{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Sympy - Python中的符号代数" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "J.R. Johansson (jrjohansson at gmail.com)\n", "\n", "这门课的最新版本[IPython notebook](http://ipython.org/notebook.html) 可以在这里获得[http://github.com/jrjohansson/scientific-python-lectures](http://github.com/jrjohansson/scientific-python-lectures).\n", "\n", "这节课其他的参考书可以在这里找到[http://jrjohansson.github.io](http://jrjohansson.github.io) 。" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "%matplotlib inline\n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 1. 简介" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "有两个值得注意的Python计算机代数系统（CAS）:\n", "\n", "* [SymPy](http://sympy.org/en/index.html) - 这是一个python模块可以被用在任意的Python程序或者是IPython会话中，它提供了强大的计算机代数系统（CAS）特征。\n", "* [Sage](http://www.sagemath.org/) - Sage 是一个全功能的且非常强大的CAS环境，它的目的是提供一个与Mathematica和Maple竞争的开源系统。Sage并不是一个常规的Python模块，而是一个CAS环境用Python作为其编程语言。\n", "\n", "Sage在某些方面比SymPy更加强大，但是他们都提供非常完善的CAS功能。SymPy的优点是，它是一个常规的Python模块而且和IPython notebook集成的很好。\n", "\n", "在这个课上我们将会学会如何运用SymPy和IPython notebooks。如果你对CAS开源环境感兴趣，我也推荐你去阅读更多关于Sage的内容。\n", "\n", "为了开始在Python程序或笔记本中使用SymPy，首先导入模块`sympy`:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [], "source": [ "from sympy import *" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "为了得到一个看起来更好的$\\LaTeX$格式输出运行：" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [], "source": [ "init_printing()\n", "\n", "# 或者使用较旧版本的sympy/ipython，加载ipython扩展\n", "#%load_ext sympy.interactive.ipythonprinting\n", "# 或者\n", "#%load_ext sympyprinting" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2. 符号变量" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "在SymPy中我们需要为需要处理的变量创造符号。我们可以创建一个新的符号用`Symbol`类：" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [], "source": [ "x = Symbol('x')" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\left(x + \\pi\\right)^{2}$" ], "text/plain": [ " 2\n", "(x + π) " ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(pi + x)**2" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [], "source": [ "# 定义符号的可替代的方式\n", "a, b, c = symbols(\"a, b, c\")" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "sympy.core.symbol.Symbol" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "type(a)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "我们可以给符号加上假设当我们创造他们的时候：" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [], "source": [ "x = Symbol('x', real=True)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "False" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x.is_imaginary" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [], "source": [ "x = Symbol('x', positive=True)" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\text{True}$" ], "text/plain": [ "True" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x > 0" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2.1 复数" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "虚数单位在Sympy中被记做`I`。" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle 1 + i$" ], "text/plain": [ "1 + ⅈ" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "1+1*I" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB4AAAASCAYAAABM8m7ZAAAABHNCSVQICAgIfAhkiAAAALNJREFUSInt1L0OQUEQBeCPqNVKUYiOVu0FFHq1UqJU3HfwAgrvofAcEhJRqUSho7C3uUEuWT8JJznZyebsnMzuzBaSJPEJFCPn62GCBfY4YXZNWIpsPEYTB2zQuCWMXfEQdZQxuCeMXfE8rzB2xbnxN165jEBeXh2VPMg21xLHB85vYxl3nk30KL7mjd+G2B9INxAqYW1jGuIdRq8wbqGf2asFwjo1jn3VCQp3WE2Fv9dcZz7QH574iD8PAAAAAElFTkSuQmCC\n", "text/latex": [ "$\\displaystyle -1$" ], "text/plain": [ "-1" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "I**2" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\left(i x + 1\\right)^{2}$" ], "text/plain": [ " 2\n", "(ⅈ⋅x + 1) " ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(x * I + 1)**2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2.2 有理数" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "在SymPy中有三个不同的数学表达形式`Real`, `Rational`, `Integer`:" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [], "source": [ "r1 = Rational(4,5)\n", "r2 = Rational(5,4)" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\frac{4}{5}$" ], "text/plain": [ "4/5" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "r1" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\frac{41}{20}$" ], "text/plain": [ "41\n", "──\n", "20" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "r1+r2" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\frac{16}{25}$" ], "text/plain": [ "16\n", "──\n", "25" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "r1/r2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 3 数值评估" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "SymPy使用任意精度的库作为数值后端，而且预定义了许多数学常数，例如：`pi`, `e`,和用`oo`代表无限。