74 1d 00 00 00 00 00 00 00 00 | k(..r.|.d.....}.d.}.nxt......... |
| 6900 | 7c 01 ab 01 00 00 00 00 00 00 64 07 6b 28 00 00 72 0b 7c 01 64 06 19 00 00 00 7d 03 7c 01 64 01 | |.........d.k(..r.|.d.....}.|.d. |
| 6920 | 19 00 00 00 7d 04 6e 5f 74 1d 00 00 00 00 00 00 00 00 7c 01 ab 01 00 00 00 00 00 00 7d 05 7c 01 | ....}.n_t.........|.........}.|. |
| 6940 | 64 08 19 00 00 00 7d 03 7c 01 64 09 19 00 00 00 7d 04 74 1f 00 00 00 00 00 00 00 00 64 0a 74 1d | d.....}.|.d.....}.t.........d.t. |
| 6960 | 00 00 00 00 00 00 00 00 7c 01 ab 01 00 00 00 00 00 00 64 01 7a 00 00 00 ab 02 00 00 00 00 00 00 | ........|.........d.z........... |
| 6980 | 44 00 5d 2f 00 00 7d 06 7c 03 7d 07 7c 05 64 01 7a 0a 00 00 7d 05 7c 01 7c 06 0b 00 19 00 00 00 | D.]/..}.|.}.|.d.z...}.|.|....... |
| 69a0 | 7c 04 7c 05 64 01 7a 0a 00 00 7c 05 7a 0b 00 00 7a 05 00 00 7a 0a 00 00 7d 03 7c 07 7c 04 7c 00 | |.|.d.z...|.z...z...z...}.|.|.|. |
| 69c0 | 7a 05 00 00 64 07 7c 05 7a 05 00 00 64 01 7a 0a 00 00 7c 05 7a 0b 00 00 7a 05 00 00 7a 00 00 00 | z...d.|.z...d.z...|.z...z...z... |
| 69e0 | 7d 04 8c 31 04 00 7c 03 7c 04 7c 00 7a 05 00 00 7a 00 00 00 53 00 29 0b 61 2a 09 00 00 0a 20 20 | }..1..|.|.|.z...z...S.).a*...... |
| 6a00 | 20 20 45 76 61 6c 75 61 74 65 20 61 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 61 74 20 | ..Evaluate.a.Legendre.series.at. |
| 6a20 | 70 6f 69 6e 74 73 20 78 2e 0a 0a 20 20 20 20 49 66 20 60 63 60 20 69 73 20 6f 66 20 6c 65 6e 67 | points.x.......If.`c`.is.of.leng |
| 6a40 | 74 68 20 60 60 6e 20 2b 20 31 60 60 2c 20 74 68 69 73 20 66 75 6e 63 74 69 6f 6e 20 72 65 74 75 | th.``n.+.1``,.this.function.retu |
| 6a60 | 72 6e 73 20 74 68 65 20 76 61 6c 75 65 3a 0a 0a 20 20 20 20 2e 2e 20 6d 61 74 68 3a 3a 20 70 28 | rns.the.value:.........math::.p( |
| 6a80 | 78 29 20 3d 20 63 5f 30 20 2a 20 4c 5f 30 28 78 29 20 2b 20 63 5f 31 20 2a 20 4c 5f 31 28 78 29 | x).=.c_0.*.L_0(x).+.c_1.*.L_1(x) |
| 6aa0 | 20 2b 20 2e 2e 2e 20 2b 20 63 5f 6e 20 2a 20 4c 5f 6e 28 78 29 0a 0a 20 20 20 20 54 68 65 20 70 | .+.....+.c_n.*.L_n(x)......The.p |
| 6ac0 | 61 72 61 6d 65 74 65 72 20 60 78 60 20 69 73 20 63 6f 6e 76 65 72 74 65 64 20 74 6f 20 61 6e 20 | arameter.`x`.is.converted.to.an. |
| 6ae0 | 61 72 72 61 79 20 6f 6e 6c 79 20 69 66 20 69 74 20 69 73 20 61 20 74 75 70 6c 65 20 6f 72 20 61 | array.only.if.it.is.a.tuple.or.a |
| 6b00 | 0a 20 20 20 20 6c 69 73 74 2c 20 6f 74 68 65 72 77 69 73 65 20 69 74 20 69 73 20 74 72 65 61 74 | .....list,.otherwise.it.is.treat |
| 6b20 | 65 64 20 61 73 20 61 20 73 63 61 6c 61 72 2e 20 49 6e 20 65 69 74 68 65 72 20 63 61 73 65 2c 20 | ed.as.a.scalar..In.either.case,. |
| 6b40 | 65 69 74 68 65 72 20 60 78 60 0a 20 20 20 20 6f 72 20 69 74 73 20 65 6c 65 6d 65 6e 74 73 20 6d | either.`x`.....or.its.elements.m |
| 6b60 | 75 73 74 20 73 75 70 70 6f 72 74 20 6d 75 6c 74 69 70 6c 69 63 61 74 69 6f 6e 20 61 6e 64 20 61 | ust.support.multiplication.and.a |
| 6b80 | 64 64 69 74 69 6f 6e 20 62 6f 74 68 20 77 69 74 68 0a 20 20 20 20 74 68 65 6d 73 65 6c 76 65 73 | ddition.both.with.....themselves |
| 6ba0 | 20 61 6e 64 20 77 69 74 68 20 74 68 65 20 65 6c 65 6d 65 6e 74 73 20 6f 66 20 60 63 60 2e 0a 0a | .and.with.the.elements.of.`c`... |
| 6bc0 | 20 20 20 20 49 66 20 60 63 60 20 69 73 20 61 20 31 2d 44 20 61 72 72 61 79 2c 20 74 68 65 6e 20 | ....If.`c`.is.a.1-D.array,.then. |
| 6be0 | 60 60 70 28 78 29 60 60 20 77 69 6c 6c 20 68 61 76 65 20 74 68 65 20 73 61 6d 65 20 73 68 61 70 | ``p(x)``.will.have.the.same.shap |
| 6c00 | 65 20 61 73 20 60 78 60 2e 20 20 49 66 0a 20 20 20 20 60 63 60 20 69 73 20 6d 75 6c 74 69 64 69 | e.as.`x`...If.....`c`.is.multidi |
| 6c20 | 6d 65 6e 73 69 6f 6e 61 6c 2c 20 74 68 65 6e 20 74 68 65 20 73 68 61 70 65 20 6f 66 20 74 68 65 | mensional,.then.the.shape.of.the |
| 6c40 | 20 72 65 73 75 6c 74 20 64 65 70 65 6e 64 73 20 6f 6e 20 74 68 65 0a 20 20 20 20 76 61 6c 75 65 | .result.depends.on.the.....value |
| 6c60 | 20 6f 66 20 60 74 65 6e 73 6f 72 60 2e 20 49 66 20 60 74 65 6e 73 6f 72 60 20 69 73 20 74 72 75 | .of.`tensor`..If.`tensor`.is.tru |
| 6c80 | 65 20 74 68 65 20 73 68 61 70 65 20 77 69 6c 6c 20 62 65 20 63 2e 73 68 61 70 65 5b 31 3a 5d 20 | e.the.shape.will.be.c.shape[1:]. |
| 6ca0 | 2b 0a 20 20 20 20 78 2e 73 68 61 70 65 2e 20 49 66 20 60 74 65 6e 73 6f 72 60 20 69 73 20 66 61 | +.....x.shape..If.`tensor`.is.fa |
| 6cc0 | 6c 73 65 20 74 68 65 20 73 68 61 70 65 20 77 69 6c 6c 20 62 65 20 63 2e 73 68 61 70 65 5b 31 3a | lse.the.shape.will.be.c.shape[1: |
| 6ce0 | 5d 2e 20 4e 6f 74 65 20 74 68 61 74 0a 20 20 20 20 73 63 61 6c 61 72 73 20 68 61 76 65 20 73 68 | ]..Note.that.....scalars.have.sh |
| 6d00 | 61 70 65 20 28 2c 29 2e 0a 0a 20 20 20 20 54 72 61 69 6c 69 6e 67 20 7a 65 72 6f 73 20 69 6e 20 | ape.(,).......Trailing.zeros.in. |
| 6d20 | 74 68 65 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 77 69 6c 6c 20 62 65 20 75 73 65 64 20 69 6e | the.coefficients.will.be.used.in |
| 6d40 | 20 74 68 65 20 65 76 61 6c 75 61 74 69 6f 6e 2c 20 73 6f 0a 20 20 20 20 74 68 65 79 20 73 68 6f | .the.evaluation,.so.....they.sho |
| 6d60 | 75 6c 64 20 62 65 20 61 76 6f 69 64 65 64 20 69 66 20 65 66 66 69 63 69 65 6e 63 79 20 69 73 20 | uld.be.avoided.if.efficiency.is. |
| 6d80 | 61 20 63 6f 6e 63 65 72 6e 2e 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 20 2d | a.concern.......Parameters.....- |
| 6da0 | 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 78 20 3a 20 61 72 72 61 79 5f 6c 69 6b 65 2c 20 63 6f | ---------.....x.:.array_like,.co |
| 6dc0 | 6d 70 61 74 69 62 6c 65 20 6f 62 6a 65 63 74 0a 20 20 20 20 20 20 20 20 49 66 20 60 78 60 20 69 | mpatible.object.........If.`x`.i |
| 6de0 | 73 20 61 20 6c 69 73 74 20 6f 72 20 74 75 70 6c 65 2c 20 69 74 20 69 73 20 63 6f 6e 76 65 72 74 | s.a.list.or.tuple,.it.is.convert |
| 6e00 | 65 64 20 74 6f 20 61 6e 20 6e 64 61 72 72 61 79 2c 20 6f 74 68 65 72 77 69 73 65 0a 20 20 20 20 | ed.to.an.ndarray,.otherwise..... |
| 6e20 | 20 20 20 20 69 74 20 69 73 20 6c 65 66 74 20 75 6e 63 68 61 6e 67 65 64 20 61 6e 64 20 74 72 65 | ....it.is.left.unchanged.and.tre |
| 6e40 | 61 74 65 64 20 61 73 20 61 20 73 63 61 6c 61 72 2e 20 49 6e 20 65 69 74 68 65 72 20 63 61 73 65 | ated.as.a.scalar..In.either.case |
| 6e60 | 2c 20 60 78 60 0a 20 20 20 20 20 20 20 20 6f 72 20 69 74 73 20 65 6c 65 6d 65 6e 74 73 20 6d 75 | ,.`x`.........or.its.elements.mu |
| 6e80 | 73 74 20 73 75 70 70 6f 72 74 20 61 64 64 69 74 69 6f 6e 20 61 6e 64 20 6d 75 6c 74 69 70 6c 69 | st.support.addition.and.multipli |
| 6ea0 | 63 61 74 69 6f 6e 20 77 69 74 68 0a 20 20 20 20 20 20 20 20 74 68 65 6d 73 65 6c 76 65 73 20 61 | cation.with.........themselves.a |
| 6ec0 | 6e 64 20 77 69 74 68 20 74 68 65 20 65 6c 65 6d 65 6e 74 73 20 6f 66 20 60 63 60 2e 0a 20 20 20 | nd.with.the.elements.of.`c`..... |
| 6ee0 | 20 63 20 3a 20 61 72 72 61 79 5f 6c 69 6b 65 0a 20 20 20 20 20 20 20 20 41 72 72 61 79 20 6f 66 | .c.:.array_like.........Array.of |
| 6f00 | 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6f 72 64 65 72 65 64 20 73 6f 20 74 68 61 74 20 74 68 | .coefficients.ordered.so.that.th |
| 6f20 | 65 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 66 6f 72 20 74 65 72 6d 73 20 6f 66 0a 20 20 20 20 | e.coefficients.for.terms.of..... |
| 6f40 | 20 20 20 20 64 65 67 72 65 65 20 6e 20 61 72 65 20 63 6f 6e 74 61 69 6e 65 64 20 69 6e 20 63 5b | ....degree.n.are.contained.in.c[ |
| 6f60 | 6e 5d 2e 20 49 66 20 60 63 60 20 69 73 20 6d 75 6c 74 69 64 69 6d 65 6e 73 69 6f 6e 61 6c 20 74 | n]..If.`c`.is.multidimensional.t |
| 6f80 | 68 65 0a 20 20 20 20 20 20 20 20 72 65 6d 61 69 6e 69 6e 67 20 69 6e 64 69 63 65 73 20 65 6e 75 | he.........remaining.indices.enu |
| 6fa0 | 6d 65 72 61 74 65 20 6d 75 6c 74 69 70 6c 65 20 70 6f 6c 79 6e 6f 6d 69 61 6c 73 2e 20 49 6e 20 | merate.multiple.polynomials..In. |
| 6fc0 | 74 68 65 20 74 77 6f 0a 20 20 20 20 20 20 20 20 64 69 6d 65 6e 73 69 6f 6e 61 6c 20 63 61 73 65 | the.two.........dimensional.case |
| 6fe0 | 20 74 68 65 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6d 61 79 20 62 65 20 74 68 6f 75 67 68 74 | .the.coefficients.may.be.thought |
| 7000 | 20 6f 66 20 61 73 20 73 74 6f 72 65 64 20 69 6e 0a 20 20 20 20 20 20 20 20 74 68 65 20 63 6f 6c | .of.as.stored.in.........the.col |
| 7020 | 75 6d 6e 73 20 6f 66 20 60 63 60 2e 0a 20 20 20 20 74 65 6e 73 6f 72 20 3a 20 62 6f 6f 6c 65 61 | umns.of.`c`......tensor.:.boolea |
| 7040 | 6e 2c 20 6f 70 74 69 6f 6e 61 6c 0a 20 20 20 20 20 20 20 20 49 66 20 54 72 75 65 2c 20 74 68 65 | n,.optional.........If.True,.the |
| 7060 | 20 73 68 61 70 65 20 6f 66 20 74 68 65 20 63 6f 65 66 66 69 63 69 65 6e 74 20 61 72 72 61 79 20 | .shape.of.the.coefficient.array. |
| 7080 | 69 73 20 65 78 74 65 6e 64 65 64 20 77 69 74 68 20 6f 6e 65 73 0a 20 20 20 20 20 20 20 20 6f 6e | is.extended.with.ones.........on |
| 70a0 | 20 74 68 65 20 72 69 67 68 74 2c 20 6f 6e 65 20 66 6f 72 20 65 61 63 68 20 64 69 6d 65 6e 73 69 | .the.right,.one.for.each.dimensi |
| 70c0 | 6f 6e 20 6f 66 20 60 78 60 2e 20 53 63 61 6c 61 72 73 20 68 61 76 65 20 64 69 6d 65 6e 73 69 6f | on.of.`x`..Scalars.have.dimensio |
| 70e0 | 6e 20 30 0a 20 20 20 20 20 20 20 20 66 6f 72 20 74 68 69 73 20 61 63 74 69 6f 6e 2e 20 54 68 65 | n.0.........for.this.action..The |
| 7100 | 20 72 65 73 75 6c 74 20 69 73 20 74 68 61 74 20 65 76 65 72 79 20 63 6f 6c 75 6d 6e 20 6f 66 20 | .result.is.that.every.column.of. |
| 7120 | 63 6f 65 66 66 69 63 69 65 6e 74 73 20 69 6e 0a 20 20 20 20 20 20 20 20 60 63 60 20 69 73 20 65 | coefficients.in.........`c`.is.e |
| 7140 | 76 61 6c 75 61 74 65 64 20 66 6f 72 20 65 76 65 72 79 20 65 6c 65 6d 65 6e 74 20 6f 66 20 60 78 | valuated.for.every.element.of.`x |
| 7160 | 60 2e 20 49 66 20 46 61 6c 73 65 2c 20 60 78 60 20 69 73 20 62 72 6f 61 64 63 61 73 74 0a 20 20 | `..If.False,.`x`.is.broadcast... |
| 7180 | 20 20 20 20 20 20 6f 76 65 72 20 74 68 65 20 63 6f 6c 75 6d 6e 73 20 6f 66 20 60 63 60 20 66 6f | ......over.the.columns.of.`c`.fo |
| 71a0 | 72 20 74 68 65 20 65 76 61 6c 75 61 74 69 6f 6e 2e 20 20 54 68 69 73 20 6b 65 79 77 6f 72 64 20 | r.the.evaluation...This.keyword. |
| 71c0 | 69 73 20 75 73 65 66 75 6c 0a 20 20 20 20 20 20 20 20 77 68 65 6e 20 60 63 60 20 69 73 20 6d 75 | is.useful.........when.`c`.is.mu |
| 71e0 | 6c 74 69 64 69 6d 65 6e 73 69 6f 6e 61 6c 2e 20 54 68 65 20 64 65 66 61 75 6c 74 20 76 61 6c 75 | ltidimensional..The.default.valu |
| 7200 | 65 20 69 73 20 54 72 75 65 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 0a 20 20 20 20 2d 2d 2d 2d | e.is.True.......Returns.....---- |
| 7220 | 2d 2d 2d 0a 20 20 20 20 76 61 6c 75 65 73 20 3a 20 6e 64 61 72 72 61 79 2c 20 61 6c 67 65 62 72 | ---.....values.:.ndarray,.algebr |
| 7240 | 61 5f 6c 69 6b 65 0a 20 20 20 20 20 20 20 20 54 68 65 20 73 68 61 70 65 20 6f 66 20 74 68 65 20 | a_like.........The.shape.of.the. |
| 7260 | 72 65 74 75 72 6e 20 76 61 6c 75 65 20 69 73 20 64 65 73 63 72 69 62 65 64 20 61 62 6f 76 65 2e | return.value.is.described.above. |
| 7280 | 0a 0a 20 20 20 20 53 65 65 20 41 6c 73 6f 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 | ......See.Also.....--------..... |
| 72a0 | 6c 65 67 76 61 6c 32 64 2c 20 6c 65 67 67 72 69 64 32 64 2c 20 6c 65 67 76 61 6c 33 64 2c 20 6c | legval2d,.leggrid2d,.legval3d,.l |
| 72c0 | 65 67 67 72 69 64 33 64 0a 0a 20 20 20 20 4e 6f 74 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 0a 20 20 | eggrid3d......Notes.....-----... |
| 72e0 | 20 20 54 68 65 20 65 76 61 6c 75 61 74 69 6f 6e 20 75 73 65 73 20 43 6c 65 6e 73 68 61 77 20 72 | ..The.evaluation.uses.Clenshaw.r |
| 7300 | 65 63 75 72 73 69 6f 6e 2c 20 61 6b 61 20 73 79 6e 74 68 65 74 69 63 20 64 69 76 69 73 69 6f 6e | ecursion,.aka.synthetic.division |
| 7320 | 2e 0a 0a 20 20 20 20 72 04 00 00 00 4e 72 60 00 00 00 72 63 00 00 00 29 01 72 04 00 00 00 72 02 | .......r....Nr`...rc...).r....r. |
| 7340 | 00 00 00 72 38 00 00 00 72 37 00 00 00 72 27 00 00 00 72 36 00 00 00 29 10 72 41 00 00 00 72 42 | ...r8...r7...r'...r6...).rA...rB |
| 7360 | 00 00 00 72 50 00 00 00 72 65 00 00 00 72 66 00 00 00 72 67 00 00 00 da 0a 69 73 69 6e 73 74 61 | ...rP...re...rf...rg.....isinsta |
| 7380 | 6e 63 65 da 05 74 75 70 6c 65 72 74 00 00 00 da 07 61 73 61 72 72 61 79 da 07 6e 64 61 72 72 61 | nce..tuplert.....asarray..ndarra |
| 73a0 | 79 da 07 72 65 73 68 61 70 65 72 6c 00 00 00 72 6a 00 00 00 72 2a 00 00 00 72 2b 00 00 00 29 08 | y..reshaperl...rj...r*...r+...). |
| 73c0 | da 01 78 72 3a 00 00 00 da 06 74 65 6e 73 6f 72 72 3c 00 00 00 72 3d 00 00 00 72 58 00 00 00 72 | ..xr:.....tensorr<...r=...rX...r |
| 73e0 | 2f 00 00 00 72 3e 00 00 00 73 08 00 00 00 20 20 20 20 20 20 20 20 72 30 00 00 00 72 12 00 00 00 | /...r>...s............r0...r.... |
| 7400 | 72 12 00 00 00 3e 03 00 00 73 4d 01 00 00 80 00 f4 72 01 00 09 0b 8f 08 89 08 90 11 98 21 a0 24 | r....>...sM......r...........!.$ |
| 7420 | d4 08 27 80 41 d8 07 08 87 77 81 77 87 7c 81 7c 90 7f d1 07 26 d8 0c 0d 8f 48 89 48 94 52 97 59 | ..'.A....w.w.|.|....&....H.H.R.Y |
| 7440 | 91 59 d3 0c 1f 88 01 dc 07 11 90 21 94 65 9c 54 90 5d d4 07 23 dc 0c 0e 8f 4a 89 4a 90 71 8b 4d | .Y.........!.e.T.]..#....J.J.q.M |
| 7460 | 88 01 dc 07 11 90 21 94 52 97 5a 91 5a d4 07 20 a1 56 d8 0c 0d 8f 49 89 49 90 61 97 67 91 67 a0 | ......!.R.Z.Z....V....I.I.a.g.g. |
| 7480 | 04 a0 71 a7 76 a1 76 a1 0d d1 16 2d d3 0c 2e 88 01 e4 07 0a 88 31 83 76 90 11 82 7b d8 0d 0e 88 | ..q.v.v....-.........1.v...{.... |
| 74a0 | 71 89 54 88 02 d8 0d 0e 89 02 dc 09 0c 88 51 8b 16 90 31 8a 1b d8 0d 0e 88 71 89 54 88 02 d8 0d | q.T...........Q...1......q.T.... |
| 74c0 | 0e 88 71 89 54 89 02 e4 0d 10 90 11 8b 56 88 02 d8 0d 0e 88 72 89 55 88 02 d8 0d 0e 88 72 89 55 | ..q.T........V......r.U......r.U |
| 74e0 | 88 02 dc 11 16 90 71 9c 23 98 61 9b 26 a0 31 99 2a d3 11 25 f2 00 04 09 34 88 41 d8 12 14 88 43 | ......q.#.a.&.1.*..%....4.A....C |
| 7500 | d8 11 13 90 61 91 16 88 42 d8 11 12 90 41 90 32 91 15 98 12 a0 02 a0 51 a1 06 a8 22 99 7d d1 19 | ....a...B....A.2.......Q...".}.. |
| 7520 | 2d d1 11 2d 88 42 d8 11 14 90 72 98 41 91 76 a0 21 a0 62 a1 26 a8 31 a1 2a b0 02 d1 21 32 d1 17 | -..-.B....r.A.v.!.b.&.1.*...!2.. |
| 7540 | 33 d1 11 33 89 42 f0 09 04 09 34 f0 0a 00 0c 0e 90 02 90 51 91 06 89 3b d0 04 16 72 31 00 00 00 | 3..3.B....4........Q...;...r1... |
| 7560 | 63 03 00 00 00 00 00 00 00 00 00 00 00 06 00 00 00 03 00 00 00 f3 3a 00 00 00 97 00 74 01 00 00 | c.....................:.....t... |
| 7580 | 00 00 00 00 00 00 6a 02 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 74 04 00 00 00 00 | ......j...................t..... |
| 75a0 | 00 00 00 00 7c 02 7c 00 7c 01 ab 04 00 00 00 00 00 00 53 00 29 01 61 13 06 00 00 0a 20 20 20 20 | ....|.|.|.........S.).a......... |
| 75c0 | 45 76 61 6c 75 61 74 65 20 61 20 32 2d 44 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 61 | Evaluate.a.2-D.Legendre.series.a |
| 75e0 | 74 20 70 6f 69 6e 74 73 20 28 78 2c 20 79 29 2e 0a 0a 20 20 20 20 54 68 69 73 20 66 75 6e 63 74 | t.points.(x,.y).......This.funct |
| 7600 | 69 6f 6e 20 72 65 74 75 72 6e 73 20 74 68 65 20 76 61 6c 75 65 73 3a 0a 0a 20 20 20 20 2e 2e 20 | ion.returns.the.values:......... |
| 7620 | 6d 61 74 68 3a 3a 20 70 28 78 2c 79 29 20 3d 20 5c 73 75 6d 5f 7b 69 2c 6a 7d 20 63 5f 7b 69 2c | math::.p(x,y).=.\sum_{i,j}.c_{i, |
