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Diffstat (limited to 'vis/math_formula.ipynb')
| -rw-r--r-- | vis/math_formula.ipynb | 762 |
1 files changed, 762 insertions, 0 deletions
diff --git a/vis/math_formula.ipynb b/vis/math_formula.ipynb new file mode 100644 index 0000000..6fd5f7e --- /dev/null +++ b/vis/math_formula.ipynb @@ -0,0 +1,762 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T03:12:26.273991Z", + "start_time": "2022-04-16T03:12:26.268549Z" + } + }, + "outputs": [], + "source": [ + "from IPython.display import Math, HTML\n", + "\n", + "def load_mathjax_in_cell_output():\n", + " display(HTML(\"<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>\"))\n", + "get_ipython().events.register('pre_run_cell', load_mathjax_in_cell_output)" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T03:12:37.881284Z", + "start_time": "2022-04-16T03:12:37.013651Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "from sympy import Function, hessian, Matrix, init_printing\n", + "init_printing()" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:22:39.068211Z", + "start_time": "2022-04-16T04:22:39.062261Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "# n=3, m = 4\n", + "from sympy.abc import x, y, z" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:23:50.884011Z", + "start_time": "2022-04-16T04:23:50.878458Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "f = Function('f')(x, y, z)\n", + "g = Function('g')(x, y, z)\n", + "h = Function('h')(x, y, z)\n", + "p = Function('p')(x, y, z)" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:24:02.186209Z", + "start_time": "2022-04-16T04:24:02.181132Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "multi_f = Matrix([f, g, h, p])" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:24:21.767987Z", + "start_time": "2022-04-16T04:24:21.745196Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$\\left[\\begin{matrix}\\frac{\\partial}{\\partial x} f{\\left (x,y,z \\right )} & \\frac{\\partial}{\\partial y} f{\\left (x,y,z \\right )} & \\frac{\\partial}{\\partial z} f{\\left (x,y,z \\right )}\\\\\\frac{\\partial}{\\partial x} g{\\left (x,y,z \\right )} & \\frac{\\partial}{\\partial y} g{\\left (x,y,z \\right )} & \\frac{\\partial}{\\partial z} g{\\left (x,y,z \\right )}\\\\\\frac{\\partial}{\\partial x} h{\\left (x,y,z \\right )} & \\frac{\\partial}{\\partial y} h{\\left (x,y,z \\right )} & \\frac{\\partial}{\\partial z} h{\\left (x,y,z \\right )}\\\\\\frac{\\partial}{\\partial x} p{\\left (x,y,z \\right )} & \\frac{\\partial}{\\partial y} p{\\left (x,y,z \\right )} & \\frac{\\partial}{\\partial z} p{\\left (x,y,z \\right )}\\end{matrix}\\right]$$" + ], + "text/plain": [ + "⎡∂ ∂ ∂ ⎤\n", + "⎢──(f(x, y, z)) ──(f(x, y, z)) ──(f(x, y, z))⎥\n", + "⎢∂x ∂y ∂z ⎥\n", + "⎢ ⎥\n", + "⎢∂ ∂ ∂ ⎥\n", + "⎢──(g(x, y, z)) ──(g(x, y, z)) ──(g(x, y, z))⎥\n", + "⎢∂x ∂y ∂z ⎥\n", + "⎢ ⎥\n", + "⎢∂ ∂ ∂ ⎥\n", + "⎢──(h(x, y, z)) ──(h(x, y, z)) ──(h(x, y, z))⎥\n", + "⎢∂x ∂y ∂z ⎥\n", + "⎢ ⎥\n", + "⎢∂ ∂ ∂ ⎥\n", + "⎢──(p(x, y, z)) ──(p(x, y, z)) ──(p(x, y, z))⎥\n", + "⎣∂x ∂y ∂z ⎦" + ] + }, + "execution_count": 17, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "multi_f.jacobian([x, y, z])" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:25:45.025603Z", + "start_time": "2022-04-16T04:25:45.020531Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "# jacobian, hessian" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:26:25.662601Z", + "start_time": "2022-04-16T04:26:25.657969Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "from sympy import sin" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:26:41.371024Z", + "start_time": "2022-04-16T04:26:41.366746Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "f1 = x**2*y\n", + "f2 = 5*x+sin(y)" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:26:54.987676Z", + "start_time": "2022-04-16T04:26:54.982735Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "multi_f = Matrix([f1, f2])" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:27:13.712719Z", + "start_time": "2022-04-16T04:27:13.699817Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$\\left[\\begin{matrix}2 x y & x^{2}\\\\5 & \\cos{\\left (y \\right )}\\end{matrix}\\right]$$" + ], + "text/plain": [ + "⎡ 2 ⎤\n", + "⎢2⋅x⋅y x ⎥\n", + "⎢ ⎥\n", + "⎣ 5 cos(y)⎦" + ] + }, + "execution_count": 23, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "multi_f.jacobian([x, y])" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:28:25.417289Z", + "start_time": "2022-04-16T04:28:25.398308Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$\\left[\\begin{matrix}2 y & 2 x\\\\2 x & 0\\end{matrix}\\right]$$" + ], + "text/plain": [ + "⎡2⋅y 2⋅x⎤\n", + "⎢ ⎥\n", + "⎣2⋅x 0 ⎦" + ] + }, + "execution_count": 26, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "hessian(Matrix([f1, ]), [x, y])" + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:29:40.067324Z", + "start_time": "2022-04-16T04:29:40.062911Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "# Jacobian 和 Gradient vector" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:29:51.701948Z", + "start_time": "2022-04-16T04:29:51.696999Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "single_f = Matrix([f1, ])" + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:30:02.285106Z", + "start_time": "2022-04-16T04:30:02.273177Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$\\left[\\begin{matrix}2 x y & x^{2}\\end{matrix}\\right]$$" + ], + "text/plain": [ + "⎡ 2⎤\n", + "⎣2⋅x⋅y x ⎦" + ] + }, + "execution_count": 30, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "single_f.jacobian([x, y])" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:30:11.565985Z", + "start_time": "2022-04-16T04:30:11.560948Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "from sympy import derive_by_array" + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:30:24.919642Z", + "start_time": "2022-04-16T04:30:24.899448Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$\\left[\\begin{matrix}\\left[\\left[\\begin{matrix}2 x y\\end{matrix}\\right]\\right] & \\left[\\left[\\begin{matrix}x^{2}\\end{matrix}\\right]\\right]\\end{matrix}\\right]$$" + ], + "text/plain": [ + "⎡ ⎡⎡ 2⎤⎤⎤\n", + "⎣[[2⋅x⋅y]] ⎣⎣x ⎦⎦⎦" + ] + }, + "execution_count": 32, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "derive_by_array(single_f, [x, y])" + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:31:46.066642Z", + "start_time": "2022-04-16T04:31:46.057182Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "f = 0.5*(1-x)**2 + (y-x**2)**2" + ] + }, + { + "cell_type": "code", + "execution_count": 34, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:32:18.968269Z", + "start_time": "2022-04-16T04:32:18.923939Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$\\left[\\begin{matrix}- 4 x \\left(- x^{2} + y\\right) + 1.0 x - 1.0 & - 2 x^{2} + 2 y\\end{matrix}\\right]$$" + ], + "text/plain": [ + "⎡ ⎛ 2 ⎞ 2 ⎤\n", + "⎣- 4⋅x⋅⎝- x + y⎠ + 1.0⋅x - 1.0 - 2⋅x + 2⋅y⎦" + ] + }, + "execution_count": 34, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Matrix([f, ]).jacobian([x, y])" + ] + }, + { + "cell_type": "code", + "execution_count": 35, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:33:17.308460Z", + "start_time": "2022-04-16T04:33:17.291752Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$\\left[\\begin{matrix}12 x^{2} - 4 y + 1.0 & - 4 x\\\\- 4 x & 2\\end{matrix}\\right]$$" + ], + "text/plain": [ + "⎡ 2 ⎤\n", + "⎢12⋅x - 4⋅y + 1.0 -4⋅x⎥\n", + "⎢ ⎥\n", + "⎣ -4⋅x 2 ⎦" + ] + }, + "execution_count": 35, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "hessian(Matrix([f, ]), [x, y])" + ] + }, + { + "cell_type": "code", + "execution_count": 44, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:38:05.525793Z", + "start_time": "2022-04-16T04:38:05.521680Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "from sympy.abc import w\n", + "g = Function('g')(x, y, z)" + ] + }, + { + "cell_type": "code", + "execution_count": 45, + "metadata": { + "ExecuteTime": { + "end_time": "2022-04-16T04:38:08.404704Z", + "start_time": "2022-04-16T04:38:08.383639Z" + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "<script src='https://www.gstatic.com/external_hosted/mathjax/latest/MathJax.js'></script>" + ], + "text/plain": [ + "<IPython.core.display.HTML object>" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$\\left[\\begin{matrix}\\frac{\\partial^{2}}{\\partial x^{2}} g{\\left (x,y,z \\right )} & \\frac{\\partial^{2}}{\\partial x\\partial y} g{\\left (x,y,z \\right )} & \\frac{\\partial^{2}}{\\partial x\\partial z} g{\\left (x,y,z \\right )}\\\\\\frac{\\partial^{2}}{\\partial x\\partial y} g{\\left (x,y,z \\right )} & \\frac{\\partial^{2}}{\\partial y^{2}} g{\\left (x,y,z \\right )} & \\frac{\\partial^{2}}{\\partial y\\partial z} g{\\left (x,y,z \\right )}\\\\\\frac{\\partial^{2}}{\\partial x\\partial z} g{\\left (x,y,z \\right )} & \\frac{\\partial^{2}}{\\partial y\\partial z} g{\\left (x,y,z \\right )} & \\frac{\\partial^{2}}{\\partial z^{2}} g{\\left (x,y,z \\right )}\\end{matrix}\\right]$$" + ], + "text/plain": [ + "⎡ 2 2 2 ⎤\n", + "⎢ ∂ ∂ ∂ ⎥\n", + "⎢ ───(g(x, y, z)) ─────(g(x, y, z)) ─────(g(x, y, z))⎥\n", + "⎢ 2 ∂y ∂x ∂z ∂x ⎥\n", + "⎢ ∂x ⎥\n", + "⎢ ⎥\n", + "⎢ 2 2 2 ⎥\n", + "⎢ ∂ ∂ ∂ ⎥\n", + "⎢─────(g(x, y, z)) ───(g(x, y, z)) ─────(g(x, y, z))⎥\n", + "⎢∂y ∂x 2 ∂z ∂y ⎥\n", + "⎢ ∂y ⎥\n", + "⎢ ⎥\n", + "⎢ 2 2 2 ⎥\n", + "⎢ ∂ ∂ ∂ ⎥\n", + "⎢─────(g(x, y, z)) ─────(g(x, y, z)) ───(g(x, y, z)) ⎥\n", + "⎢∂z ∂x ∂z ∂y 2 ⎥\n", + "⎣ ∂z ⎦" + ] + }, + "execution_count": 45, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "hessian(g, [x, y, z])" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.6.8" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} |