| 7640 | 6a 7d 20 2a 20 4c 5f 69 28 78 29 20 2a 20 4c 5f 6a 28 79 29 0a 0a 20 20 20 20 54 68 65 20 70 61 | j}.*.L_i(x).*.L_j(y)......The.pa |
| 7660 | 72 61 6d 65 74 65 72 73 20 60 78 60 20 61 6e 64 20 60 79 60 20 61 72 65 20 63 6f 6e 76 65 72 74 | rameters.`x`.and.`y`.are.convert |
| 7680 | 65 64 20 74 6f 20 61 72 72 61 79 73 20 6f 6e 6c 79 20 69 66 20 74 68 65 79 20 61 72 65 0a 20 20 | ed.to.arrays.only.if.they.are... |
| 76a0 | 20 20 74 75 70 6c 65 73 20 6f 72 20 61 20 6c 69 73 74 73 2c 20 6f 74 68 65 72 77 69 73 65 20 74 | ..tuples.or.a.lists,.otherwise.t |
| 76c0 | 68 65 79 20 61 72 65 20 74 72 65 61 74 65 64 20 61 73 20 61 20 73 63 61 6c 61 72 73 20 61 6e 64 | hey.are.treated.as.a.scalars.and |
| 76e0 | 20 74 68 65 79 0a 20 20 20 20 6d 75 73 74 20 68 61 76 65 20 74 68 65 20 73 61 6d 65 20 73 68 61 | .they.....must.have.the.same.sha |
| 7700 | 70 65 20 61 66 74 65 72 20 63 6f 6e 76 65 72 73 69 6f 6e 2e 20 49 6e 20 65 69 74 68 65 72 20 63 | pe.after.conversion..In.either.c |
| 7720 | 61 73 65 2c 20 65 69 74 68 65 72 20 60 78 60 0a 20 20 20 20 61 6e 64 20 60 79 60 20 6f 72 20 74 | ase,.either.`x`.....and.`y`.or.t |
| 7740 | 68 65 69 72 20 65 6c 65 6d 65 6e 74 73 20 6d 75 73 74 20 73 75 70 70 6f 72 74 20 6d 75 6c 74 69 | heir.elements.must.support.multi |
| 7760 | 70 6c 69 63 61 74 69 6f 6e 20 61 6e 64 20 61 64 64 69 74 69 6f 6e 20 62 6f 74 68 0a 20 20 20 20 | plication.and.addition.both..... |
| 7780 | 77 69 74 68 20 74 68 65 6d 73 65 6c 76 65 73 20 61 6e 64 20 77 69 74 68 20 74 68 65 20 65 6c 65 | with.themselves.and.with.the.ele |
| 77a0 | 6d 65 6e 74 73 20 6f 66 20 60 63 60 2e 0a 0a 20 20 20 20 49 66 20 60 63 60 20 69 73 20 61 20 31 | ments.of.`c`.......If.`c`.is.a.1 |
| 77c0 | 2d 44 20 61 72 72 61 79 20 61 20 6f 6e 65 20 69 73 20 69 6d 70 6c 69 63 69 74 6c 79 20 61 70 70 | -D.array.a.one.is.implicitly.app |
| 77e0 | 65 6e 64 65 64 20 74 6f 20 69 74 73 20 73 68 61 70 65 20 74 6f 20 6d 61 6b 65 0a 20 20 20 20 69 | ended.to.its.shape.to.make.....i |
| 7800 | 74 20 32 2d 44 2e 20 54 68 65 20 73 68 61 70 65 20 6f 66 20 74 68 65 20 72 65 73 75 6c 74 20 77 | t.2-D..The.shape.of.the.result.w |
| 7820 | 69 6c 6c 20 62 65 20 63 2e 73 68 61 70 65 5b 32 3a 5d 20 2b 20 78 2e 73 68 61 70 65 2e 0a 0a 20 | ill.be.c.shape[2:].+.x.shape.... |
| 7840 | 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 | ...Parameters.....----------.... |
| 7860 | 20 78 2c 20 79 20 3a 20 61 72 72 61 79 5f 6c 69 6b 65 2c 20 63 6f 6d 70 61 74 69 62 6c 65 20 6f | .x,.y.:.array_like,.compatible.o |
| 7880 | 62 6a 65 63 74 73 0a 20 20 20 20 20 20 20 20 54 68 65 20 74 77 6f 20 64 69 6d 65 6e 73 69 6f 6e | bjects.........The.two.dimension |
| 78a0 | 61 6c 20 73 65 72 69 65 73 20 69 73 20 65 76 61 6c 75 61 74 65 64 20 61 74 20 74 68 65 20 70 6f | al.series.is.evaluated.at.the.po |
| 78c0 | 69 6e 74 73 20 60 60 28 78 2c 20 79 29 60 60 2c 0a 20 20 20 20 20 20 20 20 77 68 65 72 65 20 60 | ints.``(x,.y)``,.........where.` |
| 78e0 | 78 60 20 61 6e 64 20 60 79 60 20 6d 75 73 74 20 68 61 76 65 20 74 68 65 20 73 61 6d 65 20 73 68 | x`.and.`y`.must.have.the.same.sh |
| 7900 | 61 70 65 2e 20 49 66 20 60 78 60 20 6f 72 20 60 79 60 20 69 73 20 61 20 6c 69 73 74 0a 20 20 20 | ape..If.`x`.or.`y`.is.a.list.... |
| 7920 | 20 20 20 20 20 6f 72 20 74 75 70 6c 65 2c 20 69 74 20 69 73 20 66 69 72 73 74 20 63 6f 6e 76 65 | .....or.tuple,.it.is.first.conve |
| 7940 | 72 74 65 64 20 74 6f 20 61 6e 20 6e 64 61 72 72 61 79 2c 20 6f 74 68 65 72 77 69 73 65 20 69 74 | rted.to.an.ndarray,.otherwise.it |
| 7960 | 20 69 73 20 6c 65 66 74 0a 20 20 20 20 20 20 20 20 75 6e 63 68 61 6e 67 65 64 20 61 6e 64 20 69 | .is.left.........unchanged.and.i |
| 7980 | 66 20 69 74 20 69 73 6e 27 74 20 61 6e 20 6e 64 61 72 72 61 79 20 69 74 20 69 73 20 74 72 65 61 | f.it.isn't.an.ndarray.it.is.trea |
| 79a0 | 74 65 64 20 61 73 20 61 20 73 63 61 6c 61 72 2e 0a 20 20 20 20 63 20 3a 20 61 72 72 61 79 5f 6c | ted.as.a.scalar......c.:.array_l |
| 79c0 | 69 6b 65 0a 20 20 20 20 20 20 20 20 41 72 72 61 79 20 6f 66 20 63 6f 65 66 66 69 63 69 65 6e 74 | ike.........Array.of.coefficient |
| 79e0 | 73 20 6f 72 64 65 72 65 64 20 73 6f 20 74 68 61 74 20 74 68 65 20 63 6f 65 66 66 69 63 69 65 6e | s.ordered.so.that.the.coefficien |
| 7a00 | 74 20 6f 66 20 74 68 65 20 74 65 72 6d 0a 20 20 20 20 20 20 20 20 6f 66 20 6d 75 6c 74 69 2d 64 | t.of.the.term.........of.multi-d |
| 7a20 | 65 67 72 65 65 20 69 2c 6a 20 69 73 20 63 6f 6e 74 61 69 6e 65 64 20 69 6e 20 60 60 63 5b 69 2c | egree.i,j.is.contained.in.``c[i, |
| 7a40 | 6a 5d 60 60 2e 20 49 66 20 60 63 60 20 68 61 73 0a 20 20 20 20 20 20 20 20 64 69 6d 65 6e 73 69 | j]``..If.`c`.has.........dimensi |
| 7a60 | 6f 6e 20 67 72 65 61 74 65 72 20 74 68 61 6e 20 74 77 6f 20 74 68 65 20 72 65 6d 61 69 6e 69 6e | on.greater.than.two.the.remainin |
| 7a80 | 67 20 69 6e 64 69 63 65 73 20 65 6e 75 6d 65 72 61 74 65 20 6d 75 6c 74 69 70 6c 65 0a 20 20 20 | g.indices.enumerate.multiple.... |
| 7aa0 | 20 20 20 20 20 73 65 74 73 20 6f 66 20 63 6f 65 66 66 69 63 69 65 6e 74 73 2e 0a 0a 20 20 20 20 | .....sets.of.coefficients....... |
| 7ac0 | 52 65 74 75 72 6e 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 76 61 6c 75 65 73 20 3a | Returns.....-------.....values.: |
| 7ae0 | 20 6e 64 61 72 72 61 79 2c 20 63 6f 6d 70 61 74 69 62 6c 65 20 6f 62 6a 65 63 74 0a 20 20 20 20 | .ndarray,.compatible.object..... |
| 7b00 | 20 20 20 20 54 68 65 20 76 61 6c 75 65 73 20 6f 66 20 74 68 65 20 74 77 6f 20 64 69 6d 65 6e 73 | ....The.values.of.the.two.dimens |
| 7b20 | 69 6f 6e 61 6c 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 61 74 20 70 6f 69 6e 74 73 20 | ional.Legendre.series.at.points. |
| 7b40 | 66 6f 72 6d 65 64 0a 20 20 20 20 20 20 20 20 66 72 6f 6d 20 70 61 69 72 73 20 6f 66 20 63 6f 72 | formed.........from.pairs.of.cor |
| 7b60 | 72 65 73 70 6f 6e 64 69 6e 67 20 76 61 6c 75 65 73 20 66 72 6f 6d 20 60 78 60 20 61 6e 64 20 60 | responding.values.from.`x`.and.` |
| 7b80 | 79 60 2e 0a 0a 20 20 20 20 53 65 65 20 41 6c 73 6f 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 | y`.......See.Also.....--------.. |
| 7ba0 | 20 20 20 6c 65 67 76 61 6c 2c 20 6c 65 67 67 72 69 64 32 64 2c 20 6c 65 67 76 61 6c 33 64 2c 20 | ...legval,.leggrid2d,.legval3d,. |
| 7bc0 | 6c 65 67 67 72 69 64 33 64 0a 20 20 20 20 a9 03 72 28 00 00 00 da 06 5f 76 61 6c 6e 64 72 12 00 | leggrid3d.......r(....._valndr.. |
| 7be0 | 00 00 a9 03 72 7e 00 00 00 da 01 79 72 3a 00 00 00 73 03 00 00 00 20 20 20 72 30 00 00 00 72 1d | ....r~.....yr:...s.......r0...r. |
| 7c00 | 00 00 00 72 1d 00 00 00 91 03 00 00 73 1a 00 00 00 80 00 f4 50 01 00 0c 0e 8f 39 89 39 94 56 98 | ...r........s.......P.....9.9.V. |
| 7c20 | 51 a0 01 a0 31 d3 0b 25 d0 04 25 72 31 00 00 00 63 03 00 00 00 00 00 00 00 00 00 00 00 06 00 00 | Q...1..%..%r1...c............... |
| 7c40 | 00 03 00 00 00 f3 3a 00 00 00 97 00 74 01 00 00 00 00 00 00 00 00 6a 02 00 00 00 00 00 00 00 00 | ......:.....t.........j......... |
| 7c60 | 00 00 00 00 00 00 00 00 00 00 74 04 00 00 00 00 00 00 00 00 7c 02 7c 00 7c 01 ab 04 00 00 00 00 | ..........t.........|.|.|....... |
| 7c80 | 00 00 53 00 29 01 61 b4 06 00 00 0a 20 20 20 20 45 76 61 6c 75 61 74 65 20 61 20 32 2d 44 20 4c | ..S.).a.........Evaluate.a.2-D.L |
| 7ca0 | 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 6f 6e 20 74 68 65 20 43 61 72 74 65 73 69 61 6e 20 | egendre.series.on.the.Cartesian. |
| 7cc0 | 70 72 6f 64 75 63 74 20 6f 66 20 78 20 61 6e 64 20 79 2e 0a 0a 20 20 20 20 54 68 69 73 20 66 75 | product.of.x.and.y.......This.fu |
| 7ce0 | 6e 63 74 69 6f 6e 20 72 65 74 75 72 6e 73 20 74 68 65 20 76 61 6c 75 65 73 3a 0a 0a 20 20 20 20 | nction.returns.the.values:...... |
| 7d00 | 2e 2e 20 6d 61 74 68 3a 3a 20 70 28 61 2c 62 29 20 3d 20 5c 73 75 6d 5f 7b 69 2c 6a 7d 20 63 5f | ...math::.p(a,b).=.\sum_{i,j}.c_ |
| 7d20 | 7b 69 2c 6a 7d 20 2a 20 4c 5f 69 28 61 29 20 2a 20 4c 5f 6a 28 62 29 0a 0a 20 20 20 20 77 68 65 | {i,j}.*.L_i(a).*.L_j(b)......whe |
| 7d40 | 72 65 20 74 68 65 20 70 6f 69 6e 74 73 20 60 60 28 61 2c 20 62 29 60 60 20 63 6f 6e 73 69 73 74 | re.the.points.``(a,.b)``.consist |
| 7d60 | 20 6f 66 20 61 6c 6c 20 70 61 69 72 73 20 66 6f 72 6d 65 64 20 62 79 20 74 61 6b 69 6e 67 0a 20 | .of.all.pairs.formed.by.taking.. |
| 7d80 | 20 20 20 60 61 60 20 66 72 6f 6d 20 60 78 60 20 61 6e 64 20 60 62 60 20 66 72 6f 6d 20 60 79 60 | ...`a`.from.`x`.and.`b`.from.`y` |
| 7da0 | 2e 20 54 68 65 20 72 65 73 75 6c 74 69 6e 67 20 70 6f 69 6e 74 73 20 66 6f 72 6d 20 61 20 67 72 | ..The.resulting.points.form.a.gr |
| 7dc0 | 69 64 20 77 69 74 68 0a 20 20 20 20 60 78 60 20 69 6e 20 74 68 65 20 66 69 72 73 74 20 64 69 6d | id.with.....`x`.in.the.first.dim |
| 7de0 | 65 6e 73 69 6f 6e 20 61 6e 64 20 60 79 60 20 69 6e 20 74 68 65 20 73 65 63 6f 6e 64 2e 0a 0a 20 | ension.and.`y`.in.the.second.... |
| 7e00 | 20 20 20 54 68 65 20 70 61 72 61 6d 65 74 65 72 73 20 60 78 60 20 61 6e 64 20 60 79 60 20 61 72 | ...The.parameters.`x`.and.`y`.ar |
| 7e20 | 65 20 63 6f 6e 76 65 72 74 65 64 20 74 6f 20 61 72 72 61 79 73 20 6f 6e 6c 79 20 69 66 20 74 68 | e.converted.to.arrays.only.if.th |
| 7e40 | 65 79 20 61 72 65 0a 20 20 20 20 74 75 70 6c 65 73 20 6f 72 20 61 20 6c 69 73 74 73 2c 20 6f 74 | ey.are.....tuples.or.a.lists,.ot |
| 7e60 | 68 65 72 77 69 73 65 20 74 68 65 79 20 61 72 65 20 74 72 65 61 74 65 64 20 61 73 20 61 20 73 63 | herwise.they.are.treated.as.a.sc |
| 7e80 | 61 6c 61 72 73 2e 20 49 6e 20 65 69 74 68 65 72 0a 20 20 20 20 63 61 73 65 2c 20 65 69 74 68 65 | alars..In.either.....case,.eithe |
| 7ea0 | 72 20 60 78 60 20 61 6e 64 20 60 79 60 20 6f 72 20 74 68 65 69 72 20 65 6c 65 6d 65 6e 74 73 20 | r.`x`.and.`y`.or.their.elements. |
| 7ec0 | 6d 75 73 74 20 73 75 70 70 6f 72 74 20 6d 75 6c 74 69 70 6c 69 63 61 74 69 6f 6e 0a 20 20 20 20 | must.support.multiplication..... |
| 7ee0 | 61 6e 64 20 61 64 64 69 74 69 6f 6e 20 62 6f 74 68 20 77 69 74 68 20 74 68 65 6d 73 65 6c 76 65 | and.addition.both.with.themselve |
| 7f00 | 73 20 61 6e 64 20 77 69 74 68 20 74 68 65 20 65 6c 65 6d 65 6e 74 73 20 6f 66 20 60 63 60 2e 0a | s.and.with.the.elements.of.`c`.. |
| 7f20 | 0a 20 20 20 20 49 66 20 60 63 60 20 68 61 73 20 66 65 77 65 72 20 74 68 61 6e 20 74 77 6f 20 64 | .....If.`c`.has.fewer.than.two.d |
| 7f40 | 69 6d 65 6e 73 69 6f 6e 73 2c 20 6f 6e 65 73 20 61 72 65 20 69 6d 70 6c 69 63 69 74 6c 79 20 61 | imensions,.ones.are.implicitly.a |
| 7f60 | 70 70 65 6e 64 65 64 20 74 6f 0a 20 20 20 20 69 74 73 20 73 68 61 70 65 20 74 6f 20 6d 61 6b 65 | ppended.to.....its.shape.to.make |
| 7f80 | 20 69 74 20 32 2d 44 2e 20 54 68 65 20 73 68 61 70 65 20 6f 66 20 74 68 65 20 72 65 73 75 6c 74 | .it.2-D..The.shape.of.the.result |
| 7fa0 | 20 77 69 6c 6c 20 62 65 20 63 2e 73 68 61 70 65 5b 32 3a 5d 20 2b 0a 20 20 20 20 78 2e 73 68 61 | .will.be.c.shape[2:].+.....x.sha |
| 7fc0 | 70 65 20 2b 20 79 2e 73 68 61 70 65 2e 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 20 | pe.+.y.shape.......Parameters... |
| 7fe0 | 20 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 78 2c 20 79 20 3a 20 61 72 72 61 79 5f 6c 69 | ..----------.....x,.y.:.array_li |
| 8000 | 6b 65 2c 20 63 6f 6d 70 61 74 69 62 6c 65 20 6f 62 6a 65 63 74 73 0a 20 20 20 20 20 20 20 20 54 | ke,.compatible.objects.........T |
| 8020 | 68 65 20 74 77 6f 20 64 69 6d 65 6e 73 69 6f 6e 61 6c 20 73 65 72 69 65 73 20 69 73 20 65 76 61 | he.two.dimensional.series.is.eva |
| 8040 | 6c 75 61 74 65 64 20 61 74 20 74 68 65 20 70 6f 69 6e 74 73 20 69 6e 20 74 68 65 0a 20 20 20 20 | luated.at.the.points.in.the..... |
| 8060 | 20 20 20 20 43 61 72 74 65 73 69 61 6e 20 70 72 6f 64 75 63 74 20 6f 66 20 60 78 60 20 61 6e 64 | ....Cartesian.product.of.`x`.and |
| 8080 | 20 60 79 60 2e 20 20 49 66 20 60 78 60 20 6f 72 20 60 79 60 20 69 73 20 61 20 6c 69 73 74 20 6f | .`y`...If.`x`.or.`y`.is.a.list.o |
| 80a0 | 72 0a 20 20 20 20 20 20 20 20 74 75 70 6c 65 2c 20 69 74 20 69 73 20 66 69 72 73 74 20 63 6f 6e | r.........tuple,.it.is.first.con |
| 80c0 | 76 65 72 74 65 64 20 74 6f 20 61 6e 20 6e 64 61 72 72 61 79 2c 20 6f 74 68 65 72 77 69 73 65 20 | verted.to.an.ndarray,.otherwise. |
| 80e0 | 69 74 20 69 73 20 6c 65 66 74 0a 20 20 20 20 20 20 20 20 75 6e 63 68 61 6e 67 65 64 20 61 6e 64 | it.is.left.........unchanged.and |
| 8100 | 2c 20 69 66 20 69 74 20 69 73 6e 27 74 20 61 6e 20 6e 64 61 72 72 61 79 2c 20 69 74 20 69 73 20 | ,.if.it.isn't.an.ndarray,.it.is. |
| 8120 | 74 72 65 61 74 65 64 20 61 73 20 61 20 73 63 61 6c 61 72 2e 0a 20 20 20 20 63 20 3a 20 61 72 72 | treated.as.a.scalar......c.:.arr |
| 8140 | 61 79 5f 6c 69 6b 65 0a 20 20 20 20 20 20 20 20 41 72 72 61 79 20 6f 66 20 63 6f 65 66 66 69 63 | ay_like.........Array.of.coeffic |
| 8160 | 69 65 6e 74 73 20 6f 72 64 65 72 65 64 20 73 6f 20 74 68 61 74 20 74 68 65 20 63 6f 65 66 66 69 | ients.ordered.so.that.the.coeffi |
| 8180 | 63 69 65 6e 74 20 6f 66 20 74 68 65 20 74 65 72 6d 20 6f 66 0a 20 20 20 20 20 20 20 20 6d 75 6c | cient.of.the.term.of.........mul |
| 81a0 | 74 69 2d 64 65 67 72 65 65 20 69 2c 6a 20 69 73 20 63 6f 6e 74 61 69 6e 65 64 20 69 6e 20 60 60 | ti-degree.i,j.is.contained.in.`` |
| 81c0 | 63 5b 69 2c 6a 5d 60 60 2e 20 49 66 20 60 63 60 20 68 61 73 20 64 69 6d 65 6e 73 69 6f 6e 0a 20 | c[i,j]``..If.`c`.has.dimension.. |
| 81e0 | 20 20 20 20 20 20 20 67 72 65 61 74 65 72 20 74 68 61 6e 20 74 77 6f 20 74 68 65 20 72 65 6d 61 | .......greater.than.two.the.rema |
| 8200 | 69 6e 69 6e 67 20 69 6e 64 69 63 65 73 20 65 6e 75 6d 65 72 61 74 65 20 6d 75 6c 74 69 70 6c 65 | ining.indices.enumerate.multiple |
| 8220 | 20 73 65 74 73 20 6f 66 0a 20 20 20 20 20 20 20 20 63 6f 65 66 66 69 63 69 65 6e 74 73 2e 0a 0a | .sets.of.........coefficients... |
| 8240 | 20 20 20 20 52 65 74 75 72 6e 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 76 61 6c 75 | ....Returns.....-------.....valu |
| 8260 | 65 73 20 3a 20 6e 64 61 72 72 61 79 2c 20 63 6f 6d 70 61 74 69 62 6c 65 20 6f 62 6a 65 63 74 0a | es.:.ndarray,.compatible.object. |
| 8280 | 20 20 20 20 20 20 20 20 54 68 65 20 76 61 6c 75 65 73 20 6f 66 20 74 68 65 20 74 77 6f 20 64 69 | ........The.values.of.the.two.di |
| 82a0 | 6d 65 6e 73 69 6f 6e 61 6c 20 43 68 65 62 79 73 68 65 76 20 73 65 72 69 65 73 20 61 74 20 70 6f | mensional.Chebyshev.series.at.po |
| 82c0 | 69 6e 74 73 20 69 6e 20 74 68 65 0a 20 20 20 20 20 20 20 20 43 61 72 74 65 73 69 61 6e 20 70 72 | ints.in.the.........Cartesian.pr |
| 82e0 | 6f 64 75 63 74 20 6f 66 20 60 78 60 20 61 6e 64 20 60 79 60 2e 0a 0a 20 20 20 20 53 65 65 20 41 | oduct.of.`x`.and.`y`.......See.A |
| 8300 | 6c 73 6f 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 6c 65 67 76 61 6c 2c 20 6c 65 67 | lso.....--------.....legval,.leg |
| 8320 | 76 61 6c 32 64 2c 20 6c 65 67 76 61 6c 33 64 2c 20 6c 65 67 67 72 69 64 33 64 0a 20 20 20 20 a9 | val2d,.legval3d,.leggrid3d...... |
| 8340 | 03 72 28 00 00 00 da 07 5f 67 72 69 64 6e 64 72 12 00 00 00 72 83 00 00 00 73 03 00 00 00 20 20 | .r(....._gridndr....r....s...... |
| 8360 | 20 72 30 00 00 00 72 1f 00 00 00 72 1f 00 00 00 bc 03 00 00 73 1a 00 00 00 80 00 f4 58 01 00 0c | .r0...r....r........s.......X... |
| 8380 | 0e 8f 3a 89 3a 94 66 98 61 a0 11 a0 41 d3 0b 26 d0 04 26 72 31 00 00 00 63 04 00 00 00 00 00 00 | ..:.:.f.a...A..&..&r1...c....... |
| 83a0 | 00 00 00 00 00 07 00 00 00 03 00 00 00 f3 3c 00 00 00 97 00 74 01 00 00 00 00 00 00 00 00 6a 02 | ..............<.....t.........j. |
| 83c0 | 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 74 04 00 00 00 00 00 00 00 00 7c 03 7c 00 | ..................t.........|.|. |
| 83e0 | 7c 01 7c 02 ab 05 00 00 00 00 00 00 53 00 29 01 61 6d 06 00 00 0a 20 20 20 20 45 76 61 6c 75 61 | |.|.........S.).am........Evalua |
| 8400 | 74 65 20 61 20 33 2d 44 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 61 74 20 70 6f 69 6e | te.a.3-D.Legendre.series.at.poin |
| 8420 | 74 73 20 28 78 2c 20 79 2c 20 7a 29 2e 0a 0a 20 20 20 20 54 68 69 73 20 66 75 6e 63 74 69 6f 6e | ts.(x,.y,.z).......This.function |
| 8440 | 20 72 65 74 75 72 6e 73 20 74 68 65 20 76 61 6c 75 65 73 3a 0a 0a 20 20 20 20 2e 2e 20 6d 61 74 | .returns.the.values:.........mat |
| 8460 | 68 3a 3a 20 70 28 78 2c 79 2c 7a 29 20 3d 20 5c 73 75 6d 5f 7b 69 2c 6a 2c 6b 7d 20 63 5f 7b 69 | h::.p(x,y,z).=.\sum_{i,j,k}.c_{i |
| 8480 | 2c 6a 2c 6b 7d 20 2a 20 4c 5f 69 28 78 29 20 2a 20 4c 5f 6a 28 79 29 20 2a 20 4c 5f 6b 28 7a 29 | ,j,k}.*.L_i(x).*.L_j(y).*.L_k(z) |
| 84a0 | 0a 0a 20 20 20 20 54 68 65 20 70 61 72 61 6d 65 74 65 72 73 20 60 78 60 2c 20 60 79 60 2c 20 61 | ......The.parameters.`x`,.`y`,.a |
| 84c0 | 6e 64 20 60 7a 60 20 61 72 65 20 63 6f 6e 76 65 72 74 65 64 20 74 6f 20 61 72 72 61 79 73 20 6f | nd.`z`.are.converted.to.arrays.o |
| 84e0 | 6e 6c 79 20 69 66 0a 20 20 20 20 74 68 65 79 20 61 72 65 20 74 75 70 6c 65 73 20 6f 72 20 61 20 | nly.if.....they.are.tuples.or.a. |
| 8500 | 6c 69 73 74 73 2c 20 6f 74 68 65 72 77 69 73 65 20 74 68 65 79 20 61 72 65 20 74 72 65 61 74 65 | lists,.otherwise.they.are.treate |
| 8520 | 64 20 61 73 20 61 20 73 63 61 6c 61 72 73 20 61 6e 64 0a 20 20 20 20 74 68 65 79 20 6d 75 73 74 | d.as.a.scalars.and.....they.must |
| 8540 | 20 68 61 76 65 20 74 68 65 20 73 61 6d 65 20 73 68 61 70 65 20 61 66 74 65 72 20 63 6f 6e 76 65 | .have.the.same.shape.after.conve |
| 8560 | 72 73 69 6f 6e 2e 20 49 6e 20 65 69 74 68 65 72 20 63 61 73 65 2c 20 65 69 74 68 65 72 0a 20 20 | rsion..In.either.case,.either... |
| 8580 | 20 20 60 78 60 2c 20 60 79 60 2c 20 61 6e 64 20 60 7a 60 20 6f 72 20 74 68 65 69 72 20 65 6c 65 | ..`x`,.`y`,.and.`z`.or.their.ele |
| 85a0 | 6d 65 6e 74 73 20 6d 75 73 74 20 73 75 70 70 6f 72 74 20 6d 75 6c 74 69 70 6c 69 63 61 74 69 6f | ments.must.support.multiplicatio |
| 85c0 | 6e 20 61 6e 64 0a 20 20 20 20 61 64 64 69 74 69 6f 6e 20 62 6f 74 68 20 77 69 74 68 20 74 68 65 | n.and.....addition.both.with.the |
| 85e0 | 6d 73 65 6c 76 65 73 20 61 6e 64 20 77 69 74 68 20 74 68 65 20 65 6c 65 6d 65 6e 74 73 20 6f 66 | mselves.and.with.the.elements.of |
| 8600 | 20 60 63 60 2e 0a 0a 20 20 20 20 49 66 20 60 63 60 20 68 61 73 20 66 65 77 65 72 20 74 68 61 6e | .`c`.......If.`c`.has.fewer.than |
| 8620 | 20 33 20 64 69 6d 65 6e 73 69 6f 6e 73 2c 20 6f 6e 65 73 20 61 72 65 20 69 6d 70 6c 69 63 69 74 | .3.dimensions,.ones.are.implicit |
| 8640 | 6c 79 20 61 70 70 65 6e 64 65 64 20 74 6f 20 69 74 73 0a 20 20 20 20 73 68 61 70 65 20 74 6f 20 | ly.appended.to.its.....shape.to. |
| 8660 | 6d 61 6b 65 20 69 74 20 33 2d 44 2e 20 54 68 65 20 73 68 61 70 65 20 6f 66 20 74 68 65 20 72 65 | make.it.3-D..The.shape.of.the.re |
| 8680 | 73 75 6c 74 20 77 69 6c 6c 20 62 65 20 63 2e 73 68 61 70 65 5b 33 3a 5d 20 2b 0a 20 20 20 20 78 | sult.will.be.c.shape[3:].+.....x |
| 86a0 | 2e 73 68 61 70 65 2e 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 20 2d 2d 2d 2d | .shape.......Parameters.....---- |
| 86c0 | 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 78 2c 20 79 2c 20 7a 20 3a 20 61 72 72 61 79 5f 6c 69 6b 65 2c | ------.....x,.y,.z.:.array_like, |
| 86e0 | 20 63 6f 6d 70 61 74 69 62 6c 65 20 6f 62 6a 65 63 74 0a 20 20 20 20 20 20 20 20 54 68 65 20 74 | .compatible.object.........The.t |
| 8700 | 68 72 65 65 20 64 69 6d 65 6e 73 69 6f 6e 61 6c 20 73 65 72 69 65 73 20 69 73 20 65 76 61 6c 75 | hree.dimensional.series.is.evalu |
| 8720 | 61 74 65 64 20 61 74 20 74 68 65 20 70 6f 69 6e 74 73 0a 20 20 20 20 20 20 20 20 60 60 28 78 2c | ated.at.the.points.........``(x, |
| 8740 | 20 79 2c 20 7a 29 60 60 2c 20 77 68 65 72 65 20 60 78 60 2c 20 60 79 60 2c 20 61 6e 64 20 60 7a | .y,.z)``,.where.`x`,.`y`,.and.`z |
| 8760 | 60 20 6d 75 73 74 20 68 61 76 65 20 74 68 65 20 73 61 6d 65 20 73 68 61 70 65 2e 20 20 49 66 0a | `.must.have.the.same.shape...If. |
| 8780 | 20 20 20 20 20 20 20 20 61 6e 79 20 6f 66 20 60 78 60 2c 20 60 79 60 2c 20 6f 72 20 60 7a 60 20 | ........any.of.`x`,.`y`,.or.`z`. |
| 87a0 | 69 73 20 61 20 6c 69 73 74 20 6f 72 20 74 75 70 6c 65 2c 20 69 74 20 69 73 20 66 69 72 73 74 20 | is.a.list.or.tuple,.it.is.first. |
| 87c0 | 63 6f 6e 76 65 72 74 65 64 0a 20 20 20 20 20 20 20 20 74 6f 20 61 6e 20 6e 64 61 72 72 61 79 2c | converted.........to.an.ndarray, |
| 87e0 | 20 6f 74 68 65 72 77 69 73 65 20 69 74 20 69 73 20 6c 65 66 74 20 75 6e 63 68 61 6e 67 65 64 20 | .otherwise.it.is.left.unchanged. |
| 8800 | 61 6e 64 20 69 66 20 69 74 20 69 73 6e 27 74 20 61 6e 0a 20 20 20 20 20 20 20 20 6e 64 61 72 72 | and.if.it.isn't.an.........ndarr |
| 8820 | 61 79 20 69 74 20 69 73 20 20 74 72 65 61 74 65 64 20 61 73 20 61 20 73 63 61 6c 61 72 2e 0a 20 | ay.it.is..treated.as.a.scalar... |
| 8840 | 20 20 20 63 20 3a 20 61 72 72 61 79 5f 6c 69 6b 65 0a 20 20 20 20 20 20 20 20 41 72 72 61 79 20 | ...c.:.array_like.........Array. |
| 8860 | 6f 66 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6f 72 64 65 72 65 64 20 73 6f 20 74 68 61 74 20 | of.coefficients.ordered.so.that. |
| 8880 | 74 68 65 20 63 6f 65 66 66 69 63 69 65 6e 74 20 6f 66 20 74 68 65 20 74 65 72 6d 20 6f 66 0a 20 | the.coefficient.of.the.term.of.. |
| 88a0 | 20 20 20 20 20 20 20 6d 75 6c 74 69 2d 64 65 67 72 65 65 20 69 2c 6a 2c 6b 20 69 73 20 63 6f 6e | .......multi-degree.i,j,k.is.con |
| 88c0 | 74 61 69 6e 65 64 20 69 6e 20 60 60 63 5b 69 2c 6a 2c 6b 5d 60 60 2e 20 49 66 20 60 63 60 20 68 | tained.in.``c[i,j,k]``..If.`c`.h |
| 88e0 | 61 73 20 64 69 6d 65 6e 73 69 6f 6e 0a 20 20 20 20 20 20 20 20 67 72 65 61 74 65 72 20 74 68 61 | as.dimension.........greater.tha |
| 8900 | 6e 20 33 20 74 68 65 20 72 65 6d 61 69 6e 69 6e 67 20 69 6e 64 69 63 65 73 20 65 6e 75 6d 65 72 | n.3.the.remaining.indices.enumer |
| 8920 | 61 74 65 20 6d 75 6c 74 69 70 6c 65 20 73 65 74 73 20 6f 66 0a 20 20 20 20 20 20 20 20 63 6f 65 | ate.multiple.sets.of.........coe |
| 8940 | 66 66 69 63 69 65 6e 74 73 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 0a 20 20 20 20 2d 2d 2d 2d | fficients.......Returns.....---- |
| 8960 | 2d 2d 2d 0a 20 20 20 20 76 61 6c 75 65 73 20 3a 20 6e 64 61 72 72 61 79 2c 20 63 6f 6d 70 61 74 | ---.....values.:.ndarray,.compat |
| 8980 | 69 62 6c 65 20 6f 62 6a 65 63 74 0a 20 20 20 20 20 20 20 20 54 68 65 20 76 61 6c 75 65 73 20 6f | ible.object.........The.values.o |
| 89a0 | 66 20 74 68 65 20 6d 75 6c 74 69 64 69 6d 65 6e 73 69 6f 6e 61 6c 20 70 6f 6c 79 6e 6f 6d 69 61 | f.the.multidimensional.polynomia |
| 89c0 | 6c 20 6f 6e 20 70 6f 69 6e 74 73 20 66 6f 72 6d 65 64 20 77 69 74 68 0a 20 20 20 20 20 20 20 20 | l.on.points.formed.with......... |
| 89e0 | 74 72 69 70 6c 65 73 20 6f 66 20 63 6f 72 72 65 73 70 6f 6e 64 69 6e 67 20 76 61 6c 75 65 73 20 | triples.of.corresponding.values. |
| 8a00 | 66 72 6f 6d 20 60 78 60 2c 20 60 79 60 2c 20 61 6e 64 20 60 7a 60 2e 0a 0a 20 20 20 20 53 65 65 | from.`x`,.`y`,.and.`z`.......See |
| 8a20 | 20 41 6c 73 6f 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 6c 65 67 76 61 6c 2c 20 6c | .Also.....--------.....legval,.l |
| 8a40 | 65 67 76 61 6c 32 64 2c 20 6c 65 67 67 72 69 64 32 64 2c 20 6c 65 67 67 72 69 64 33 64 0a 20 20 | egval2d,.leggrid2d,.leggrid3d... |
| 8a60 | 20 20 72 81 00 00 00 a9 04 72 7e 00 00 00 72 84 00 00 00 da 01 7a 72 3a 00 00 00 73 04 00 00 00 | ..r......r~...r......zr:...s.... |
| 8a80 | 20 20 20 20 72 30 00 00 00 72 1e 00 00 00 72 1e 00 00 00 eb 03 00 00 73 1c 00 00 00 80 00 f4 54 | ....r0...r....r........s.......T |
| 8aa0 | 01 00 0c 0e 8f 39 89 39 94 56 98 51 a0 01 a0 31 a0 61 d3 0b 28 d0 04 28 72 31 00 00 00 63 04 00 | .....9.9.V.Q...1.a..(..(r1...c.. |
| 8ac0 | 00 00 00 00 00 00 00 00 00 00 07 00 00 00 03 00 00 00 f3 3c 00 00 00 97 00 74 01 00 00 00 00 00 | ...................<.....t...... |
| 8ae0 | 00 00 00 6a 02 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 74 04 00 00 00 00 00 00 00 | ...j...................t........ |
| 8b00 | 00 7c 03 7c 00 7c 01 7c 02 ab 05 00 00 00 00 00 00 53 00 29 01 61 1b 07 00 00 0a 20 20 20 20 45 | .|.|.|.|.........S.).a.........E |
| 8b20 | 76 61 6c 75 61 74 65 20 61 20 33 2d 44 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 6f 6e | valuate.a.3-D.Legendre.series.on |
| 8b40 | 20 74 68 65 20 43 61 72 74 65 73 69 61 6e 20 70 72 6f 64 75 63 74 20 6f 66 20 78 2c 20 79 2c 20 | .the.Cartesian.product.of.x,.y,. |
| 8b60 | 61 6e 64 20 7a 2e 0a 0a 20 20 20 20 54 68 69 73 20 66 75 6e 63 74 69 6f 6e 20 72 65 74 75 72 6e | and.z.......This.function.return |
| 8b80 | 73 20 74 68 65 20 76 61 6c 75 65 73 3a 0a 0a 20 20 20 20 2e 2e 20 6d 61 74 68 3a 3a 20 70 28 61 | s.the.values:.........math::.p(a |
| 8ba0 | 2c 62 2c 63 29 20 3d 20 5c 73 75 6d 5f 7b 69 2c 6a 2c 6b 7d 20 63 5f 7b 69 2c 6a 2c 6b 7d 20 2a | ,b,c).=.\sum_{i,j,k}.c_{i,j,k}.* |
| 8bc0 | 20 4c 5f 69 28 61 29 20 2a 20 4c 5f 6a 28 62 29 20 2a 20 4c 5f 6b 28 63 29 0a 0a 20 20 20 20 77 | .L_i(a).*.L_j(b).*.L_k(c)......w |
| 8be0 | 68 65 72 65 20 74 68 65 20 70 6f 69 6e 74 73 20 60 60 28 61 2c 20 62 2c 20 63 29 60 60 20 63 6f | here.the.points.``(a,.b,.c)``.co |
| 8c00 | 6e 73 69 73 74 20 6f 66 20 61 6c 6c 20 74 72 69 70 6c 65 73 20 66 6f 72 6d 65 64 20 62 79 20 74 | nsist.of.all.triples.formed.by.t |
| 8c20 | 61 6b 69 6e 67 0a 20 20 20 20 60 61 60 20 66 72 6f 6d 20 60 78 60 2c 20 60 62 60 20 66 72 6f 6d | aking.....`a`.from.`x`,.`b`.from |
| 8c40 | 20 60 79 60 2c 20 61 6e 64 20 60 63 60 20 66 72 6f 6d 20 60 7a 60 2e 20 54 68 65 20 72 65 73 75 | .`y`,.and.`c`.from.`z`..The.resu |
| 8c60 | 6c 74 69 6e 67 20 70 6f 69 6e 74 73 20 66 6f 72 6d 0a 20 20 20 20 61 20 67 72 69 64 20 77 69 74 | lting.points.form.....a.grid.wit |
| 8c80 | 68 20 60 78 60 20 69 6e 20 74 68 65 20 66 69 72 73 74 20 64 69 6d 65 6e 73 69 6f 6e 2c 20 60 79 | h.`x`.in.the.first.dimension,.`y |
| 8ca0 | 60 20 69 6e 20 74 68 65 20 73 65 63 6f 6e 64 2c 20 61 6e 64 20 60 7a 60 20 69 6e 0a 20 20 20 20 | `.in.the.second,.and.`z`.in..... |
| 8cc0 | 74 68 65 20 74 68 69 72 64 2e 0a 0a 20 20 20 20 54 68 65 20 70 61 72 61 6d 65 74 65 72 73 20 60 | the.third.......The.parameters.` |
| 8ce0 | 78 60 2c 20 60 79 60 2c 20 61 6e 64 20 60 7a 60 20 61 72 65 20 63 6f 6e 76 65 72 74 65 64 20 74 | x`,.`y`,.and.`z`.are.converted.t |
| 8d00 | 6f 20 61 72 72 61 79 73 20 6f 6e 6c 79 20 69 66 20 74 68 65 79 0a 20 20 20 20 61 72 65 20 74 75 | o.arrays.only.if.they.....are.tu |
| 8d20 | 70 6c 65 73 20 6f 72 20 61 20 6c 69 73 74 73 2c 20 6f 74 68 65 72 77 69 73 65 20 74 68 65 79 20 | ples.or.a.lists,.otherwise.they. |
| 8d40 | 61 72 65 20 74 72 65 61 74 65 64 20 61 73 20 61 20 73 63 61 6c 61 72 73 2e 20 49 6e 0a 20 20 20 | are.treated.as.a.scalars..In.... |
| 8d60 | 20 65 69 74 68 65 72 20 63 61 73 65 2c 20 65 69 74 68 65 72 20 60 78 60 2c 20 60 79 60 2c 20 61 | .either.case,.either.`x`,.`y`,.a |
| 8d80 | 6e 64 20 60 7a 60 20 6f 72 20 74 68 65 69 72 20 65 6c 65 6d 65 6e 74 73 20 6d 75 73 74 20 73 75 | nd.`z`.or.their.elements.must.su |
| 8da0 | 70 70 6f 72 74 0a 20 20 20 20 6d 75 6c 74 69 70 6c 69 63 61 74 69 6f 6e 20 61 6e 64 20 61 64 64 | pport.....multiplication.and.add |
| 8dc0 | 69 74 69 6f 6e 20 62 6f 74 68 20 77 69 74 68 20 74 68 65 6d 73 65 6c 76 65 73 20 61 6e 64 20 77 | ition.both.with.themselves.and.w |
| 8de0 | 69 74 68 20 74 68 65 20 65 6c 65 6d 65 6e 74 73 0a 20 20 20 20 6f 66 20 60 63 60 2e 0a 0a 20 20 | ith.the.elements.....of.`c`..... |
| 8e00 | 20 20 49 66 20 60 63 60 20 68 61 73 20 66 65 77 65 72 20 74 68 61 6e 20 74 68 72 65 65 20 64 69 | ..If.`c`.has.fewer.than.three.di |
| 8e20 | 6d 65 6e 73 69 6f 6e 73 2c 20 6f 6e 65 73 20 61 72 65 20 69 6d 70 6c 69 63 69 74 6c 79 20 61 70 | mensions,.ones.are.implicitly.ap |
| 8e40 | 70 65 6e 64 65 64 20 74 6f 0a 20 20 20 20 69 74 73 20 73 68 61 70 65 20 74 6f 20 6d 61 6b 65 20 | pended.to.....its.shape.to.make. |
| 8e60 | 69 74 20 33 2d 44 2e 20 54 68 65 20 73 68 61 70 65 20 6f 66 20 74 68 65 20 72 65 73 75 6c 74 20 | it.3-D..The.shape.of.the.result. |
| 8e80 | 77 69 6c 6c 20 62 65 20 63 2e 73 68 61 70 65 5b 33 3a 5d 20 2b 0a 20 20 20 20 78 2e 73 68 61 70 | will.be.c.shape[3:].+.....x.shap |
| 8ea0 | 65 20 2b 20 79 2e 73 68 61 70 65 20 2b 20 7a 2e 73 68 61 70 65 2e 0a 0a 20 20 20 20 50 61 72 61 | e.+.y.shape.+.z.shape.......Para |
| 8ec0 | 6d 65 74 65 72 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 78 2c 20 79 2c 20 | meters.....----------.....x,.y,. |
| 8ee0 | 7a 20 3a 20 61 72 72 61 79 5f 6c 69 6b 65 2c 20 63 6f 6d 70 61 74 69 62 6c 65 20 6f 62 6a 65 63 | z.:.array_like,.compatible.objec |
| 8f00 | 74 73 0a 20 20 20 20 20 20 20 20 54 68 65 20 74 68 72 65 65 20 64 69 6d 65 6e 73 69 6f 6e 61 6c | ts.........The.three.dimensional |
| 8f20 | 20 73 65 72 69 65 73 20 69 73 20 65 76 61 6c 75 61 74 65 64 20 61 74 20 74 68 65 20 70 6f 69 6e | .series.is.evaluated.at.the.poin |
| 8f40 | 74 73 20 69 6e 20 74 68 65 0a 20 20 20 20 20 20 20 20 43 61 72 74 65 73 69 61 6e 20 70 72 6f 64 | ts.in.the.........Cartesian.prod |
| 8f60 | 75 63 74 20 6f 66 20 60 78 60 2c 20 60 79 60 2c 20 61 6e 64 20 60 7a 60 2e 20 20 49 66 20 60 78 | uct.of.`x`,.`y`,.and.`z`...If.`x |
| 8f80 | 60 2c 20 60 79 60 2c 20 6f 72 20 60 7a 60 20 69 73 20 61 0a 20 20 20 20 20 20 20 20 6c 69 73 74 | `,.`y`,.or.`z`.is.a.........list |
| 8fa0 | 20 6f 72 20 74 75 70 6c 65 2c 20 69 74 20 69 73 20 66 69 72 73 74 20 63 6f 6e 76 65 72 74 65 64 | .or.tuple,.it.is.first.converted |
| 8fc0 | 20 74 6f 20 61 6e 20 6e 64 61 72 72 61 79 2c 20 6f 74 68 65 72 77 69 73 65 20 69 74 20 69 73 0a | .to.an.ndarray,.otherwise.it.is. |
| 8fe0 | 20 20 20 20 20 20 20 20 6c 65 66 74 20 75 6e 63 68 61 6e 67 65 64 20 61 6e 64 2c 20 69 66 20 69 | ........left.unchanged.and,.if.i |
| 9000 | 74 20 69 73 6e 27 74 20 61 6e 20 6e 64 61 72 72 61 79 2c 20 69 74 20 69 73 20 74 72 65 61 74 65 | t.isn't.an.ndarray,.it.is.treate |
| 9020 | 64 20 61 73 20 61 0a 20 20 20 20 20 20 20 20 73 63 61 6c 61 72 2e 0a 20 20 20 20 63 20 3a 20 61 | d.as.a.........scalar......c.:.a |
| 9040 | 72 72 61 79 5f 6c 69 6b 65 0a 20 20 20 20 20 20 20 20 41 72 72 61 79 20 6f 66 20 63 6f 65 66 66 | rray_like.........Array.of.coeff |
| 9060 | 69 63 69 65 6e 74 73 20 6f 72 64 65 72 65 64 20 73 6f 20 74 68 61 74 20 74 68 65 20 63 6f 65 66 | icients.ordered.so.that.the.coef |
| 9080 | 66 69 63 69 65 6e 74 73 20 66 6f 72 20 74 65 72 6d 73 20 6f 66 0a 20 20 20 20 20 20 20 20 64 65 | ficients.for.terms.of.........de |
| 90a0 | 67 72 65 65 20 69 2c 6a 20 61 72 65 20 63 6f 6e 74 61 69 6e 65 64 20 69 6e 20 60 60 63 5b 69 2c | gree.i,j.are.contained.in.``c[i, |
| 90c0 | 6a 5d 60 60 2e 20 49 66 20 60 63 60 20 68 61 73 20 64 69 6d 65 6e 73 69 6f 6e 0a 20 20 20 20 20 | j]``..If.`c`.has.dimension...... |
| 90e0 | 20 20 20 67 72 65 61 74 65 72 20 74 68 61 6e 20 74 77 6f 20 74 68 65 20 72 65 6d 61 69 6e 69 6e | ...greater.than.two.the.remainin |
| 9100 | 67 20 69 6e 64 69 63 65 73 20 65 6e 75 6d 65 72 61 74 65 20 6d 75 6c 74 69 70 6c 65 20 73 65 74 | g.indices.enumerate.multiple.set |
| 9120 | 73 20 6f 66 0a 20 20 20 20 20 20 20 20 63 6f 65 66 66 69 63 69 65 6e 74 73 2e 0a 0a 20 20 20 20 | s.of.........coefficients....... |
| 9140 | 52 65 74 75 72 6e 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 76 61 6c 75 65 73 20 3a | Returns.....-------.....values.: |
| 9160 | 20 6e 64 61 72 72 61 79 2c 20 63 6f 6d 70 61 74 69 62 6c 65 20 6f 62 6a 65 63 74 0a 20 20 20 20 | .ndarray,.compatible.object..... |
| 9180 | 20 20 20 20 54 68 65 20 76 61 6c 75 65 73 20 6f 66 20 74 68 65 20 74 77 6f 20 64 69 6d 65 6e 73 | ....The.values.of.the.two.dimens |
| 91a0 | 69 6f 6e 61 6c 20 70 6f 6c 79 6e 6f 6d 69 61 6c 20 61 74 20 70 6f 69 6e 74 73 20 69 6e 20 74 68 | ional.polynomial.at.points.in.th |
| 91c0 | 65 20 43 61 72 74 65 73 69 61 6e 0a 20 20 20 20 20 20 20 20 70 72 6f 64 75 63 74 20 6f 66 20 60 | e.Cartesian.........product.of.` |
| 91e0 | 78 60 20 61 6e 64 20 60 79 60 2e 0a 0a 20 20 20 20 53 65 65 20 41 6c 73 6f 0a 20 20 20 20 2d 2d | x`.and.`y`.......See.Also.....-- |
| 9200 | 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 6c 65 67 76 61 6c 2c 20 6c 65 67 76 61 6c 32 64 2c 20 6c 65 67 | ------.....legval,.legval2d,.leg |
| 9220 | 67 72 69 64 32 64 2c 20 6c 65 67 76 61 6c 33 64 0a 20 20 20 20 72 86 00 00 00 72 89 00 00 00 73 | grid2d,.legval3d.....r....r....s |
| 9240 | 04 00 00 00 20 20 20 20 72 30 00 00 00 72 20 00 00 00 72 20 00 00 00 18 04 00 00 73 1c 00 00 00 | ........r0...r....r........s.... |
| 9260 | 80 00 f4 5e 01 00 0c 0e 8f 3a 89 3a 94 66 98 61 a0 11 a0 41 a0 71 d3 0b 29 d0 04 29 72 31 00 00 | ...^.....:.:.f.a...A.q..)..)r1.. |
| 9280 | 00 63 02 00 00 00 00 00 00 00 00 00 00 00 05 00 00 00 03 00 00 00 f3 c2 01 00 00 97 00 74 01 00 | .c...........................t.. |
| 92a0 | 00 00 00 00 00 00 00 6a 02 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 01 64 01 ab | .......j...................|.d.. |
| 92c0 | 02 00 00 00 00 00 00 7d 02 7c 02 64 02 6b 02 00 00 72 0b 74 05 00 00 00 00 00 00 00 00 64 03 ab | .......}.|.d.k...r.t.........d.. |
| 92e0 | 01 00 00 00 00 00 00 82 01 74 07 00 00 00 00 00 00 00 00 6a 08 00 00 00 00 00 00 00 00 00 00 00 | .........t.........j............ |
| 9300 | 00 00 00 00 00 00 00 7c 00 64 04 64 05 ac 06 ab 03 00 00 00 00 00 00 64 07 7a 00 00 00 7d 00 7c | .......|.d.d...........d.z...}.| |
| 9320 | 02 64 05 7a 00 00 00 66 01 7c 00 6a 0a 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7a | .d.z...f.|.j...................z |
| 9340 | 00 00 00 7d 03 7c 00 6a 0c 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7d 04 74 07 00 | ...}.|.j...................}.t.. |
| 9360 | 00 00 00 00 00 00 00 6a 0e 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 03 7c 04 ac | .......j...................|.|.. |
| 9380 | 08 ab 02 00 00 00 00 00 00 7d 05 7c 00 64 02 7a 05 00 00 64 05 7a 00 00 00 7c 05 64 02 3c 00 00 | .........}.|.d.z...d.z...|.d.<.. |
| 93a0 | 00 7c 02 64 02 6b 44 00 00 72 42 7c 00 7c 05 64 05 3c 00 00 00 74 11 00 00 00 00 00 00 00 00 64 | .|.d.kD..rB|.|.d.<...t.........d |
| 93c0 | 09 7c 02 64 05 7a 00 00 00 ab 02 00 00 00 00 00 00 44 00 5d 2b 00 00 7d 06 7c 05 7c 06 64 05 7a | .|.d.z...........D.]+..}.|.|.d.z |
| 93e0 | 0a 00 00 19 00 00 00 7c 00 7a 05 00 00 64 09 7c 06 7a 05 00 00 64 05 7a 0a 00 00 7a 05 00 00 7c | .......|.z...d.|.z...d.z...z...| |
| 9400 | 05 7c 06 64 09 7a 0a 00 00 19 00 00 00 7c 06 64 05 7a 0a 00 00 7a 05 00 00 7a 0a 00 00 7c 06 7a | .|.d.z.......|.d.z...z...z...|.z |
| 9420 | 0b 00 00 7c 05 7c 06 3c 00 00 00 8c 2d 04 00 74 07 00 00 00 00 00 00 00 00 6a 12 00 00 00 00 00 | ...|.|.<....-..t.........j...... |
| 9440 | 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 05 64 02 64 0a ab 03 00 00 00 00 00 00 53 00 29 0b 61 | .............|.d.d.........S.).a |
| 9460 | 1b 05 00 00 50 73 65 75 64 6f 2d 56 61 6e 64 65 72 6d 6f 6e 64 65 20 6d 61 74 72 69 78 20 6f 66 | ....Pseudo-Vandermonde.matrix.of |
| 9480 | 20 67 69 76 65 6e 20 64 65 67 72 65 65 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 20 74 68 65 20 | .given.degree.......Returns.the. |
| 94a0 | 70 73 65 75 64 6f 2d 56 61 6e 64 65 72 6d 6f 6e 64 65 20 6d 61 74 72 69 78 20 6f 66 20 64 65 67 | pseudo-Vandermonde.matrix.of.deg |
| 94c0 | 72 65 65 20 60 64 65 67 60 20 61 6e 64 20 73 61 6d 70 6c 65 20 70 6f 69 6e 74 73 0a 20 20 20 20 | ree.`deg`.and.sample.points..... |
| 94e0 | 60 78 60 2e 20 54 68 65 20 70 73 65 75 64 6f 2d 56 61 6e 64 65 72 6d 6f 6e 64 65 20 6d 61 74 72 | `x`..The.pseudo-Vandermonde.matr |
| 9500 | 69 78 20 69 73 20 64 65 66 69 6e 65 64 20 62 79 0a 0a 20 20 20 20 2e 2e 20 6d 61 74 68 3a 3a 20 | ix.is.defined.by.........math::. |
| 9520 | 56 5b 2e 2e 2e 2c 20 69 5d 20 3d 20 4c 5f 69 28 78 29 0a 0a 20 20 20 20 77 68 65 72 65 20 60 60 | V[...,.i].=.L_i(x)......where.`` |
| 9540 | 30 20 3c 3d 20 69 20 3c 3d 20 64 65 67 60 60 2e 20 54 68 65 20 6c 65 61 64 69 6e 67 20 69 6e 64 | 0.<=.i.<=.deg``..The.leading.ind |
| 9560 | 69 63 65 73 20 6f 66 20 60 56 60 20 69 6e 64 65 78 20 74 68 65 20 65 6c 65 6d 65 6e 74 73 20 6f | ices.of.`V`.index.the.elements.o |
| 9580 | 66 0a 20 20 20 20 60 78 60 20 61 6e 64 20 74 68 65 20 6c 61 73 74 20 69 6e 64 65 78 20 69 73 20 | f.....`x`.and.the.last.index.is. |
| 95a0 | 74 68 65 20 64 65 67 72 65 65 20 6f 66 20 74 68 65 20 4c 65 67 65 6e 64 72 65 20 70 6f 6c 79 6e | the.degree.of.the.Legendre.polyn |
| 95c0 | 6f 6d 69 61 6c 2e 0a 0a 20 20 20 20 49 66 20 60 63 60 20 69 73 20 61 20 31 2d 44 20 61 72 72 61 | omial.......If.`c`.is.a.1-D.arra |
| 95e0 | 79 20 6f 66 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6f 66 20 6c 65 6e 67 74 68 20 60 60 6e 20 | y.of.coefficients.of.length.``n. |
| 9600 | 2b 20 31 60 60 20 61 6e 64 20 60 56 60 20 69 73 20 74 68 65 0a 20 20 20 20 61 72 72 61 79 20 60 | +.1``.and.`V`.is.the.....array.` |
| 9620 | 60 56 20 3d 20 6c 65 67 76 61 6e 64 65 72 28 78 2c 20 6e 29 60 60 2c 20 74 68 65 6e 20 60 60 6e | `V.=.legvander(x,.n)``,.then.``n |
| 9640 | 70 2e 64 6f 74 28 56 2c 20 63 29 60 60 20 61 6e 64 0a 20 20 20 20 60 60 6c 65 67 76 61 6c 28 78 | p.dot(V,.c)``.and.....``legval(x |
| 9660 | 2c 20 63 29 60 60 20 61 72 65 20 74 68 65 20 73 61 6d 65 20 75 70 20 74 6f 20 72 6f 75 6e 64 6f | ,.c)``.are.the.same.up.to.roundo |
| 9680 | 66 66 2e 20 54 68 69 73 20 65 71 75 69 76 61 6c 65 6e 63 65 20 69 73 0a 20 20 20 20 75 73 65 66 | ff..This.equivalence.is.....usef |
| 96a0 | 75 6c 20 62 6f 74 68 20 66 6f 72 20 6c 65 61 73 74 20 73 71 75 61 72 65 73 20 66 69 74 74 69 6e | ul.both.for.least.squares.fittin |
| 96c0 | 67 20 61 6e 64 20 66 6f 72 20 74 68 65 20 65 76 61 6c 75 61 74 69 6f 6e 20 6f 66 20 61 20 6c 61 | g.and.for.the.evaluation.of.a.la |
| 96e0 | 72 67 65 0a 20 20 20 20 6e 75 6d 62 65 72 20 6f 66 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 | rge.....number.of.Legendre.serie |
| 9700 | 73 20 6f 66 20 74 68 65 20 73 61 6d 65 20 64 65 67 72 65 65 20 61 6e 64 20 73 61 6d 70 6c 65 20 | s.of.the.same.degree.and.sample. |
| 9720 | 70 6f 69 6e 74 73 2e 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 20 2d 2d 2d 2d | points.......Parameters.....---- |
| 9740 | 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 78 20 3a 20 61 72 72 61 79 5f 6c 69 6b 65 0a 20 20 20 20 20 20 | ------.....x.:.array_like....... |
| 9760 | 20 20 41 72 72 61 79 20 6f 66 20 70 6f 69 6e 74 73 2e 20 54 68 65 20 64 74 79 70 65 20 69 73 20 | ..Array.of.points..The.dtype.is. |
| 9780 | 63 6f 6e 76 65 72 74 65 64 20 74 6f 20 66 6c 6f 61 74 36 34 20 6f 72 20 63 6f 6d 70 6c 65 78 31 | converted.to.float64.or.complex1 |
| 97a0 | 32 38 0a 20 20 20 20 20 20 20 20 64 65 70 65 6e 64 69 6e 67 20 6f 6e 20 77 68 65 74 68 65 72 20 | 28.........depending.on.whether. |
| 97c0 | 61 6e 79 20 6f 66 20 74 68 65 20 65 6c 65 6d 65 6e 74 73 20 61 72 65 20 63 6f 6d 70 6c 65 78 2e | any.of.the.elements.are.complex. |
| 97e0 | 20 49 66 20 60 78 60 20 69 73 0a 20 20 20 20 20 20 20 20 73 63 61 6c 61 72 20 69 74 20 69 73 20 | .If.`x`.is.........scalar.it.is. |
| 9800 | 63 6f 6e 76 65 72 74 65 64 20 74 6f 20 61 20 31 2d 44 20 61 72 72 61 79 2e 0a 20 20 20 20 64 65 | converted.to.a.1-D.array......de |
| 9820 | 67 20 3a 20 69 6e 74 0a 20 20 20 20 20 20 20 20 44 65 67 72 65 65 20 6f 66 20 74 68 65 20 72 65 | g.:.int.........Degree.of.the.re |
| 9840 | 73 75 6c 74 69 6e 67 20 6d 61 74 72 69 78 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 0a 20 20 20 | sulting.matrix.......Returns.... |
| 9860 | 20 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 76 61 6e 64 65 72 20 3a 20 6e 64 61 72 72 61 79 0a 20 20 | .-------.....vander.:.ndarray... |
| 9880 | 20 20 20 20 20 20 54 68 65 20 70 73 65 75 64 6f 2d 56 61 6e 64 65 72 6d 6f 6e 64 65 20 6d 61 74 | ......The.pseudo-Vandermonde.mat |
| 98a0 | 72 69 78 2e 20 54 68 65 20 73 68 61 70 65 20 6f 66 20 74 68 65 20 72 65 74 75 72 6e 65 64 20 6d | rix..The.shape.of.the.returned.m |
| 98c0 | 61 74 72 69 78 20 69 73 0a 20 20 20 20 20 20 20 20 60 60 78 2e 73 68 61 70 65 20 2b 20 28 64 65 | atrix.is.........``x.shape.+.(de |
| 98e0 | 67 20 2b 20 31 2c 29 60 60 2c 20 77 68 65 72 65 20 54 68 65 20 6c 61 73 74 20 69 6e 64 65 78 20 | g.+.1,)``,.where.The.last.index. |
| 9900 | 69 73 20 74 68 65 20 64 65 67 72 65 65 20 6f 66 20 74 68 65 0a 20 20 20 20 20 20 20 20 63 6f 72 | is.the.degree.of.the.........cor |
| 9920 | 72 65 73 70 6f 6e 64 69 6e 67 20 4c 65 67 65 6e 64 72 65 20 70 6f 6c 79 6e 6f 6d 69 61 6c 2e 20 | responding.Legendre.polynomial.. |
| 9940 | 20 54 68 65 20 64 74 79 70 65 20 77 69 6c 6c 20 62 65 20 74 68 65 20 73 61 6d 65 20 61 73 0a 20 | .The.dtype.will.be.the.same.as.. |
| 9960 | 20 20 20 20 20 20 20 74 68 65 20 63 6f 6e 76 65 72 74 65 64 20 60 78 60 2e 0a 0a 20 20 20 20 72 | .......the.converted.`x`.......r |
| 9980 | 2d 00 00 00 72 02 00 00 00 7a 18 64 65 67 20 6d 75 73 74 20 62 65 20 6e 6f 6e 2d 6e 65 67 61 74 | -...r....z.deg.must.be.non-negat |
| 99a0 | 69 76 65 4e 72 04 00 00 00 29 02 72 62 00 00 00 72 61 00 00 00 e7 00 00 00 00 00 00 00 00 72 4f | iveNr....).rb...ra............rO |
| 99c0 | 00 00 00 72 38 00 00 00 72 27 00 00 00 29 0a 72 28 00 00 00 72 68 00 00 00 72 69 00 00 00 72 41 | ...r8...r'...).r(...rh...ri...rA |
| 99e0 | 00 00 00 72 42 00 00 00 72 6c 00 00 00 72 50 00 00 00 72 51 00 00 00 72 2b 00 00 00 72 6b 00 00 | ...rB...rl...rP...rQ...r+...rk.. |
| 9a00 | 00 29 07 72 7e 00 00 00 72 2d 00 00 00 da 04 69 64 65 67 da 04 64 69 6d 73 da 04 64 74 79 70 da | .).r~...r-.....ideg..dims..dtyp. |
| 9a20 | 01 76 72 2f 00 00 00 73 07 00 00 00 20 20 20 20 20 20 20 72 30 00 00 00 72 18 00 00 00 72 18 00 | .vr/...s...........r0...r....r.. |
| 9a40 | 00 00 4a 04 00 00 73 fb 00 00 00 80 00 f4 46 01 00 0c 0e 8f 3a 89 3a 90 63 98 35 d3 0b 21 80 44 | ..J...s.......F.....:.:.c.5..!.D |
| 9a60 | d8 07 0b 88 61 82 78 dc 0e 18 d0 19 33 d3 0e 34 d0 08 34 e4 08 0a 8f 08 89 08 90 11 98 14 a0 51 | ....a.x.....3..4..4............Q |
| 9a80 | d4 08 27 a8 23 d1 08 2d 80 41 d8 0c 10 90 31 89 48 88 3b 98 11 9f 17 99 17 d1 0b 20 80 44 d8 0b | ..'.#..-.A....1.H.;..........D.. |
| 9aa0 | 0c 8f 37 89 37 80 44 dc 08 0a 8f 08 89 08 90 14 98 54 d4 08 22 80 41 f0 06 00 0c 0d 88 71 89 35 | ..7.7.D..........T..".A......q.5 |
| 9ac0 | 90 31 89 39 80 41 80 61 81 44 d8 07 0b 88 61 82 78 d8 0f 10 88 01 88 21 89 04 dc 11 16 90 71 98 | .1.9.A.a.D....a.x......!......q. |
| 9ae0 | 24 a0 11 99 28 d3 11 23 f2 00 01 09 49 01 88 41 d8 14 15 90 61 98 21 91 65 91 48 98 71 91 4c a0 | $...(..#....I..A....a.!.e.H.q.L. |
| 9b00 | 41 a8 01 a1 45 a8 41 a1 49 d1 14 2e b0 11 b0 31 b0 71 b1 35 b1 18 b8 51 c0 11 b9 55 d1 31 43 d1 | A...E.A.I......1.q.5...Q...U.1C. |
| 9b20 | 14 43 c0 71 d1 13 48 88 41 88 61 8a 44 f0 03 01 09 49 01 e4 0b 0d 8f 3b 89 3b 90 71 98 21 98 52 | .C.q..H.A.a.D....I.....;.;.q.!.R |
| 9b40 | d3 0b 20 d0 04 20 72 31 00 00 00 63 03 00 00 00 00 00 00 00 00 00 00 00 05 00 00 00 03 00 00 00 | ......r1...c.................... |
| 9b60 | f3 48 00 00 00 97 00 74 01 00 00 00 00 00 00 00 00 6a 02 00 00 00 00 00 00 00 00 00 00 00 00 00 | .H.....t.........j.............. |
| 9b80 | 00 00 00 00 00 74 04 00 00 00 00 00 00 00 00 74 04 00 00 00 00 00 00 00 00 66 02 7c 00 7c 01 66 | .....t.........t.........f.|.|.f |
| 9ba0 | 02 7c 02 ab 03 00 00 00 00 00 00 53 00 29 01 61 61 06 00 00 50 73 65 75 64 6f 2d 56 61 6e 64 65 | .|.........S.).aa...Pseudo-Vande |
| 9bc0 | 72 6d 6f 6e 64 65 20 6d 61 74 72 69 78 20 6f 66 20 67 69 76 65 6e 20 64 65 67 72 65 65 73 2e 0a | rmonde.matrix.of.given.degrees.. |
| 9be0 | 0a 20 20 20 20 52 65 74 75 72 6e 73 20 74 68 65 20 70 73 65 75 64 6f 2d 56 61 6e 64 65 72 6d 6f | .....Returns.the.pseudo-Vandermo |
| 9c00 | 6e 64 65 20 6d 61 74 72 69 78 20 6f 66 20 64 65 67 72 65 65 73 20 60 64 65 67 60 20 61 6e 64 20 | nde.matrix.of.degrees.`deg`.and. |
| 9c20 | 73 61 6d 70 6c 65 0a 20 20 20 20 70 6f 69 6e 74 73 20 60 60 28 78 2c 20 79 29 60 60 2e 20 54 68 | sample.....points.``(x,.y)``..Th |
| 9c40 | 65 20 70 73 65 75 64 6f 2d 56 61 6e 64 65 72 6d 6f 6e 64 65 20 6d 61 74 72 69 78 20 69 73 20 64 | e.pseudo-Vandermonde.matrix.is.d |
| 9c60 | 65 66 69 6e 65 64 20 62 79 0a 0a 20 20 20 20 2e 2e 20 6d 61 74 68 3a 3a 20 56 5b 2e 2e 2e 2c 20 | efined.by.........math::.V[...,. |
| 9c80 | 28 64 65 67 5b 31 5d 20 2b 20 31 29 2a 69 20 2b 20 6a 5d 20 3d 20 4c 5f 69 28 78 29 20 2a 20 4c | (deg[1].+.1)*i.+.j].=.L_i(x).*.L |
| 9ca0 | 5f 6a 28 79 29 2c 0a 0a 20 20 20 20 77 68 65 72 65 20 60 60 30 20 3c 3d 20 69 20 3c 3d 20 64 65 | _j(y),......where.``0.<=.i.<=.de |
| 9cc0 | 67 5b 30 5d 60 60 20 61 6e 64 20 60 60 30 20 3c 3d 20 6a 20 3c 3d 20 64 65 67 5b 31 5d 60 60 2e | g[0]``.and.``0.<=.j.<=.deg[1]``. |
| 9ce0 | 20 54 68 65 20 6c 65 61 64 69 6e 67 20 69 6e 64 69 63 65 73 20 6f 66 0a 20 20 20 20 60 56 60 20 | .The.leading.indices.of.....`V`. |
| 9d00 | 69 6e 64 65 78 20 74 68 65 20 70 6f 69 6e 74 73 20 60 60 28 78 2c 20 79 29 60 60 20 61 6e 64 20 | index.the.points.``(x,.y)``.and. |
| 9d20 | 74 68 65 20 6c 61 73 74 20 69 6e 64 65 78 20 65 6e 63 6f 64 65 73 20 74 68 65 20 64 65 67 72 65 | the.last.index.encodes.the.degre |
| 9d40 | 65 73 20 6f 66 0a 20 20 20 20 74 68 65 20 4c 65 67 65 6e 64 72 65 20 70 6f 6c 79 6e 6f 6d 69 61 | es.of.....the.Legendre.polynomia |
| 9d60 | 6c 73 2e 0a 0a 20 20 20 20 49 66 20 60 60 56 20 3d 20 6c 65 67 76 61 6e 64 65 72 32 64 28 78 2c | ls.......If.``V.=.legvander2d(x, |
| 9d80 | 20 79 2c 20 5b 78 64 65 67 2c 20 79 64 65 67 5d 29 60 60 2c 20 74 68 65 6e 20 74 68 65 20 63 6f | .y,.[xdeg,.ydeg])``,.then.the.co |
| 9da0 | 6c 75 6d 6e 73 20 6f 66 20 60 56 60 0a 20 20 20 20 63 6f 72 72 65 73 70 6f 6e 64 20 74 6f 20 74 | lumns.of.`V`.....correspond.to.t |
| 9dc0 | 68 65 20 65 6c 65 6d 65 6e 74 73 20 6f 66 20 61 20 32 2d 44 20 63 6f 65 66 66 69 63 69 65 6e 74 | he.elements.of.a.2-D.coefficient |
| 9de0 | 20 61 72 72 61 79 20 60 63 60 20 6f 66 20 73 68 61 70 65 0a 20 20 20 20 28 78 64 65 67 20 2b 20 | .array.`c`.of.shape.....(xdeg.+. |
| 9e00 | 31 2c 20 79 64 65 67 20 2b 20 31 29 20 69 6e 20 74 68 65 20 6f 72 64 65 72 0a 0a 20 20 20 20 2e | 1,.ydeg.+.1).in.the.order....... |
| 9e20 | 2e 20 6d 61 74 68 3a 3a 20 63 5f 7b 30 30 7d 2c 20 63 5f 7b 30 31 7d 2c 20 63 5f 7b 30 32 7d 20 | ..math::.c_{00},.c_{01},.c_{02}. |
| 9e40 | 2e 2e 2e 20 2c 20 63 5f 7b 31 30 7d 2c 20 63 5f 7b 31 31 7d 2c 20 63 5f 7b 31 32 7d 20 2e 2e 2e | ....,.c_{10},.c_{11},.c_{12}.... |
| 9e60 | 0a 0a 20 20 20 20 61 6e 64 20 60 60 6e 70 2e 64 6f 74 28 56 2c 20 63 2e 66 6c 61 74 29 60 60 20 | ......and.``np.dot(V,.c.flat)``. |
| 9e80 | 61 6e 64 20 60 60 6c 65 67 76 61 6c 32 64 28 78 2c 20 79 2c 20 63 29 60 60 20 77 69 6c 6c 20 62 | and.``legval2d(x,.y,.c)``.will.b |
| 9ea0 | 65 20 74 68 65 20 73 61 6d 65 0a 20 20 20 20 75 70 20 74 6f 20 72 6f 75 6e 64 6f 66 66 2e 20 54 | e.the.same.....up.to.roundoff..T |
| 9ec0 | 68 69 73 20 65 71 75 69 76 61 6c 65 6e 63 65 20 69 73 20 75 73 65 66 75 6c 20 62 6f 74 68 20 66 | his.equivalence.is.useful.both.f |
| 9ee0 | 6f 72 20 6c 65 61 73 74 20 73 71 75 61 72 65 73 0a 20 20 20 20 66 69 74 74 69 6e 67 20 61 6e 64 | or.least.squares.....fitting.and |
| 9f00 | 20 66 6f 72 20 74 68 65 20 65 76 61 6c 75 61 74 69 6f 6e 20 6f 66 20 61 20 6c 61 72 67 65 20 6e | .for.the.evaluation.of.a.large.n |
| 9f20 | 75 6d 62 65 72 20 6f 66 20 32 2d 44 20 4c 65 67 65 6e 64 72 65 0a 20 20 20 20 73 65 72 69 65 73 | umber.of.2-D.Legendre.....series |
| 9f40 | 20 6f 66 20 74 68 65 20 73 61 6d 65 20 64 65 67 72 65 65 73 20 61 6e 64 20 73 61 6d 70 6c 65 20 | .of.the.same.degrees.and.sample. |
| 9f60 | 70 6f 69 6e 74 73 2e 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 20 2d 2d 2d 2d | points.......Parameters.....---- |
| 9f80 | 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 78 2c 20 79 20 3a 20 61 72 72 61 79 5f 6c 69 6b 65 0a 20 20 20 | ------.....x,.y.:.array_like.... |
| 9fa0 | 20 20 20 20 20 41 72 72 61 79 73 20 6f 66 20 70 6f 69 6e 74 20 63 6f 6f 72 64 69 6e 61 74 65 73 | .....Arrays.of.point.coordinates |
| 9fc0 | 2c 20 61 6c 6c 20 6f 66 20 74 68 65 20 73 61 6d 65 20 73 68 61 70 65 2e 20 54 68 65 20 64 74 79 | ,.all.of.the.same.shape..The.dty |
| 9fe0 | 70 65 73 0a 20 20 20 20 20 20 20 20 77 69 6c 6c 20 62 65 20 63 6f 6e 76 65 72 74 65 64 20 74 6f | pes.........will.be.converted.to |
| a000 | 20 65 69 74 68 65 72 20 66 6c 6f 61 74 36 34 20 6f 72 20 63 6f 6d 70 6c 65 78 31 32 38 20 64 65 | .either.float64.or.complex128.de |
| a020 | 70 65 6e 64 69 6e 67 20 6f 6e 0a 20 20 20 20 20 20 20 20 77 68 65 74 68 65 72 20 61 6e 79 20 6f | pending.on.........whether.any.o |
| a040 | 66 20 74 68 65 20 65 6c 65 6d 65 6e 74 73 20 61 72 65 20 63 6f 6d 70 6c 65 78 2e 20 53 63 61 6c | f.the.elements.are.complex..Scal |
| a060 | 61 72 73 20 61 72 65 20 63 6f 6e 76 65 72 74 65 64 20 74 6f 0a 20 20 20 20 20 20 20 20 31 2d 44 | ars.are.converted.to.........1-D |
| a080 | 20 61 72 72 61 79 73 2e 0a 20 20 20 20 64 65 67 20 3a 20 6c 69 73 74 20 6f 66 20 69 6e 74 73 0a | .arrays......deg.:.list.of.ints. |
| a0a0 | 20 20 20 20 20 20 20 20 4c 69 73 74 20 6f 66 20 6d 61 78 69 6d 75 6d 20 64 65 67 72 65 65 73 20 | ........List.of.maximum.degrees. |
| a0c0 | 6f 66 20 74 68 65 20 66 6f 72 6d 20 5b 78 5f 64 65 67 2c 20 79 5f 64 65 67 5d 2e 0a 0a 20 20 20 | of.the.form.[x_deg,.y_deg]...... |
| a0e0 | 20 52 65 74 75 72 6e 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 76 61 6e 64 65 72 32 | .Returns.....-------.....vander2 |
| a100 | 64 20 3a 20 6e 64 61 72 72 61 79 0a 20 20 20 20 20 20 20 20 54 68 65 20 73 68 61 70 65 20 6f 66 | d.:.ndarray.........The.shape.of |
| a120 | 20 74 68 65 20 72 65 74 75 72 6e 65 64 20 6d 61 74 72 69 78 20 69 73 20 60 60 78 2e 73 68 61 70 | .the.returned.matrix.is.``x.shap |
| a140 | 65 20 2b 20 28 6f 72 64 65 72 2c 29 60 60 2c 20 77 68 65 72 65 0a 20 20 20 20 20 20 20 20 3a 6d | e.+.(order,)``,.where.........:m |
| a160 | 61 74 68 3a 60 6f 72 64 65 72 20 3d 20 28 64 65 67 5b 30 5d 2b 31 29 2a 28 64 65 67 5b 31 5d 2b | ath:`order.=.(deg[0]+1)*(deg[1]+ |
| a180 | 31 29 60 2e 20 20 54 68 65 20 64 74 79 70 65 20 77 69 6c 6c 20 62 65 20 74 68 65 20 73 61 6d 65 | 1)`...The.dtype.will.be.the.same |
| a1a0 | 0a 20 20 20 20 20 20 20 20 61 73 20 74 68 65 20 63 6f 6e 76 65 72 74 65 64 20 60 78 60 20 61 6e | .........as.the.converted.`x`.an |
| a1c0 | 64 20 60 79 60 2e 0a 0a 20 20 20 20 53 65 65 20 41 6c 73 6f 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d | d.`y`.......See.Also.....------- |
| a1e0 | 2d 0a 20 20 20 20 6c 65 67 76 61 6e 64 65 72 2c 20 6c 65 67 76 61 6e 64 65 72 33 64 2c 20 6c 65 | -.....legvander,.legvander3d,.le |
| a200 | 67 76 61 6c 32 64 2c 20 6c 65 67 76 61 6c 33 64 0a 20 20 20 20 a9 03 72 28 00 00 00 da 0f 5f 76 | gval2d,.legval3d.......r(....._v |
| a220 | 61 6e 64 65 72 5f 6e 64 5f 66 6c 61 74 72 18 00 00 00 29 03 72 7e 00 00 00 72 84 00 00 00 72 2d | ander_nd_flatr....).r~...r....r- |
| a240 | 00 00 00 73 03 00 00 00 20 20 20 72 30 00 00 00 72 21 00 00 00 72 21 00 00 00 7f 04 00 00 73 23 | ...s.......r0...r!...r!.......s# |
| a260 | 00 00 00 80 00 f4 58 01 00 0c 0e d7 0b 1d d1 0b 1d 9c 79 ac 29 d0 1e 34 b0 71 b8 21 b0 66 b8 63 | ......X...........y.)..4.q.!.f.c |
| a280 | d3 0b 42 d0 04 42 72 31 00 00 00 63 04 00 00 00 00 00 00 00 00 00 00 00 06 00 00 00 03 00 00 00 | ..B..Br1...c.................... |
| a2a0 | f3 54 00 00 00 97 00 74 01 00 00 00 00 00 00 00 00 6a 02 00 00 00 00 00 00 00 00 00 00 00 00 00 | .T.....t.........j.............. |
| a2c0 | 00 00 00 00 00 74 04 00 00 00 00 00 00 00 00 74 04 00 00 00 00 00 00 00 00 74 04 00 00 00 00 00 | .....t.........t.........t...... |
| a2e0 | 00 00 00 66 03 7c 00 7c 01 7c 02 66 03 7c 03 ab 03 00 00 00 00 00 00 53 00 29 01 61 f4 06 00 00 | ...f.|.|.|.f.|.........S.).a.... |
| a300 | 50 73 65 75 64 6f 2d 56 61 6e 64 65 72 6d 6f 6e 64 65 20 6d 61 74 72 69 78 20 6f 66 20 67 69 76 | Pseudo-Vandermonde.matrix.of.giv |
| a320 | 65 6e 20 64 65 67 72 65 65 73 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 20 74 68 65 20 70 73 65 | en.degrees.......Returns.the.pse |
| a340 | 75 64 6f 2d 56 61 6e 64 65 72 6d 6f 6e 64 65 20 6d 61 74 72 69 78 20 6f 66 20 64 65 67 72 65 65 | udo-Vandermonde.matrix.of.degree |
| a360 | 73 20 60 64 65 67 60 20 61 6e 64 20 73 61 6d 70 6c 65 0a 20 20 20 20 70 6f 69 6e 74 73 20 60 60 | s.`deg`.and.sample.....points.`` |
| a380 | 28 78 2c 20 79 2c 20 7a 29 60 60 2e 20 49 66 20 60 6c 60 2c 20 60 6d 60 2c 20 60 6e 60 20 61 72 | (x,.y,.z)``..If.`l`,.`m`,.`n`.ar |
| a3a0 | 65 20 74 68 65 20 67 69 76 65 6e 20 64 65 67 72 65 65 73 20 69 6e 20 60 78 60 2c 20 60 79 60 2c | e.the.given.degrees.in.`x`,.`y`, |
| a3c0 | 20 60 7a 60 2c 0a 20 20 20 20 74 68 65 6e 20 54 68 65 20 70 73 65 75 64 6f 2d 56 61 6e 64 65 72 | .`z`,.....then.The.pseudo-Vander |
| a3e0 | 6d 6f 6e 64 65 20 6d 61 74 72 69 78 20 69 73 20 64 65 66 69 6e 65 64 20 62 79 0a 0a 20 20 20 20 | monde.matrix.is.defined.by...... |
| a400 | 2e 2e 20 6d 61 74 68 3a 3a 20 56 5b 2e 2e 2e 2c 20 28 6d 2b 31 29 28 6e 2b 31 29 69 20 2b 20 28 | ...math::.V[...,.(m+1)(n+1)i.+.( |
| a420 | 6e 2b 31 29 6a 20 2b 20 6b 5d 20 3d 20 4c 5f 69 28 78 29 2a 4c 5f 6a 28 79 29 2a 4c 5f 6b 28 7a | n+1)j.+.k].=.L_i(x)*L_j(y)*L_k(z |
| a440 | 29 2c 0a 0a 20 20 20 20 77 68 65 72 65 20 60 60 30 20 3c 3d 20 69 20 3c 3d 20 6c 60 60 2c 20 60 | ),......where.``0.<=.i.<=.l``,.` |
| a460 | 60 30 20 3c 3d 20 6a 20 3c 3d 20 6d 60 60 2c 20 61 6e 64 20 60 60 30 20 3c 3d 20 6a 20 3c 3d 20 | `0.<=.j.<=.m``,.and.``0.<=.j.<=. |
| a480 | 6e 60 60 2e 20 20 54 68 65 20 6c 65 61 64 69 6e 67 0a 20 20 20 20 69 6e 64 69 63 65 73 20 6f 66 | n``...The.leading.....indices.of |
| a4a0 | 20 60 56 60 20 69 6e 64 65 78 20 74 68 65 20 70 6f 69 6e 74 73 20 60 60 28 78 2c 20 79 2c 20 7a | .`V`.index.the.points.``(x,.y,.z |
| a4c0 | 29 60 60 20 61 6e 64 20 74 68 65 20 6c 61 73 74 20 69 6e 64 65 78 20 65 6e 63 6f 64 65 73 0a 20 | )``.and.the.last.index.encodes.. |
| a4e0 | 20 20 20 74 68 65 20 64 65 67 72 65 65 73 20 6f 66 20 74 68 65 20 4c 65 67 65 6e 64 72 65 20 70 | ...the.degrees.of.the.Legendre.p |
| a500 | 6f 6c 79 6e 6f 6d 69 61 6c 73 2e 0a 0a 20 20 20 20 49 66 20 60 60 56 20 3d 20 6c 65 67 76 61 6e | olynomials.......If.``V.=.legvan |
| a520 | 64 65 72 33 64 28 78 2c 20 79 2c 20 7a 2c 20 5b 78 64 65 67 2c 20 79 64 65 67 2c 20 7a 64 65 67 | der3d(x,.y,.z,.[xdeg,.ydeg,.zdeg |
| a540 | 5d 29 60 60 2c 20 74 68 65 6e 20 74 68 65 20 63 6f 6c 75 6d 6e 73 0a 20 20 20 20 6f 66 20 60 56 | ])``,.then.the.columns.....of.`V |
| a560 | 60 20 63 6f 72 72 65 73 70 6f 6e 64 20 74 6f 20 74 68 65 20 65 6c 65 6d 65 6e 74 73 20 6f 66 20 | `.correspond.to.the.elements.of. |
| a580 | 61 20 33 2d 44 20 63 6f 65 66 66 69 63 69 65 6e 74 20 61 72 72 61 79 20 60 63 60 20 6f 66 0a 20 | a.3-D.coefficient.array.`c`.of.. |
| a5a0 | 20 20 20 73 68 61 70 65 20 28 78 64 65 67 20 2b 20 31 2c 20 79 64 65 67 20 2b 20 31 2c 20 7a 64 | ...shape.(xdeg.+.1,.ydeg.+.1,.zd |
| a5c0 | 65 67 20 2b 20 31 29 20 69 6e 20 74 68 65 20 6f 72 64 65 72 0a 0a 20 20 20 20 2e 2e 20 6d 61 74 | eg.+.1).in.the.order.........mat |
| a5e0 | 68 3a 3a 20 63 5f 7b 30 30 30 7d 2c 20 63 5f 7b 30 30 31 7d 2c 20 63 5f 7b 30 30 32 7d 2c 2e 2e | h::.c_{000},.c_{001},.c_{002},.. |
| a600 | 2e 20 2c 20 63 5f 7b 30 31 30 7d 2c 20 63 5f 7b 30 31 31 7d 2c 20 63 5f 7b 30 31 32 7d 2c 2e 2e | ..,.c_{010},.c_{011},.c_{012},.. |
| a620 | 2e 0a 0a 20 20 20 20 61 6e 64 20 60 60 6e 70 2e 64 6f 74 28 56 2c 20 63 2e 66 6c 61 74 29 60 60 | .......and.``np.dot(V,.c.flat)`` |
| a640 | 20 61 6e 64 20 60 60 6c 65 67 76 61 6c 33 64 28 78 2c 20 79 2c 20 7a 2c 20 63 29 60 60 20 77 69 | .and.``legval3d(x,.y,.z,.c)``.wi |
| a660 | 6c 6c 20 62 65 20 74 68 65 0a 20 20 20 20 73 61 6d 65 20 75 70 20 74 6f 20 72 6f 75 6e 64 6f 66 | ll.be.the.....same.up.to.roundof |
| a680 | 66 2e 20 54 68 69 73 20 65 71 75 69 76 61 6c 65 6e 63 65 20 69 73 20 75 73 65 66 75 6c 20 62 6f | f..This.equivalence.is.useful.bo |
| a6a0 | 74 68 20 66 6f 72 20 6c 65 61 73 74 20 73 71 75 61 72 65 73 0a 20 20 20 20 66 69 74 74 69 6e 67 | th.for.least.squares.....fitting |
| a6c0 | 20 61 6e 64 20 66 6f 72 20 74 68 65 20 65 76 61 6c 75 61 74 69 6f 6e 20 6f 66 20 61 20 6c 61 72 | .and.for.the.evaluation.of.a.lar |
| a6e0 | 67 65 20 6e 75 6d 62 65 72 20 6f 66 20 33 2d 44 20 4c 65 67 65 6e 64 72 65 0a 20 20 20 20 73 65 | ge.number.of.3-D.Legendre.....se |
| a700 | 72 69 65 73 20 6f 66 20 74 68 65 20 73 61 6d 65 20 64 65 67 72 65 65 73 20 61 6e 64 20 73 61 6d | ries.of.the.same.degrees.and.sam |
| a720 | 70 6c 65 20 70 6f 69 6e 74 73 2e 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 20 | ple.points.......Parameters..... |
| a740 | 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 78 2c 20 79 2c 20 7a 20 3a 20 61 72 72 61 79 5f 6c | ----------.....x,.y,.z.:.array_l |
| a760 | 69 6b 65 0a 20 20 20 20 20 20 20 20 41 72 72 61 79 73 20 6f 66 20 70 6f 69 6e 74 20 63 6f 6f 72 | ike.........Arrays.of.point.coor |
| a780 | 64 69 6e 61 74 65 73 2c 20 61 6c 6c 20 6f 66 20 74 68 65 20 73 61 6d 65 20 73 68 61 70 65 2e 20 | dinates,.all.of.the.same.shape.. |
| a7a0 | 54 68 65 20 64 74 79 70 65 73 20 77 69 6c 6c 0a 20 20 20 20 20 20 20 20 62 65 20 63 6f 6e 76 65 | The.dtypes.will.........be.conve |
| a7c0 | 72 74 65 64 20 74 6f 20 65 69 74 68 65 72 20 66 6c 6f 61 74 36 34 20 6f 72 20 63 6f 6d 70 6c 65 | rted.to.either.float64.or.comple |
| a7e0 | 78 31 32 38 20 64 65 70 65 6e 64 69 6e 67 20 6f 6e 20 77 68 65 74 68 65 72 0a 20 20 20 20 20 20 | x128.depending.on.whether....... |
| a800 | 20 20 61 6e 79 20 6f 66 20 74 68 65 20 65 6c 65 6d 65 6e 74 73 20 61 72 65 20 63 6f 6d 70 6c 65 | ..any.of.the.elements.are.comple |
| a820 | 78 2e 20 53 63 61 6c 61 72 73 20 61 72 65 20 63 6f 6e 76 65 72 74 65 64 20 74 6f 20 31 2d 44 0a | x..Scalars.are.converted.to.1-D. |
| a840 | 20 20 20 20 20 20 20 20 61 72 72 61 79 73 2e 0a 20 20 20 20 64 65 67 20 3a 20 6c 69 73 74 20 6f | ........arrays......deg.:.list.o |
| a860 | 66 20 69 6e 74 73 0a 20 20 20 20 20 20 20 20 4c 69 73 74 20 6f 66 20 6d 61 78 69 6d 75 6d 20 64 | f.ints.........List.of.maximum.d |
| a880 | 65 67 72 65 65 73 20 6f 66 20 74 68 65 20 66 6f 72 6d 20 5b 78 5f 64 65 67 2c 20 79 5f 64 65 67 | egrees.of.the.form.[x_deg,.y_deg |
| a8a0 | 2c 20 7a 5f 64 65 67 5d 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 0a 20 20 20 20 2d 2d 2d 2d 2d | ,.z_deg].......Returns.....----- |
| a8c0 | 2d 2d 0a 20 20 20 20 76 61 6e 64 65 72 33 64 20 3a 20 6e 64 61 72 72 61 79 0a 20 20 20 20 20 20 | --.....vander3d.:.ndarray....... |
| a8e0 | 20 20 54 68 65 20 73 68 61 70 65 20 6f 66 20 74 68 65 20 72 65 74 75 72 6e 65 64 20 6d 61 74 72 | ..The.shape.of.the.returned.matr |
| a900 | 69 78 20 69 73 20 60 60 78 2e 73 68 61 70 65 20 2b 20 28 6f 72 64 65 72 2c 29 60 60 2c 20 77 68 | ix.is.``x.shape.+.(order,)``,.wh |
| a920 | 65 72 65 0a 20 20 20 20 20 20 20 20 3a 6d 61 74 68 3a 60 6f 72 64 65 72 20 3d 20 28 64 65 67 5b | ere.........:math:`order.=.(deg[ |
| a940 | 30 5d 2b 31 29 2a 28 64 65 67 5b 31 5d 2b 31 29 2a 28 64 65 67 5b 32 5d 2b 31 29 60 2e 20 20 54 | 0]+1)*(deg[1]+1)*(deg[2]+1)`...T |
| a960 | 68 65 20 64 74 79 70 65 20 77 69 6c 6c 0a 20 20 20 20 20 20 20 20 62 65 20 74 68 65 20 73 61 6d | he.dtype.will.........be.the.sam |
| a980 | 65 20 61 73 20 74 68 65 20 63 6f 6e 76 65 72 74 65 64 20 60 78 60 2c 20 60 79 60 2c 20 61 6e 64 | e.as.the.converted.`x`,.`y`,.and |
| a9a0 | 20 60 7a 60 2e 0a 0a 20 20 20 20 53 65 65 20 41 6c 73 6f 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d | .`z`.......See.Also.....-------- |
| a9c0 | 0a 20 20 20 20 6c 65 67 76 61 6e 64 65 72 2c 20 6c 65 67 76 61 6e 64 65 72 33 64 2c 20 6c 65 67 | .....legvander,.legvander3d,.leg |
| a9e0 | 76 61 6c 32 64 2c 20 6c 65 67 76 61 6c 33 64 0a 20 20 20 20 72 93 00 00 00 29 04 72 7e 00 00 00 | val2d,.legval3d.....r....).r~... |
| aa00 | 72 84 00 00 00 72 8a 00 00 00 72 2d 00 00 00 73 04 00 00 00 20 20 20 20 72 30 00 00 00 72 22 00 | r....r....r-...s........r0...r". |
| aa20 | 00 00 72 22 00 00 00 ae 04 00 00 73 27 00 00 00 80 00 f4 5a 01 00 0c 0e d7 0b 1d d1 0b 1d 9c 79 | ..r".......s'......Z...........y |
| aa40 | ac 29 b4 59 d0 1e 3f c0 21 c0 51 c8 01 c0 19 c8 43 d3 0b 50 d0 04 50 72 31 00 00 00 63 06 00 00 | .).Y..?.!.Q.....C..P..Pr1...c... |
| aa60 | 00 00 00 00 00 00 00 00 00 09 00 00 00 03 00 00 00 f3 40 00 00 00 97 00 74 01 00 00 00 00 00 00 | ..................@.....t....... |
| aa80 | 00 00 6a 02 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 74 04 00 00 00 00 00 00 00 00 | ..j...................t......... |
| aaa0 | 7c 00 7c 01 7c 02 7c 03 7c 04 7c 05 ab 07 00 00 00 00 00 00 53 00 29 01 61 a7 14 00 00 0a 20 20 | |.|.|.|.|.|.........S.).a....... |
| aac0 | 20 20 4c 65 61 73 74 20 73 71 75 61 72 65 73 20 66 69 74 20 6f 66 20 4c 65 67 65 6e 64 72 65 20 | ..Least.squares.fit.of.Legendre. |
| aae0 | 73 65 72 69 65 73 20 74 6f 20 64 61 74 61 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 20 74 68 65 20 | series.to.data.......Return.the. |
| ab00 | 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6f 66 20 61 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 | coefficients.of.a.Legendre.serie |
| ab20 | 73 20 6f 66 20 64 65 67 72 65 65 20 60 64 65 67 60 20 74 68 61 74 20 69 73 20 74 68 65 0a 20 20 | s.of.degree.`deg`.that.is.the... |
| ab40 | 20 20 6c 65 61 73 74 20 73 71 75 61 72 65 73 20 66 69 74 20 74 6f 20 74 68 65 20 64 61 74 61 20 | ..least.squares.fit.to.the.data. |
| ab60 | 76 61 6c 75 65 73 20 60 79 60 20 67 69 76 65 6e 20 61 74 20 70 6f 69 6e 74 73 20 60 78 60 2e 20 | values.`y`.given.at.points.`x`.. |
| ab80 | 49 66 20 60 79 60 20 69 73 0a 20 20 20 20 31 2d 44 20 74 68 65 20 72 65 74 75 72 6e 65 64 20 63 | If.`y`.is.....1-D.the.returned.c |
| aba0 | 6f 65 66 66 69 63 69 65 6e 74 73 20 77 69 6c 6c 20 61 6c 73 6f 20 62 65 20 31 2d 44 2e 20 49 66 | oefficients.will.also.be.1-D..If |
| abc0 | 20 60 79 60 20 69 73 20 32 2d 44 20 6d 75 6c 74 69 70 6c 65 0a 20 20 20 20 66 69 74 73 20 61 72 | .`y`.is.2-D.multiple.....fits.ar |
| abe0 | 65 20 64 6f 6e 65 2c 20 6f 6e 65 20 66 6f 72 20 65 61 63 68 20 63 6f 6c 75 6d 6e 20 6f 66 20 60 | e.done,.one.for.each.column.of.` |
| ac00 | 79 60 2c 20 61 6e 64 20 74 68 65 20 72 65 73 75 6c 74 69 6e 67 0a 20 20 20 20 63 6f 65 66 66 69 | y`,.and.the.resulting.....coeffi |
| ac20 | 63 69 65 6e 74 73 20 61 72 65 20 73 74 6f 72 65 64 20 69 6e 20 74 68 65 20 63 6f 72 72 65 73 70 | cients.are.stored.in.the.corresp |
| ac40 | 6f 6e 64 69 6e 67 20 63 6f 6c 75 6d 6e 73 20 6f 66 20 61 20 32 2d 44 20 72 65 74 75 72 6e 2e 0a | onding.columns.of.a.2-D.return.. |
| ac60 | 20 20 20 20 54 68 65 20 66 69 74 74 65 64 20 70 6f 6c 79 6e 6f 6d 69 61 6c 28 73 29 20 61 72 65 | ....The.fitted.polynomial(s).are |
| ac80 | 20 69 6e 20 74 68 65 20 66 6f 72 6d 0a 0a 20 20 20 20 2e 2e 20 6d 61 74 68 3a 3a 20 20 70 28 78 | .in.the.form.........math::..p(x |
| aca0 | 29 20 3d 20 63 5f 30 20 2b 20 63 5f 31 20 2a 20 4c 5f 31 28 78 29 20 2b 20 2e 2e 2e 20 2b 20 63 | ).=.c_0.+.c_1.*.L_1(x).+.....+.c |
| acc0 | 5f 6e 20 2a 20 4c 5f 6e 28 78 29 2c 0a 0a 20 20 20 20 77 68 65 72 65 20 60 6e 60 20 69 73 20 60 | _n.*.L_n(x),......where.`n`.is.` |
| ace0 | 64 65 67 60 2e 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d | deg`.......Parameters.....------ |
| ad00 | 2d 2d 2d 2d 0a 20 20 20 20 78 20 3a 20 61 72 72 61 79 5f 6c 69 6b 65 2c 20 73 68 61 70 65 20 28 | ----.....x.:.array_like,.shape.( |
| ad20 | 4d 2c 29 0a 20 20 20 20 20 20 20 20 78 2d 63 6f 6f 72 64 69 6e 61 74 65 73 20 6f 66 20 74 68 65 | M,).........x-coordinates.of.the |
| ad40 | 20 4d 20 73 61 6d 70 6c 65 20 70 6f 69 6e 74 73 20 60 60 28 78 5b 69 5d 2c 20 79 5b 69 5d 29 60 | .M.sample.points.``(x[i],.y[i])` |
| ad60 | 60 2e 0a 20 20 20 20 79 20 3a 20 61 72 72 61 79 5f 6c 69 6b 65 2c 20 73 68 61 70 65 20 28 4d 2c | `......y.:.array_like,.shape.(M, |
| ad80 | 29 20 6f 72 20 28 4d 2c 20 4b 29 0a 20 20 20 20 20 20 20 20 79 2d 63 6f 6f 72 64 69 6e 61 74 65 | ).or.(M,.K).........y-coordinate |
| ada0 | 73 20 6f 66 20 74 68 65 20 73 61 6d 70 6c 65 20 70 6f 69 6e 74 73 2e 20 53 65 76 65 72 61 6c 20 | s.of.the.sample.points..Several. |
| adc0 | 64 61 74 61 20 73 65 74 73 20 6f 66 20 73 61 6d 70 6c 65 0a 20 20 20 20 20 20 20 20 70 6f 69 6e | data.sets.of.sample.........poin |
| ade0 | 74 73 20 73 68 61 72 69 6e 67 20 74 68 65 20 73 61 6d 65 20 78 2d 63 6f 6f 72 64 69 6e 61 74 65 | ts.sharing.the.same.x-coordinate |
| ae00 | 73 20 63 61 6e 20 62 65 20 66 69 74 74 65 64 20 61 74 20 6f 6e 63 65 20 62 79 0a 20 20 20 20 20 | s.can.be.fitted.at.once.by...... |
| ae20 | 20 20 20 70 61 73 73 69 6e 67 20 69 6e 20 61 20 32 44 2d 61 72 72 61 79 20 74 68 61 74 20 63 6f | ...passing.in.a.2D-array.that.co |
| ae40 | 6e 74 61 69 6e 73 20 6f 6e 65 20 64 61 74 61 73 65 74 20 70 65 72 20 63 6f 6c 75 6d 6e 2e 0a 20 | ntains.one.dataset.per.column... |
| ae60 | 20 20 20 64 65 67 20 3a 20 69 6e 74 20 6f 72 20 31 2d 44 20 61 72 72 61 79 5f 6c 69 6b 65 0a 20 | ...deg.:.int.or.1-D.array_like.. |
| ae80 | 20 20 20 20 20 20 20 44 65 67 72 65 65 28 73 29 20 6f 66 20 74 68 65 20 66 69 74 74 69 6e 67 20 | .......Degree(s).of.the.fitting. |
| aea0 | 70 6f 6c 79 6e 6f 6d 69 61 6c 73 2e 20 49 66 20 60 64 65 67 60 20 69 73 20 61 20 73 69 6e 67 6c | polynomials..If.`deg`.is.a.singl |
| aec0 | 65 20 69 6e 74 65 67 65 72 0a 20 20 20 20 20 20 20 20 61 6c 6c 20 74 65 72 6d 73 20 75 70 20 74 | e.integer.........all.terms.up.t |
| aee0 | 6f 20 61 6e 64 20 69 6e 63 6c 75 64 69 6e 67 20 74 68 65 20 60 64 65 67 60 27 74 68 20 74 65 72 | o.and.including.the.`deg`'th.ter |
| af00 | 6d 20 61 72 65 20 69 6e 63 6c 75 64 65 64 20 69 6e 20 74 68 65 0a 20 20 20 20 20 20 20 20 66 69 | m.are.included.in.the.........fi |
| af20 | 74 2e 20 46 6f 72 20 4e 75 6d 50 79 20 76 65 72 73 69 6f 6e 73 20 3e 3d 20 31 2e 31 31 2e 30 20 | t..For.NumPy.versions.>=.1.11.0. |
| af40 | 61 20 6c 69 73 74 20 6f 66 20 69 6e 74 65 67 65 72 73 20 73 70 65 63 69 66 79 69 6e 67 20 74 68 | a.list.of.integers.specifying.th |
| af60 | 65 0a 20 20 20 20 20 20 20 20 64 65 67 72 65 65 73 20 6f 66 20 74 68 65 20 74 65 72 6d 73 20 74 | e.........degrees.of.the.terms.t |
| af80 | 6f 20 69 6e 63 6c 75 64 65 20 6d 61 79 20 62 65 20 75 73 65 64 20 69 6e 73 74 65 61 64 2e 0a 20 | o.include.may.be.used.instead... |
| afa0 | 20 20 20 72 63 6f 6e 64 20 3a 20 66 6c 6f 61 74 2c 20 6f 70 74 69 6f 6e 61 6c 0a 20 20 20 20 20 | ...rcond.:.float,.optional...... |
| afc0 | 20 20 20 52 65 6c 61 74 69 76 65 20 63 6f 6e 64 69 74 69 6f 6e 20 6e 75 6d 62 65 72 20 6f 66 20 | ...Relative.condition.number.of. |
| afe0 | 74 68 65 20 66 69 74 2e 20 53 69 6e 67 75 6c 61 72 20 76 61 6c 75 65 73 20 73 6d 61 6c 6c 65 72 | the.fit..Singular.values.smaller |
| b000 | 20 74 68 61 6e 0a 20 20 20 20 20 20 20 20 74 68 69 73 20 72 65 6c 61 74 69 76 65 20 74 6f 20 74 | .than.........this.relative.to.t |
| b020 | 68 65 20 6c 61 72 67 65 73 74 20 73 69 6e 67 75 6c 61 72 20 76 61 6c 75 65 20 77 69 6c 6c 20 62 | he.largest.singular.value.will.b |
| b040 | 65 20 69 67 6e 6f 72 65 64 2e 20 54 68 65 0a 20 20 20 20 20 20 20 20 64 65 66 61 75 6c 74 20 76 | e.ignored..The.........default.v |
| b060 | 61 6c 75 65 20 69 73 20 6c 65 6e 28 78 29 2a 65 70 73 2c 20 77 68 65 72 65 20 65 70 73 20 69 73 | alue.is.len(x)*eps,.where.eps.is |
| b080 | 20 74 68 65 20 72 65 6c 61 74 69 76 65 20 70 72 65 63 69 73 69 6f 6e 20 6f 66 0a 20 20 20 20 20 | .the.relative.precision.of...... |
| b0a0 | 20 20 20 74 68 65 20 66 6c 6f 61 74 20 74 79 70 65 2c 20 61 62 6f 75 74 20 32 65 2d 31 36 20 69 | ...the.float.type,.about.2e-16.i |
| b0c0 | 6e 20 6d 6f 73 74 20 63 61 73 65 73 2e 0a 20 20 20 20 66 75 6c 6c 20 3a 20 62 6f 6f 6c 2c 20 6f | n.most.cases......full.:.bool,.o |
| b0e0 | 70 74 69 6f 6e 61 6c 0a 20 20 20 20 20 20 20 20 53 77 69 74 63 68 20 64 65 74 65 72 6d 69 6e 69 | ptional.........Switch.determini |
| b100 | 6e 67 20 6e 61 74 75 72 65 20 6f 66 20 72 65 74 75 72 6e 20 76 61 6c 75 65 2e 20 57 68 65 6e 20 | ng.nature.of.return.value..When. |
| b120 | 69 74 20 69 73 20 46 61 6c 73 65 20 28 74 68 65 0a 20 20 20 20 20 20 20 20 64 65 66 61 75 6c 74 | it.is.False.(the.........default |
| b140 | 29 20 6a 75 73 74 20 74 68 65 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 61 72 65 20 72 65 74 75 | ).just.the.coefficients.are.retu |
| b160 | 72 6e 65 64 2c 20 77 68 65 6e 20 54 72 75 65 20 64 69 61 67 6e 6f 73 74 69 63 0a 20 20 20 20 20 | rned,.when.True.diagnostic...... |
| b180 | 20 20 20 69 6e 66 6f 72 6d 61 74 69 6f 6e 20 66 72 6f 6d 20 74 68 65 20 73 69 6e 67 75 6c 61 72 | ...information.from.the.singular |
| b1a0 | 20 76 61 6c 75 65 20 64 65 63 6f 6d 70 6f 73 69 74 69 6f 6e 20 69 73 20 61 6c 73 6f 20 72 65 74 | .value.decomposition.is.also.ret |
| b1c0 | 75 72 6e 65 64 2e 0a 20 20 20 20 77 20 3a 20 61 72 72 61 79 5f 6c 69 6b 65 2c 20 73 68 61 70 65 | urned......w.:.array_like,.shape |
| b1e0 | 20 28 60 4d 60 2c 29 2c 20 6f 70 74 69 6f 6e 61 6c 0a 20 20 20 20 20 20 20 20 57 65 69 67 68 74 | .(`M`,),.optional.........Weight |
| b200 | 73 2e 20 49 66 20 6e 6f 74 20 4e 6f 6e 65 2c 20 74 68 65 20 77 65 69 67 68 74 20 60 60 77 5b 69 | s..If.not.None,.the.weight.``w[i |
| b220 | 5d 60 60 20 61 70 70 6c 69 65 73 20 74 6f 20 74 68 65 20 75 6e 73 71 75 61 72 65 64 0a 20 20 20 | ]``.applies.to.the.unsquared.... |
| b240 | 20 20 20 20 20 72 65 73 69 64 75 61 6c 20 60 60 79 5b 69 5d 20 2d 20 79 5f 68 61 74 5b 69 5d 60 | .....residual.``y[i].-.y_hat[i]` |
| b260 | 60 20 61 74 20 60 60 78 5b 69 5d 60 60 2e 20 49 64 65 61 6c 6c 79 20 74 68 65 20 77 65 69 67 68 | `.at.``x[i]``..Ideally.the.weigh |
| b280 | 74 73 20 61 72 65 0a 20 20 20 20 20 20 20 20 63 68 6f 73 65 6e 20 73 6f 20 74 68 61 74 20 74 68 | ts.are.........chosen.so.that.th |
| b2a0 | 65 20 65 72 72 6f 72 73 20 6f 66 20 74 68 65 20 70 72 6f 64 75 63 74 73 20 60 60 77 5b 69 5d 2a | e.errors.of.the.products.``w[i]* |
| b2c0 | 79 5b 69 5d 60 60 20 61 6c 6c 20 68 61 76 65 20 74 68 65 0a 20 20 20 20 20 20 20 20 73 61 6d 65 | y[i]``.all.have.the.........same |
| b2e0 | 20 76 61 72 69 61 6e 63 65 2e 20 20 57 68 65 6e 20 75 73 69 6e 67 20 69 6e 76 65 72 73 65 2d 76 | .variance...When.using.inverse-v |
| b300 | 61 72 69 61 6e 63 65 20 77 65 69 67 68 74 69 6e 67 2c 20 75 73 65 0a 20 20 20 20 20 20 20 20 60 | ariance.weighting,.use.........` |
| b320 | 60 77 5b 69 5d 20 3d 20 31 2f 73 69 67 6d 61 28 79 5b 69 5d 29 60 60 2e 20 20 54 68 65 20 64 65 | `w[i].=.1/sigma(y[i])``...The.de |
| b340 | 66 61 75 6c 74 20 76 61 6c 75 65 20 69 73 20 4e 6f 6e 65 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e | fault.value.is.None.......Return |
| b360 | 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 63 6f 65 66 20 3a 20 6e 64 61 72 72 61 79 | s.....-------.....coef.:.ndarray |
| b380 | 2c 20 73 68 61 70 65 20 28 4d 2c 29 20 6f 72 20 28 4d 2c 20 4b 29 0a 20 20 20 20 20 20 20 20 4c | ,.shape.(M,).or.(M,.K).........L |
| b3a0 | 65 67 65 6e 64 72 65 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6f 72 64 65 72 65 64 20 66 72 6f | egendre.coefficients.ordered.fro |
| b3c0 | 6d 20 6c 6f 77 20 74 6f 20 68 69 67 68 2e 20 49 66 20 60 79 60 20 77 61 73 0a 20 20 20 20 20 20 | m.low.to.high..If.`y`.was....... |
| b3e0 | 20 20 32 2d 44 2c 20 74 68 65 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 66 6f 72 20 74 68 65 20 | ..2-D,.the.coefficients.for.the. |
| b400 | 64 61 74 61 20 69 6e 20 63 6f 6c 75 6d 6e 20 6b 20 6f 66 20 60 79 60 20 61 72 65 20 69 6e 0a 20 | data.in.column.k.of.`y`.are.in.. |
| b420 | 20 20 20 20 20 20 20 63 6f 6c 75 6d 6e 20 60 6b 60 2e 20 49 66 20 60 64 65 67 60 20 69 73 20 73 | .......column.`k`..If.`deg`.is.s |
| b440 | 70 65 63 69 66 69 65 64 20 61 73 20 61 20 6c 69 73 74 2c 20 63 6f 65 66 66 69 63 69 65 6e 74 73 | pecified.as.a.list,.coefficients |
| b460 | 20 66 6f 72 0a 20 20 20 20 20 20 20 20 74 65 72 6d 73 20 6e 6f 74 20 69 6e 63 6c 75 64 65 64 20 | .for.........terms.not.included. |
| b480 | 69 6e 20 74 68 65 20 66 69 74 20 61 72 65 20 73 65 74 20 65 71 75 61 6c 20 74 6f 20 7a 65 72 6f | in.the.fit.are.set.equal.to.zero |
| b4a0 | 20 69 6e 20 74 68 65 0a 20 20 20 20 20 20 20 20 72 65 74 75 72 6e 65 64 20 60 63 6f 65 66 60 2e | .in.the.........returned.`coef`. |
| b4c0 | 0a 0a 20 20 20 20 5b 72 65 73 69 64 75 61 6c 73 2c 20 72 61 6e 6b 2c 20 73 69 6e 67 75 6c 61 72 | ......[residuals,.rank,.singular |
| b4e0 | 5f 76 61 6c 75 65 73 2c 20 72 63 6f 6e 64 5d 20 3a 20 6c 69 73 74 0a 20 20 20 20 20 20 20 20 54 | _values,.rcond].:.list.........T |
| b500 | 68 65 73 65 20 76 61 6c 75 65 73 20 61 72 65 20 6f 6e 6c 79 20 72 65 74 75 72 6e 65 64 20 69 66 | hese.values.are.only.returned.if |
| b520 | 20 60 60 66 75 6c 6c 20 3d 3d 20 54 72 75 65 60 60 0a 0a 20 20 20 20 20 20 20 20 2d 20 72 65 73 | .``full.==.True``..........-.res |
| b540 | 69 64 75 61 6c 73 20 2d 2d 20 73 75 6d 20 6f 66 20 73 71 75 61 72 65 64 20 72 65 73 69 64 75 61 | iduals.--.sum.of.squared.residua |
| b560 | 6c 73 20 6f 66 20 74 68 65 20 6c 65 61 73 74 20 73 71 75 61 72 65 73 20 66 69 74 0a 20 20 20 20 | ls.of.the.least.squares.fit..... |
| b580 | 20 20 20 20 2d 20 72 61 6e 6b 20 2d 2d 20 74 68 65 20 6e 75 6d 65 72 69 63 61 6c 20 72 61 6e 6b | ....-.rank.--.the.numerical.rank |
| b5a0 | 20 6f 66 20 74 68 65 20 73 63 61 6c 65 64 20 56 61 6e 64 65 72 6d 6f 6e 64 65 20 6d 61 74 72 69 | .of.the.scaled.Vandermonde.matri |
| b5c0 | 78 0a 20 20 20 20 20 20 20 20 2d 20 73 69 6e 67 75 6c 61 72 5f 76 61 6c 75 65 73 20 2d 2d 20 73 | x.........-.singular_values.--.s |
| b5e0 | 69 6e 67 75 6c 61 72 20 76 61 6c 75 65 73 20 6f 66 20 74 68 65 20 73 63 61 6c 65 64 20 56 61 6e | ingular.values.of.the.scaled.Van |
| b600 | 64 65 72 6d 6f 6e 64 65 20 6d 61 74 72 69 78 0a 20 20 20 20 20 20 20 20 2d 20 72 63 6f 6e 64 20 | dermonde.matrix.........-.rcond. |
| b620 | 2d 2d 20 76 61 6c 75 65 20 6f 66 20 60 72 63 6f 6e 64 60 2e 0a 0a 20 20 20 20 20 20 20 20 46 6f | --.value.of.`rcond`...........Fo |
| b640 | 72 20 6d 6f 72 65 20 64 65 74 61 69 6c 73 2c 20 73 65 65 20 60 6e 75 6d 70 79 2e 6c 69 6e 61 6c | r.more.details,.see.`numpy.linal |
| b660 | 67 2e 6c 73 74 73 71 60 2e 0a 0a 20 20 20 20 57 61 72 6e 73 0a 20 20 20 20 2d 2d 2d 2d 2d 0a 20 | g.lstsq`.......Warns.....-----.. |
| b680 | 20 20 20 52 61 6e 6b 57 61 72 6e 69 6e 67 0a 20 20 20 20 20 20 20 20 54 68 65 20 72 61 6e 6b 20 | ...RankWarning.........The.rank. |
| b6a0 | 6f 66 20 74 68 65 20 63 6f 65 66 66 69 63 69 65 6e 74 20 6d 61 74 72 69 78 20 69 6e 20 74 68 65 | of.the.coefficient.matrix.in.the |
| b6c0 | 20 6c 65 61 73 74 2d 73 71 75 61 72 65 73 20 66 69 74 20 69 73 0a 20 20 20 20 20 20 20 20 64 65 | .least-squares.fit.is.........de |
| b6e0 | 66 69 63 69 65 6e 74 2e 20 54 68 65 20 77 61 72 6e 69 6e 67 20 69 73 20 6f 6e 6c 79 20 72 61 69 | ficient..The.warning.is.only.rai |
| b700 | 73 65 64 20 69 66 20 60 60 66 75 6c 6c 20 3d 3d 20 46 61 6c 73 65 60 60 2e 20 20 54 68 65 0a 20 | sed.if.``full.==.False``...The.. |
| b720 | 20 20 20 20 20 20 20 77 61 72 6e 69 6e 67 73 20 63 61 6e 20 62 65 20 74 75 72 6e 65 64 20 6f 66 | .......warnings.can.be.turned.of |
| b740 | 66 20 62 79 0a 0a 20 20 20 20 20 20 20 20 3e 3e 3e 20 69 6d 70 6f 72 74 20 77 61 72 6e 69 6e 67 | f.by..........>>>.import.warning |
| b760 | 73 0a 20 20 20 20 20 20 20 20 3e 3e 3e 20 77 61 72 6e 69 6e 67 73 2e 73 69 6d 70 6c 65 66 69 6c | s.........>>>.warnings.simplefil |
| b780 | 74 65 72 28 27 69 67 6e 6f 72 65 27 2c 20 6e 70 2e 65 78 63 65 70 74 69 6f 6e 73 2e 52 61 6e 6b | ter('ignore',.np.exceptions.Rank |
| b7a0 | 57 61 72 6e 69 6e 67 29 0a 0a 20 20 20 20 53 65 65 20 41 6c 73 6f 0a 20 20 20 20 2d 2d 2d 2d 2d | Warning)......See.Also.....----- |
| b7c0 | 2d 2d 2d 0a 20 20 20 20 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 2e 70 6f 6c 79 6e 6f 6d | ---.....numpy.polynomial.polynom |
| b7e0 | 69 61 6c 2e 70 6f 6c 79 66 69 74 0a 20 20 20 20 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c | ial.polyfit.....numpy.polynomial |
| b800 | 2e 63 68 65 62 79 73 68 65 76 2e 63 68 65 62 66 69 74 0a 20 20 20 20 6e 75 6d 70 79 2e 70 6f 6c | .chebyshev.chebfit.....numpy.pol |
| b820 | 79 6e 6f 6d 69 61 6c 2e 6c 61 67 75 65 72 72 65 2e 6c 61 67 66 69 74 0a 20 20 20 20 6e 75 6d 70 | ynomial.laguerre.lagfit.....nump |
| b840 | 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 2e 68 65 72 6d 69 74 65 2e 68 65 72 6d 66 69 74 0a 20 20 20 | y.polynomial.hermite.hermfit.... |
| b860 | 20 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 2e 68 65 72 6d 69 74 65 5f 65 2e 68 65 72 6d | .numpy.polynomial.hermite_e.herm |
| b880 | 65 66 69 74 0a 20 20 20 20 6c 65 67 76 61 6c 20 3a 20 45 76 61 6c 75 61 74 65 73 20 61 20 4c 65 | efit.....legval.:.Evaluates.a.Le |
| b8a0 | 67 65 6e 64 72 65 20 73 65 72 69 65 73 2e 0a 20 20 20 20 6c 65 67 76 61 6e 64 65 72 20 3a 20 56 | gendre.series......legvander.:.V |
| b8c0 | 61 6e 64 65 72 6d 6f 6e 64 65 20 6d 61 74 72 69 78 20 6f 66 20 4c 65 67 65 6e 64 72 65 20 73 65 | andermonde.matrix.of.Legendre.se |
| b8e0 | 72 69 65 73 2e 0a 20 20 20 20 6c 65 67 77 65 69 67 68 74 20 3a 20 4c 65 67 65 6e 64 72 65 20 77 | ries......legweight.:.Legendre.w |
| b900 | 65 69 67 68 74 20 66 75 6e 63 74 69 6f 6e 20 28 3d 20 31 29 2e 0a 20 20 20 20 6e 75 6d 70 79 2e | eight.function.(=.1)......numpy. |
| b920 | 6c 69 6e 61 6c 67 2e 6c 73 74 73 71 20 3a 20 43 6f 6d 70 75 74 65 73 20 61 20 6c 65 61 73 74 2d | linalg.lstsq.:.Computes.a.least- |
| b940 | 73 71 75 61 72 65 73 20 66 69 74 20 66 72 6f 6d 20 74 68 65 20 6d 61 74 72 69 78 2e 0a 20 20 20 | squares.fit.from.the.matrix..... |
| b960 | 20 73 63 69 70 79 2e 69 6e 74 65 72 70 6f 6c 61 74 65 2e 55 6e 69 76 61 72 69 61 74 65 53 70 6c | .scipy.interpolate.UnivariateSpl |
| b980 | 69 6e 65 20 3a 20 43 6f 6d 70 75 74 65 73 20 73 70 6c 69 6e 65 20 66 69 74 73 2e 0a 0a 20 20 20 | ine.:.Computes.spline.fits...... |
| b9a0 | 20 4e 6f 74 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 0a 20 20 20 20 54 68 65 20 73 6f 6c 75 74 69 6f | .Notes.....-----.....The.solutio |
| b9c0 | 6e 20 69 73 20 74 68 65 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6f 66 20 74 68 65 20 4c 65 67 | n.is.the.coefficients.of.the.Leg |
| b9e0 | 65 6e 64 72 65 20 73 65 72 69 65 73 20 60 70 60 20 74 68 61 74 0a 20 20 20 20 6d 69 6e 69 6d 69 | endre.series.`p`.that.....minimi |
| ba00 | 7a 65 73 20 74 68 65 20 73 75 6d 20 6f 66 20 74 68 65 20 77 65 69 67 68 74 65 64 20 73 71 75 61 | zes.the.sum.of.the.weighted.squa |
| ba20 | 72 65 64 20 65 72 72 6f 72 73 0a 0a 20 20 20 20 2e 2e 20 6d 61 74 68 3a 3a 20 45 20 3d 20 5c 73 | red.errors.........math::.E.=.\s |
| ba40 | 75 6d 5f 6a 20 77 5f 6a 5e 32 20 2a 20 7c 79 5f 6a 20 2d 20 70 28 78 5f 6a 29 7c 5e 32 2c 0a 0a | um_j.w_j^2.*.|y_j.-.p(x_j)|^2,.. |
| ba60 | 20 20 20 20 77 68 65 72 65 20 3a 6d 61 74 68 3a 60 77 5f 6a 60 20 61 72 65 20 74 68 65 20 77 65 | ....where.:math:`w_j`.are.the.we |
| ba80 | 69 67 68 74 73 2e 20 54 68 69 73 20 70 72 6f 62 6c 65 6d 20 69 73 20 73 6f 6c 76 65 64 20 62 79 | ights..This.problem.is.solved.by |
| baa0 | 20 73 65 74 74 69 6e 67 20 75 70 0a 20 20 20 20 61 73 20 74 68 65 20 28 74 79 70 69 63 61 6c 6c | .setting.up.....as.the.(typicall |
| bac0 | 79 29 20 6f 76 65 72 64 65 74 65 72 6d 69 6e 65 64 20 6d 61 74 72 69 78 20 65 71 75 61 74 69 6f | y).overdetermined.matrix.equatio |
| bae0 | 6e 0a 0a 20 20 20 20 2e 2e 20 6d 61 74 68 3a 3a 20 56 28 78 29 20 2a 20 63 20 3d 20 77 20 2a 20 | n.........math::.V(x).*.c.=.w.*. |
| bb00 | 79 2c 0a 0a 20 20 20 20 77 68 65 72 65 20 60 56 60 20 69 73 20 74 68 65 20 77 65 69 67 68 74 65 | y,......where.`V`.is.the.weighte |
| bb20 | 64 20 70 73 65 75 64 6f 20 56 61 6e 64 65 72 6d 6f 6e 64 65 20 6d 61 74 72 69 78 20 6f 66 20 60 | d.pseudo.Vandermonde.matrix.of.` |
| bb40 | 78 60 2c 20 60 63 60 20 61 72 65 20 74 68 65 0a 20 20 20 20 63 6f 65 66 66 69 63 69 65 6e 74 73 | x`,.`c`.are.the.....coefficients |
| bb60 | 20 74 6f 20 62 65 20 73 6f 6c 76 65 64 20 66 6f 72 2c 20 60 77 60 20 61 72 65 20 74 68 65 20 77 | .to.be.solved.for,.`w`.are.the.w |
| bb80 | 65 69 67 68 74 73 2c 20 61 6e 64 20 60 79 60 20 61 72 65 20 74 68 65 0a 20 20 20 20 6f 62 73 65 | eights,.and.`y`.are.the.....obse |
| bba0 | 72 76 65 64 20 76 61 6c 75 65 73 2e 20 20 54 68 69 73 20 65 71 75 61 74 69 6f 6e 20 69 73 20 74 | rved.values...This.equation.is.t |
| bbc0 | 68 65 6e 20 73 6f 6c 76 65 64 20 75 73 69 6e 67 20 74 68 65 20 73 69 6e 67 75 6c 61 72 20 76 61 | hen.solved.using.the.singular.va |
| bbe0 | 6c 75 65 0a 20 20 20 20 64 65 63 6f 6d 70 6f 73 69 74 69 6f 6e 20 6f 66 20 60 56 60 2e 0a 0a 20 | lue.....decomposition.of.`V`.... |
| bc00 | 20 20 20 49 66 20 73 6f 6d 65 20 6f 66 20 74 68 65 20 73 69 6e 67 75 6c 61 72 20 76 61 6c 75 65 | ...If.some.of.the.singular.value |
| bc20 | 73 20 6f 66 20 60 56 60 20 61 72 65 20 73 6f 20 73 6d 61 6c 6c 20 74 68 61 74 20 74 68 65 79 20 | s.of.`V`.are.so.small.that.they. |
| bc40 | 61 72 65 0a 20 20 20 20 6e 65 67 6c 65 63 74 65 64 2c 20 74 68 65 6e 20 61 20 60 7e 65 78 63 65 | are.....neglected,.then.a.`~exce |
| bc60 | 70 74 69 6f 6e 73 2e 52 61 6e 6b 57 61 72 6e 69 6e 67 60 20 77 69 6c 6c 20 62 65 20 69 73 73 75 | ptions.RankWarning`.will.be.issu |
| bc80 | 65 64 2e 20 54 68 69 73 20 6d 65 61 6e 73 20 74 68 61 74 0a 20 20 20 20 74 68 65 20 63 6f 65 66 | ed..This.means.that.....the.coef |
| bca0 | 66 69 63 69 65 6e 74 20 76 61 6c 75 65 73 20 6d 61 79 20 62 65 20 70 6f 6f 72 6c 79 20 64 65 74 | ficient.values.may.be.poorly.det |
| bcc0 | 65 72 6d 69 6e 65 64 2e 20 55 73 69 6e 67 20 61 20 6c 6f 77 65 72 20 6f 72 64 65 72 20 66 69 74 | ermined..Using.a.lower.order.fit |
| bce0 | 0a 20 20 20 20 77 69 6c 6c 20 75 73 75 61 6c 6c 79 20 67 65 74 20 72 69 64 20 6f 66 20 74 68 65 | .....will.usually.get.rid.of.the |
| bd00 | 20 77 61 72 6e 69 6e 67 2e 20 20 54 68 65 20 60 72 63 6f 6e 64 60 20 70 61 72 61 6d 65 74 65 72 | .warning...The.`rcond`.parameter |
| bd20 | 20 63 61 6e 20 61 6c 73 6f 20 62 65 0a 20 20 20 20 73 65 74 20 74 6f 20 61 20 76 61 6c 75 65 20 | .can.also.be.....set.to.a.value. |
| bd40 | 73 6d 61 6c 6c 65 72 20 74 68 61 6e 20 69 74 73 20 64 65 66 61 75 6c 74 2c 20 62 75 74 20 74 68 | smaller.than.its.default,.but.th |
| bd60 | 65 20 72 65 73 75 6c 74 69 6e 67 20 66 69 74 20 6d 61 79 20 62 65 0a 20 20 20 20 73 70 75 72 69 | e.resulting.fit.may.be.....spuri |
| bd80 | 6f 75 73 20 61 6e 64 20 68 61 76 65 20 6c 61 72 67 65 20 63 6f 6e 74 72 69 62 75 74 69 6f 6e 73 | ous.and.have.large.contributions |
| bda0 | 20 66 72 6f 6d 20 72 6f 75 6e 64 6f 66 66 20 65 72 72 6f 72 2e 0a 0a 20 20 20 20 46 69 74 73 20 | .from.roundoff.error.......Fits. |
| bdc0 | 75 73 69 6e 67 20 4c 65 67 65 6e 64 72 65 20 73 65 72 69 65 73 20 61 72 65 20 75 73 75 61 6c 6c | using.Legendre.series.are.usuall |
| bde0 | 79 20 62 65 74 74 65 72 20 63 6f 6e 64 69 74 69 6f 6e 65 64 20 74 68 61 6e 20 66 69 74 73 0a 20 | y.better.conditioned.than.fits.. |
| be00 | 20 20 20 75 73 69 6e 67 20 70 6f 77 65 72 20 73 65 72 69 65 73 2c 20 62 75 74 20 6d 75 63 68 20 | ...using.power.series,.but.much. |
| be20 | 63 61 6e 20 64 65 70 65 6e 64 20 6f 6e 20 74 68 65 20 64 69 73 74 72 69 62 75 74 69 6f 6e 20 6f | can.depend.on.the.distribution.o |
| be40 | 66 20 74 68 65 0a 20 20 20 20 73 61 6d 70 6c 65 20 70 6f 69 6e 74 73 20 61 6e 64 20 74 68 65 20 | f.the.....sample.points.and.the. |
| be60 | 73 6d 6f 6f 74 68 6e 65 73 73 20 6f 66 20 74 68 65 20 64 61 74 61 2e 20 49 66 20 74 68 65 20 71 | smoothness.of.the.data..If.the.q |
| be80 | 75 61 6c 69 74 79 20 6f 66 20 74 68 65 20 66 69 74 0a 20 20 20 20 69 73 20 69 6e 61 64 65 71 75 | uality.of.the.fit.....is.inadequ |
| bea0 | 61 74 65 20 73 70 6c 69 6e 65 73 20 6d 61 79 20 62 65 20 61 20 67 6f 6f 64 20 61 6c 74 65 72 6e | ate.splines.may.be.a.good.altern |
| bec0 | 61 74 69 76 65 2e 0a 0a 20 20 20 20 52 65 66 65 72 65 6e 63 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d | ative.......References.....----- |
| bee0 | 2d 2d 2d 2d 2d 0a 20 20 20 20 2e 2e 20 5b 31 5d 20 57 69 6b 69 70 65 64 69 61 2c 20 22 43 75 72 | -----........[1].Wikipedia,."Cur |
| bf00 | 76 65 20 66 69 74 74 69 6e 67 22 2c 0a 20 20 20 20 20 20 20 20 20 20 20 68 74 74 70 73 3a 2f 2f | ve.fitting",............https:// |
| bf20 | 65 6e 2e 77 69 6b 69 70 65 64 69 61 2e 6f 72 67 2f 77 69 6b 69 2f 43 75 72 76 65 5f 66 69 74 74 | en.wikipedia.org/wiki/Curve_fitt |
| bf40 | 69 6e 67 0a 0a 20 20 20 20 45 78 61 6d 70 6c 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 0a | ing......Examples.....--------.. |
| bf60 | 20 20 20 20 29 03 72 28 00 00 00 da 04 5f 66 69 74 72 18 00 00 00 29 06 72 7e 00 00 00 72 84 00 | ....).r(....._fitr....).r~...r.. |
| bf80 | 00 00 72 2d 00 00 00 da 05 72 63 6f 6e 64 da 04 66 75 6c 6c da 01 77 73 06 00 00 00 20 20 20 20 | ..r-.....rcond..full..ws........ |
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| bfc0 | 0c 0e 8f 37 89 37 94 39 98 61 a0 11 a0 43 a8 15 b0 04 b0 61 d3 0b 38 d0 04 38 72 31 00 00 00 63 | ...7.7.9.a...C.....a..8..8r1...c |
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| c140 | 09 00 00 00 00 00 00 00 00 6a 12 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 01 ab | .........j...................|.. |
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| c180 | 02 6a 15 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 64 07 ab 01 00 00 00 00 00 00 64 | .j...................d.........d |
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| c1e0 | 03 19 00 00 00 7d 05 74 09 00 00 00 00 00 00 00 00 6a 12 00 00 00 00 00 00 00 00 00 00 00 00 00 | .....}.t.........j.............. |
| c200 | 00 00 00 00 00 64 04 7c 01 ab 02 00 00 00 00 00 00 7c 03 64 08 7c 01 64 04 7a 0a 00 00 1a 00 7a | .....d.|.........|.d.|.d.z.....z |
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| c2a0 | 02 53 00 29 0a 61 61 02 00 00 52 65 74 75 72 6e 20 74 68 65 20 73 63 61 6c 65 64 20 63 6f 6d 70 | .S.).aa...Return.the.scaled.comp |
| c2c0 | 61 6e 69 6f 6e 20 6d 61 74 72 69 78 20 6f 66 20 63 2e 0a 0a 20 20 20 20 54 68 65 20 62 61 73 69 | anion.matrix.of.c.......The.basi |
| c2e0 | 73 20 70 6f 6c 79 6e 6f 6d 69 61 6c 73 20 61 72 65 20 73 63 61 6c 65 64 20 73 6f 20 74 68 61 74 | s.polynomials.are.scaled.so.that |
| c300 | 20 74 68 65 20 63 6f 6d 70 61 6e 69 6f 6e 20 6d 61 74 72 69 78 20 69 73 0a 20 20 20 20 73 79 6d | .the.companion.matrix.is.....sym |
| c320 | 6d 65 74 72 69 63 20 77 68 65 6e 20 60 63 60 20 69 73 20 61 6e 20 4c 65 67 65 6e 64 72 65 20 62 | metric.when.`c`.is.an.Legendre.b |
| c340 | 61 73 69 73 20 70 6f 6c 79 6e 6f 6d 69 61 6c 2e 20 54 68 69 73 20 70 72 6f 76 69 64 65 73 0a 20 | asis.polynomial..This.provides.. |
| c360 | 20 20 20 62 65 74 74 65 72 20 65 69 67 65 6e 76 61 6c 75 65 20 65 73 74 69 6d 61 74 65 73 20 74 | ...better.eigenvalue.estimates.t |
| c380 | 68 61 6e 20 74 68 65 20 75 6e 73 63 61 6c 65 64 20 63 61 73 65 20 61 6e 64 20 66 6f 72 20 62 61 | han.the.unscaled.case.and.for.ba |
| c3a0 | 73 69 73 0a 20 20 20 20 70 6f 6c 79 6e 6f 6d 69 61 6c 73 20 74 68 65 20 65 69 67 65 6e 76 61 6c | sis.....polynomials.the.eigenval |
| c3c0 | 75 65 73 20 61 72 65 20 67 75 61 72 61 6e 74 65 65 64 20 74 6f 20 62 65 20 72 65 61 6c 20 69 66 | ues.are.guaranteed.to.be.real.if |
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| c400 | 75 73 65 64 20 74 6f 20 6f 62 74 61 69 6e 20 74 68 65 6d 2e 0a 0a 20 20 20 20 50 61 72 61 6d 65 | used.to.obtain.them.......Parame |
| c420 | 74 65 72 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 63 20 3a 20 61 72 72 61 | ters.....----------.....c.:.arra |
| c440 | 79 5f 6c 69 6b 65 0a 20 20 20 20 20 20 20 20 31 2d 44 20 61 72 72 61 79 20 6f 66 20 4c 65 67 65 | y_like.........1-D.array.of.Lege |
| c460 | 6e 64 72 65 20 73 65 72 69 65 73 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6f 72 64 65 72 65 64 | ndre.series.coefficients.ordered |
| c480 | 20 66 72 6f 6d 20 6c 6f 77 20 74 6f 20 68 69 67 68 0a 20 20 20 20 20 20 20 20 64 65 67 72 65 65 | .from.low.to.high.........degree |
| c4a0 | 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 6d | .......Returns.....-------.....m |
| c4c0 | 61 74 20 3a 20 6e 64 61 72 72 61 79 0a 20 20 20 20 20 20 20 20 53 63 61 6c 65 64 20 63 6f 6d 70 | at.:.ndarray.........Scaled.comp |
| c4e0 | 61 6e 69 6f 6e 20 6d 61 74 72 69 78 20 6f 66 20 64 69 6d 65 6e 73 69 6f 6e 73 20 28 64 65 67 2c | anion.matrix.of.dimensions.(deg, |
| c500 | 20 64 65 67 29 2e 0a 20 20 20 20 72 38 00 00 00 7a 2e 53 65 72 69 65 73 20 6d 75 73 74 20 68 61 | .deg)......r8...z.Series.must.ha |
| c520 | 76 65 20 6d 61 78 69 6d 75 6d 20 64 65 67 72 65 65 20 6f 66 20 61 74 20 6c 65 61 73 74 20 31 2e | ve.maximum.degree.of.at.least.1. |
| c540 | 72 02 00 00 00 72 04 00 00 00 72 4f 00 00 00 72 3f 00 00 00 72 27 00 00 00 4e 2e 29 0b 72 28 00 | r....r....rO...r?...r'...N.).r(. |
| c560 | 00 00 72 29 00 00 00 72 2a 00 00 00 72 69 00 00 00 72 41 00 00 00 72 42 00 00 00 da 05 7a 65 72 | ..r)...r*...ri...rA...rB.....zer |
| c580 | 6f 73 72 50 00 00 00 da 04 73 71 72 74 da 06 61 72 61 6e 67 65 72 7d 00 00 00 29 06 72 3a 00 00 | osrP.....sqrt..aranger}...).r:.. |
| c5a0 | 00 72 3b 00 00 00 da 03 6d 61 74 72 44 00 00 00 da 03 74 6f 70 da 03 62 6f 74 73 06 00 00 00 20 | .r;.....matrD.....top..bots..... |
| c5c0 | 20 20 20 20 20 72 30 00 00 00 72 23 00 00 00 72 23 00 00 00 5d 05 00 00 73 55 01 00 00 80 00 f4 | .....r0...r#...r#...]...sU...... |
| c5e0 | 2a 00 0b 0d 8f 2c 89 2c 98 01 90 73 d3 0a 1b 81 43 80 51 dc 07 0a 88 31 83 76 90 01 82 7a dc 0e | *....,.,...s....C.Q....1.v...z.. |
| c600 | 18 d0 19 49 d3 0e 4a d0 08 4a dc 07 0a 88 31 83 76 90 11 82 7b dc 0f 11 8f 78 89 78 98 31 98 51 | ...I..J..J....1.v...{....x.x.1.Q |
| c620 | 99 34 98 25 a0 21 a0 41 a1 24 99 2c 98 1e d0 18 28 d3 0f 29 d0 08 29 e4 08 0b 88 41 8b 06 90 11 | .4.%.!.A.$.,....(..)..)....A.... |
| c640 | 89 0a 80 41 dc 0a 0c 8f 28 89 28 90 41 90 71 90 36 a0 11 a7 17 a1 17 d4 0a 29 80 43 d8 0a 0c 8c | ...A....(.(.A.q.6........).C.... |
| c660 | 72 8f 77 89 77 90 71 9c 32 9f 39 99 39 a0 51 9b 3c d1 17 27 a8 21 d1 17 2b d3 0f 2c d1 0a 2c 80 | r.w.w.q.2.9.9.Q.<..'.!..+..,..,. |
| c680 | 43 d8 0a 0d 8f 2b 89 2b 90 62 8b 2f 98 21 98 28 98 51 a0 11 99 55 98 28 d1 0a 23 80 43 d8 0a 0d | C....+.+.b./.!.(.Q...U.(..#.C... |
| c6a0 | 8f 2b 89 2b 90 62 8b 2f 98 21 98 28 98 51 a0 11 99 55 98 28 d1 0a 23 80 43 dc 0f 11 8f 79 89 79 | .+.+.b./.!.(.Q...U.(..#.C....y.y |
| c6c0 | 98 11 98 41 8b 7f a0 13 a0 56 a0 61 a8 21 a1 65 a0 1b d1 0f 2c a8 73 b0 31 b0 51 a8 78 d1 0f 37 | ...A.....V.a.!.e....,.s.1.Q.x..7 |
| c6e0 | 80 43 88 03 81 48 d8 0f 12 80 43 88 03 81 48 d8 04 07 8a 01 88 32 88 05 83 4a 90 31 90 53 90 62 | .C...H....C...H......2...J.1.S.b |
| c700 | 90 36 98 41 98 62 99 45 91 3e a0 63 a8 43 b0 02 a9 47 a1 6d d1 12 34 b8 01 b8 51 c0 11 b9 55 c0 | .6.A.b.E.>.c.C...G.m..4...Q...U. |
| c720 | 51 b9 59 b9 0f d1 12 48 d1 04 48 83 4a d8 0b 0e 80 4a 72 31 00 00 00 63 01 00 00 00 00 00 00 00 | Q.Y....H..H.J....Jr1...c........ |
| c740 | 00 00 00 00 05 00 00 00 03 00 00 00 f3 66 01 00 00 97 00 74 01 00 00 00 00 00 00 00 00 6a 02 00 | .............f.....t.........j.. |
| c760 | 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 00 67 01 ab 01 00 00 00 00 00 00 5c 01 00 | .................|.g.........\.. |
| c780 | 00 7d 00 74 05 00 00 00 00 00 00 00 00 7c 00 ab 01 00 00 00 00 00 00 64 01 6b 02 00 00 72 21 74 | .}.t.........|.........d.k...r!t |
| c7a0 | 07 00 00 00 00 00 00 00 00 6a 08 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 67 00 7c | .........j...................g.| |
| c7c0 | 00 6a 0a 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ac 02 ab 02 00 00 00 00 00 00 53 | .j.............................S |
| c7e0 | 00 74 05 00 00 00 00 00 00 00 00 7c 00 ab 01 00 00 00 00 00 00 64 01 6b 28 00 00 72 20 74 07 00 | .t.........|.........d.k(..r.t.. |
| c800 | 00 00 00 00 00 00 00 6a 08 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 00 64 03 19 | .......j...................|.d.. |
| c820 | 00 00 00 0b 00 7c 00 64 04 19 00 00 00 7a 0b 00 00 67 01 ab 01 00 00 00 00 00 00 53 00 74 0d 00 | .....|.d.....z...g.........S.t.. |
| c840 | 00 00 00 00 00 00 00 7c 00 ab 01 00 00 00 00 00 00 64 05 64 05 64 06 85 03 64 05 64 05 64 06 85 | .......|.........d.d.d...d.d.d.. |
| c860 | 03 66 02 19 00 00 00 7d 01 74 0f 00 00 00 00 00 00 00 00 6a 10 00 00 00 00 00 00 00 00 00 00 00 | .f.....}.t.........j............ |
| c880 | 00 00 00 00 00 00 00 7c 01 ab 01 00 00 00 00 00 00 7d 02 7c 02 6a 13 00 00 00 00 00 00 00 00 00 | .......|.........}.|.j.......... |
| c8a0 | 00 00 00 00 00 00 00 00 00 ab 00 00 00 00 00 00 00 01 00 7c 02 53 00 29 07 61 cf 05 00 00 0a 20 | ...................|.S.).a...... |
| c8c0 | 20 20 20 43 6f 6d 70 75 74 65 20 74 68 65 20 72 6f 6f 74 73 20 6f 66 20 61 20 4c 65 67 65 6e 64 | ...Compute.the.roots.of.a.Legend |
| c8e0 | 72 65 20 73 65 72 69 65 73 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 20 74 68 65 20 72 6f 6f 74 73 | re.series.......Return.the.roots |
| c900 | 20 28 61 2e 6b 2e 61 2e 20 22 7a 65 72 6f 73 22 29 20 6f 66 20 74 68 65 20 70 6f 6c 79 6e 6f 6d | .(a.k.a.."zeros").of.the.polynom |
| c920 | 69 61 6c 0a 0a 20 20 20 20 2e 2e 20 6d 61 74 68 3a 3a 20 70 28 78 29 20 3d 20 5c 73 75 6d 5f 69 | ial.........math::.p(x).=.\sum_i |
| c940 | 20 63 5b 69 5d 20 2a 20 4c 5f 69 28 78 29 2e 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a | .c[i].*.L_i(x).......Parameters. |
| c960 | 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 63 20 3a 20 31 2d 44 20 61 72 72 61 79 | ....----------.....c.:.1-D.array |
| c980 | 5f 6c 69 6b 65 0a 20 20 20 20 20 20 20 20 31 2d 44 20 61 72 72 61 79 20 6f 66 20 63 6f 65 66 66 | _like.........1-D.array.of.coeff |
| c9a0 | 69 63 69 65 6e 74 73 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d | icients.......Returns.....------ |
| c9c0 | 2d 0a 20 20 20 20 6f 75 74 20 3a 20 6e 64 61 72 72 61 79 0a 20 20 20 20 20 20 20 20 41 72 72 61 | -.....out.:.ndarray.........Arra |
| c9e0 | 79 20 6f 66 20 74 68 65 20 72 6f 6f 74 73 20 6f 66 20 74 68 65 20 73 65 72 69 65 73 2e 20 49 66 | y.of.the.roots.of.the.series..If |
| ca00 | 20 61 6c 6c 20 74 68 65 20 72 6f 6f 74 73 20 61 72 65 20 72 65 61 6c 2c 0a 20 20 20 20 20 20 20 | .all.the.roots.are.real,........ |
| ca20 | 20 74 68 65 6e 20 60 6f 75 74 60 20 69 73 20 61 6c 73 6f 20 72 65 61 6c 2c 20 6f 74 68 65 72 77 | .then.`out`.is.also.real,.otherw |
| ca40 | 69 73 65 20 69 74 20 69 73 20 63 6f 6d 70 6c 65 78 2e 0a 0a 20 20 20 20 53 65 65 20 41 6c 73 6f | ise.it.is.complex.......See.Also |
| ca60 | 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 | .....--------.....numpy.polynomi |
| ca80 | 61 6c 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 2e 70 6f 6c 79 72 6f 6f 74 73 0a 20 20 20 20 6e 75 6d 70 | al.polynomial.polyroots.....nump |
| caa0 | 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 2e 63 68 65 62 79 73 68 65 76 2e 63 68 65 62 72 6f 6f 74 73 | y.polynomial.chebyshev.chebroots |
| cac0 | 0a 20 20 20 20 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 2e 6c 61 67 75 65 72 72 65 2e 6c | .....numpy.polynomial.laguerre.l |
| cae0 | 61 67 72 6f 6f 74 73 0a 20 20 20 20 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d 69 61 6c 2e 68 65 72 | agroots.....numpy.polynomial.her |
| cb00 | 6d 69 74 65 2e 68 65 72 6d 72 6f 6f 74 73 0a 20 20 20 20 6e 75 6d 70 79 2e 70 6f 6c 79 6e 6f 6d | mite.hermroots.....numpy.polynom |
| cb20 | 69 61 6c 2e 68 65 72 6d 69 74 65 5f 65 2e 68 65 72 6d 65 72 6f 6f 74 73 0a 0a 20 20 20 20 4e 6f | ial.hermite_e.hermeroots......No |
| cb40 | 74 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 0a 20 20 20 20 54 68 65 20 72 6f 6f 74 20 65 73 74 69 6d | tes.....-----.....The.root.estim |
| cb60 | 61 74 65 73 20 61 72 65 20 6f 62 74 61 69 6e 65 64 20 61 73 20 74 68 65 20 65 69 67 65 6e 76 61 | ates.are.obtained.as.the.eigenva |
| cb80 | 6c 75 65 73 20 6f 66 20 74 68 65 20 63 6f 6d 70 61 6e 69 6f 6e 0a 20 20 20 20 6d 61 74 72 69 78 | lues.of.the.companion.....matrix |
| cba0 | 2c 20 52 6f 6f 74 73 20 66 61 72 20 66 72 6f 6d 20 74 68 65 20 6f 72 69 67 69 6e 20 6f 66 20 74 | ,.Roots.far.from.the.origin.of.t |
| cbc0 | 68 65 20 63 6f 6d 70 6c 65 78 20 70 6c 61 6e 65 20 6d 61 79 20 68 61 76 65 20 6c 61 72 67 65 0a | he.complex.plane.may.have.large. |
| cbe0 | 20 20 20 20 65 72 72 6f 72 73 20 64 75 65 20 74 6f 20 74 68 65 20 6e 75 6d 65 72 69 63 61 6c 20 | ....errors.due.to.the.numerical. |
| cc00 | 69 6e 73 74 61 62 69 6c 69 74 79 20 6f 66 20 74 68 65 20 73 65 72 69 65 73 20 66 6f 72 20 73 75 | instability.of.the.series.for.su |
| cc20 | 63 68 20 76 61 6c 75 65 73 2e 0a 20 20 20 20 52 6f 6f 74 73 20 77 69 74 68 20 6d 75 6c 74 69 70 | ch.values......Roots.with.multip |
| cc40 | 6c 69 63 69 74 79 20 67 72 65 61 74 65 72 20 74 68 61 6e 20 31 20 77 69 6c 6c 20 61 6c 73 6f 20 | licity.greater.than.1.will.also. |
| cc60 | 73 68 6f 77 20 6c 61 72 67 65 72 20 65 72 72 6f 72 73 20 61 73 0a 20 20 20 20 74 68 65 20 76 61 | show.larger.errors.as.....the.va |
| cc80 | 6c 75 65 20 6f 66 20 74 68 65 20 73 65 72 69 65 73 20 6e 65 61 72 20 73 75 63 68 20 70 6f 69 6e | lue.of.the.series.near.such.poin |
| cca0 | 74 73 20 69 73 20 72 65 6c 61 74 69 76 65 6c 79 20 69 6e 73 65 6e 73 69 74 69 76 65 20 74 6f 0a | ts.is.relatively.insensitive.to. |
| ccc0 | 20 20 20 20 65 72 72 6f 72 73 20 69 6e 20 74 68 65 20 72 6f 6f 74 73 2e 20 49 73 6f 6c 61 74 65 | ....errors.in.the.roots..Isolate |
| cce0 | 64 20 72 6f 6f 74 73 20 6e 65 61 72 20 74 68 65 20 6f 72 69 67 69 6e 20 63 61 6e 20 62 65 20 69 | d.roots.near.the.origin.can.be.i |
| cd00 | 6d 70 72 6f 76 65 64 20 62 79 0a 20 20 20 20 61 20 66 65 77 20 69 74 65 72 61 74 69 6f 6e 73 20 | mproved.by.....a.few.iterations. |
| cd20 | 6f 66 20 4e 65 77 74 6f 6e 27 73 20 6d 65 74 68 6f 64 2e 0a 0a 20 20 20 20 54 68 65 20 4c 65 67 | of.Newton's.method.......The.Leg |
| cd40 | 65 6e 64 72 65 20 73 65 72 69 65 73 20 62 61 73 69 73 20 70 6f 6c 79 6e 6f 6d 69 61 6c 73 20 61 | endre.series.basis.polynomials.a |
| cd60 | 72 65 6e 27 74 20 70 6f 77 65 72 73 20 6f 66 20 60 60 78 60 60 20 73 6f 20 74 68 65 0a 20 20 20 | ren't.powers.of.``x``.so.the.... |
| cd80 | 20 72 65 73 75 6c 74 73 20 6f 66 20 74 68 69 73 20 66 75 6e 63 74 69 6f 6e 20 6d 61 79 20 73 65 | .results.of.this.function.may.se |
| cda0 | 65 6d 20 75 6e 69 6e 74 75 69 74 69 76 65 2e 0a 0a 20 20 20 20 45 78 61 6d 70 6c 65 73 0a 20 20 | em.unintuitive.......Examples... |
| cdc0 | 20 20 2d 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 3e 3e 3e 20 69 6d 70 6f 72 74 20 6e 75 6d 70 79 2e | ..--------.....>>>.import.numpy. |
| cde0 | 70 6f 6c 79 6e 6f 6d 69 61 6c 2e 6c 65 67 65 6e 64 72 65 20 61 73 20 6c 65 67 0a 20 20 20 20 3e | polynomial.legendre.as.leg.....> |
| ce00 | 3e 3e 20 6c 65 67 2e 6c 65 67 72 6f 6f 74 73 28 28 31 2c 20 32 2c 20 33 2c 20 34 29 29 20 23 20 | >>.leg.legroots((1,.2,.3,.4)).#. |
| ce20 | 34 4c 5f 33 20 2b 20 33 4c 5f 32 20 2b 20 32 4c 5f 31 20 2b 20 31 4c 5f 30 2c 20 61 6c 6c 20 72 | 4L_3.+.3L_2.+.2L_1.+.1L_0,.all.r |
| ce40 | 65 61 6c 20 72 6f 6f 74 73 0a 20 20 20 20 61 72 72 61 79 28 5b 2d 30 2e 38 35 30 39 39 35 34 33 | eal.roots.....array([-0.85099543 |
| ce60 | 2c 20 2d 30 2e 31 31 34 30 37 31 39 32 2c 20 20 30 2e 35 31 35 30 36 37 33 35 5d 29 20 23 20 6d | ,.-0.11407192,..0.51506735]).#.m |
| ce80 | 61 79 20 76 61 72 79 0a 0a 20 20 20 20 72 38 00 00 00 72 4f 00 00 00 72 02 00 00 00 72 04 00 00 | ay.vary......r8...rO...r....r... |
| cea0 | 00 4e 72 27 00 00 00 29 0a 72 28 00 00 00 72 29 00 00 00 72 2a 00 00 00 72 41 00 00 00 72 42 00 | .Nr'...).r(...r)...r*...rA...rB. |
| cec0 | 00 00 72 50 00 00 00 72 23 00 00 00 da 02 6c 61 da 07 65 69 67 76 61 6c 73 da 04 73 6f 72 74 29 | ..rP...r#.....la..eigvals..sort) |
| cee0 | 03 72 3a 00 00 00 72 6d 00 00 00 da 01 72 73 03 00 00 00 20 20 20 72 30 00 00 00 72 1b 00 00 00 | .r:...rm.....rs.......r0...r.... |
| cf00 | 72 1b 00 00 00 83 05 00 00 73 98 00 00 00 80 00 f4 60 01 00 0b 0d 8f 2c 89 2c 98 01 90 73 d3 0a | r........s.......`.....,.,...s.. |
| cf20 | 1b 81 43 80 51 dc 07 0a 88 31 83 76 90 01 82 7a dc 0f 11 8f 78 89 78 98 02 a0 21 a7 27 a1 27 d4 | ..C.Q....1.v...z....x.x...!.'.'. |
| cf40 | 0f 2a d0 08 2a dc 07 0a 88 31 83 76 90 11 82 7b dc 0f 11 8f 78 89 78 98 21 98 41 99 24 98 15 a0 | .*..*....1.v...{....x.x.!.A.$... |
| cf60 | 11 a0 31 a1 14 99 1c 98 0e d3 0f 27 d0 08 27 f4 06 00 09 15 90 51 8b 0f 99 04 98 22 98 04 99 64 | ..1........'..'......Q....."...d |
| cf80 | a0 02 98 64 98 0a d1 08 23 80 41 dc 08 0a 8f 0a 89 0a 90 31 8b 0d 80 41 d8 04 05 87 46 81 46 84 | ...d....#.A........1...A....F.F. |
| cfa0 | 48 d8 0b 0c 80 48 72 31 00 00 00 63 01 00 00 00 00 00 00 00 00 00 00 00 06 00 00 00 03 00 00 00 | H....Hr1...c.................... |
| cfc0 | f3 4c 02 00 00 97 00 74 01 00 00 00 00 00 00 00 00 6a 02 00 00 00 00 00 00 00 00 00 00 00 00 00 | .L.....t.........j.............. |
| cfe0 | 00 00 00 00 00 7c 00 64 01 ab 02 00 00 00 00 00 00 7d 01 7c 01 64 02 6b 1a 00 00 72 0b 74 05 00 | .....|.d.........}.|.d.k...r.t.. |
| d000 | 00 00 00 00 00 00 00 64 03 ab 01 00 00 00 00 00 00 82 01 74 07 00 00 00 00 00 00 00 00 6a 08 00 | .......d...........t.........j.. |
| d020 | 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 64 02 67 01 7c 00 7a 05 00 00 64 04 67 01 7a | .................d.g.|.z...d.g.z |
| d040 | 00 00 00 ab 01 00 00 00 00 00 00 7d 02 74 0b 00 00 00 00 00 00 00 00 7c 02 ab 01 00 00 00 00 00 | ...........}.t.........|........ |
| d060 | 00 7d 03 74 0d 00 00 00 00 00 00 00 00 6a 0e 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 | .}.t.........j.................. |
| d080 | 00 7c 03 ab 01 00 00 00 00 00 00 7d 04 74 11 00 00 00 00 00 00 00 00 7c 04 7c 02 ab 02 00 00 00 | .|.........}.t.........|.|...... |
| d0a0 | 00 00 00 7d 05 74 11 00 00 00 00 00 00 00 00 7c 04 74 13 00 00 00 00 00 00 00 00 7c 02 ab 01 00 | ...}.t.........|.t.........|.... |
| d0c0 | 00 00 00 00 00 ab 02 00 00 00 00 00 00 7d 06 7c 04 7c 05 7c 06 7a 0b 00 00 7a 17 00 00 7d 04 74 | .............}.|.|.|.z...z...}.t |
| d0e0 | 11 00 00 00 00 00 00 00 00 7c 04 7c 02 64 04 64 05 1a 00 ab 02 00 00 00 00 00 00 7d 07 7c 07 74 | .........|.|.d.d...........}.|.t |
| d100 | 07 00 00 00 00 00 00 00 00 6a 14 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 7c 07 ab | .........j...................|.. |
| d120 | 01 00 00 00 00 00 00 6a 17 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ab 00 00 00 00 | .......j........................ |
| d140 | 00 00 00 7a 18 00 00 7d 07 7c 06 74 07 00 00 00 00 00 00 00 00 6a 14 00 00 00 00 00 00 00 00 00 | ...z...}.|.t.........j.......... |
| d160 | 00 00 00 00 00 00 00 00 00 7c 06 ab 01 00 00 00 00 00 00 6a 17 00 00 00 00 00 00 00 00 00 00 00 | .........|.........j............ |
| d180 | 00 00 00 00 00 00 00 ab 00 00 00 00 00 00 00 7a 18 00 00 7d 06 64 04 7c 07 7c 06 7a 05 00 00 7a | ...............z...}.d.|.|.z...z |
| d1a0 | 0b 00 00 7d 08 7c 08 7c 08 64 05 64 05 64 06 85 03 19 00 00 00 7a 00 00 00 64 07 7a 0b 00 00 7d | ...}.|.|.d.d.d.......z...d.z...} |
| d1c0 | 08 7c 04 7c 04 64 05 64 05 64 06 85 03 19 00 00 00 7a 0a 00 00 64 07 7a 0b 00 00 7d 04 7c 08 64 | .|.|.d.d.d.......z...d.z...}.|.d |
| d1e0 | 08 7c 08 6a 19 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ab 00 00 00 00 00 00 00 7a | .|.j...........................z |
| d200 | 0b 00 00 7a 12 00 00 7d 08 7c 04 7c 08 66 02 53 00 29 09 61 cb 03 00 00 0a 20 20 20 20 47 61 75 | ...z...}.|.|.f.S.).a.........Gau |
| d220 | 73 73 2d 4c 65 67 65 6e 64 72 65 20 71 75 61 64 72 61 74 75 72 65 2e 0a 0a 20 20 20 20 43 6f 6d | ss-Legendre.quadrature.......Com |
| d240 | 70 75 74 65 73 20 74 68 65 20 73 61 6d 70 6c 65 20 70 6f 69 6e 74 73 20 61 6e 64 20 77 65 69 67 | putes.the.sample.points.and.weig |
| d260 | 68 74 73 20 66 6f 72 20 47 61 75 73 73 2d 4c 65 67 65 6e 64 72 65 20 71 75 61 64 72 61 74 75 72 | hts.for.Gauss-Legendre.quadratur |
| d280 | 65 2e 0a 20 20 20 20 54 68 65 73 65 20 73 61 6d 70 6c 65 20 70 6f 69 6e 74 73 20 61 6e 64 20 77 | e......These.sample.points.and.w |
| d2a0 | 65 69 67 68 74 73 20 77 69 6c 6c 20 63 6f 72 72 65 63 74 6c 79 20 69 6e 74 65 67 72 61 74 65 20 | eights.will.correctly.integrate. |
| d2c0 | 70 6f 6c 79 6e 6f 6d 69 61 6c 73 20 6f 66 0a 20 20 20 20 64 65 67 72 65 65 20 3a 6d 61 74 68 3a | polynomials.of.....degree.:math: |
| d2e0 | 60 32 2a 64 65 67 20 2d 20 31 60 20 6f 72 20 6c 65 73 73 20 6f 76 65 72 20 74 68 65 20 69 6e 74 | `2*deg.-.1`.or.less.over.the.int |
| d300 | 65 72 76 61 6c 20 3a 6d 61 74 68 3a 60 5b 2d 31 2c 20 31 5d 60 20 77 69 74 68 0a 20 20 20 20 74 | erval.:math:`[-1,.1]`.with.....t |
| d320 | 68 65 20 77 65 69 67 68 74 20 66 75 6e 63 74 69 6f 6e 20 3a 6d 61 74 68 3a 60 66 28 78 29 20 3d | he.weight.function.:math:`f(x).= |
| d340 | 20 31 60 2e 0a 0a 20 20 20 20 50 61 72 61 6d 65 74 65 72 73 0a 20 20 20 20 2d 2d 2d 2d 2d 2d 2d | .1`.......Parameters.....------- |
| d360 | 2d 2d 2d 0a 20 20 20 20 64 65 67 20 3a 20 69 6e 74 0a 20 20 20 20 20 20 20 20 4e 75 6d 62 65 72 | ---.....deg.:.int.........Number |
| d380 | 20 6f 66 20 73 61 6d 70 6c 65 20 70 6f 69 6e 74 73 20 61 6e 64 20 77 65 69 67 68 74 73 2e 20 49 | .of.sample.points.and.weights..I |
| d3a0 | 74 20 6d 75 73 74 20 62 65 20 3e 3d 20 31 2e 0a 0a 20 20 20 20 52 65 74 75 72 6e 73 0a 20 20 20 | t.must.be.>=.1.......Returns.... |
| d3c0 | 20 2d 2d 2d 2d 2d 2d 2d 0a 20 20 20 20 78 20 3a 20 6e 64 61 72 72 61 79 0a 20 20 20 20 20 20 20 | .-------.....x.:.ndarray........ |
| d3e0 | 20 31 2d 44 20 6e 64 61 72 72 61 79 20 63 6f 6e 74 61 69 6e 69 6e 67 20 74 68 65 20 73 61 6d 70 | .1-D.ndarray.containing.the.samp |
| d400 | 6c 65 20 70 6f 69 6e 74 73 2e 0a 20 20 20 20 79 20 3a 20 6e 64 61 72 72 61 79 0a 20 20 20 20 20 | le.points......y.:.ndarray...... |
| d420 | 20 20 20 31 2d 44 20 6e 64 61 72 72 61 79 20 63 6f 6e 74 61 69 6e 69 6e 67 20 74 68 65 20 77 65 | ...1-D.ndarray.containing.the.we |
| d440 | 69 67 68 74 73 2e 0a 0a 20 20 20 20 4e 6f 74 65 73 0a 20 20 20 20 2d 2d 2d 2d 2d 0a 20 20 20 20 | ights.......Notes.....-----..... |
| d460 | 54 68 65 20 72 65 73 75 6c 74 73 20 68 61 76 65 20 6f 6e 6c 79 20 62 65 65 6e 20 74 65 73 74 65 | The.results.have.only.been.teste |
| d480 | 64 20 75 70 20 74 6f 20 64 65 67 72 65 65 20 31 30 30 2c 20 68 69 67 68 65 72 20 64 65 67 72 65 | d.up.to.degree.100,.higher.degre |
| d4a0 | 65 73 20 6d 61 79 0a 20 20 20 20 62 65 20 70 72 6f 62 6c 65 6d 61 74 69 63 2e 20 54 68 65 20 77 | es.may.....be.problematic..The.w |
| d4c0 | 65 69 67 68 74 73 20 61 72 65 20 64 65 74 65 72 6d 69 6e 65 64 20 62 79 20 75 73 69 6e 67 20 74 | eights.are.determined.by.using.t |
| d4e0 | 68 65 20 66 61 63 74 20 74 68 61 74 0a 0a 20 20 20 20 2e 2e 20 6d 61 74 68 3a 3a 20 77 5f 6b 20 | he.fact.that.........math::.w_k. |
| d500 | 3d 20 63 20 2f 20 28 4c 27 5f 6e 28 78 5f 6b 29 20 2a 20 4c 5f 7b 6e 2d 31 7d 28 78 5f 6b 29 29 | =.c./.(L'_n(x_k).*.L_{n-1}(x_k)) |
| d520 | 0a 0a 20 20 20 20 77 68 65 72 65 20 3a 6d 61 74 68 3a 60 63 60 20 69 73 20 61 20 63 6f 6e 73 74 | ......where.:math:`c`.is.a.const |
| d540 | 61 6e 74 20 69 6e 64 65 70 65 6e 64 65 6e 74 20 6f 66 20 3a 6d 61 74 68 3a 60 6b 60 20 61 6e 64 | ant.independent.of.:math:`k`.and |
| d560 | 20 3a 6d 61 74 68 3a 60 78 5f 6b 60 0a 20 20 20 20 69 73 20 74 68 65 20 6b 27 74 68 20 72 6f 6f | .:math:`x_k`.....is.the.k'th.roo |
| d580 | 74 20 6f 66 20 3a 6d 61 74 68 3a 60 4c 5f 6e 60 2c 20 61 6e 64 20 74 68 65 6e 20 73 63 61 6c 69 | t.of.:math:`L_n`,.and.then.scali |
| d5a0 | 6e 67 20 74 68 65 20 72 65 73 75 6c 74 73 20 74 6f 20 67 65 74 0a 20 20 20 20 74 68 65 20 72 69 | ng.the.results.to.get.....the.ri |
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| dea0 | 72 69 6e 67 0a 20 20 20 20 20 20 20 20 72 65 70 72 65 73 65 6e 74 61 74 69 6f 6e 73 20 6f 66 20 | ring.........representations.of. |
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 |
index : uiuc-course-graph.git | |
| Unnamed repository; edit this file 'description' to name the repository. | Ubuntu |
| summaryrefslogtreecommitdiff |
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