\n", "\n", "为了数值地衡量一个表达，我们可以用`evalf`方程（或者`N`）。它使用了一个参数`n`确定了有效数字的数量。" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle 3.1415926535897932384626433832795028841971693993751$" ], "text/plain": [ "3.1415926535897932384626433832795028841971693993751" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pi.evalf(n=50)" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [], "source": [ "y = (x + pi)**2" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle 9.8696 \\left(0.31831 x + 1\\right)^{2}$" ], "text/plain": [ " 2\n", "9.8696⋅(0.31831⋅x + 1) " ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N(y, 5) # 和evalf 一样" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "当我们数值地衡量代数表达式，我们通常希望带入一个具有数值的符号。在SymPy中我们用`subs`方程来这样做：" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\left(1.5 + \\pi\\right)^{2}$" ], "text/plain": [ " 2\n", "(1.5 + π) " ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "y.subs(x, 1.5)" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle 21.5443823618587$" ], "text/plain": [ "21.5443823618587" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N(y.subs(x, 1.5))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "`subs`当然也可以被用来代入符号和表达：" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\left(a + 2 \\pi\\right)^{2}$" ], "text/plain": [ " 2\n", "(a + 2⋅π) " ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "y.subs(x, a+pi)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "我们也可以将表达式的数值演化和NumPy数组结合：" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [], "source": [ "import numpy" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [], "source": [ "x_vec = numpy.arange(0, 10, 0.1)" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [], "source": [ "y_vec = numpy.array([N(((x + pi)**2).subs(x, xx)) for xx in x_vec])" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "fig, ax = plt.subplots()\n", "ax.plot(x_vec, y_vec);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "然而，这个类型的数值演化会是非常慢的，有一种非常有效的方式去做：使用函数`lambdify`去“编译”一个SymPy表达并代入函数再进行数值计算是非常高效的。" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [], "source": [ "f = lambdify([x], (x + pi)**2, 'numpy') # 第一个参数是一个变量列表\n", " # f将是以下函数的函数:本例中仅为x -> f(x)" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [], "source": [ "y_vec = f(x_vec) # 现在我们可以之间传入一个numpy数组而f(x)会被更有效地计算" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "在使用“speedup”函数而不是直接的进行数值计算时的加速是非常重要的，通常相差几个数量级。即使实在简单的例子中我们也会得到一个明显的加速：" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "10 loops, best of 3: 28.2 ms per loop\n" ] } ], "source": [ "%%timeit\n", "\n", "y_vec = numpy.array([N(((x + pi)**2).subs(x, xx)) for xx in x_vec])" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The slowest run took 8.86 times longer than the fastest. This could mean that an intermediate result is being cached \n", "100000 loops, best of 3: 2.93 µs per loop\n" ] } ], "source": [ "%%timeit\n", "\n", "y_vec = f(x_vec)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 4. 代数操作" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "CAS的一个主要用处是执行表达的代数操作。例如，我们可能想要展开一个乘积，一个表达式的因式，或者只是一个表达式。在本节中演示了在SymPy中进行这些基本操作的功能。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 4.1 展开和乘积" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "代数操作中的第一步：" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\left(x + 1\\right) \\left(x + 2\\right) \\left(x + 3\\right)$$" ], "text/plain": [ "(x + 1)⋅(x + 2)⋅(x + 3)" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(x+1)*(x+2)*(x+3)" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$x^{3} + 6 x^{2} + 11 x + 6$$" ], "text/plain": [ " 3 2 \n", "x + 6⋅x + 11⋅x + 6" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "expand((x+1)*(x+2)*(x+3))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "`expand`函数有一些关键字参数，我们可以告诉函数我们想要执行什么样的扩展。例如，为了扩展三角函数表达式，用`trig=True`的关键字参数：" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\sin{\\left (a + b \\right )}$$" ], "text/plain": [ "sin(a + b)" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sin(a+b)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\sin{\\left (a \\right )} \\cos{\\left (b \\right )} + \\sin{\\left (b \\right )} \\cos{\\left (a \\right )}$$" ], "text/plain": [ "sin(a)⋅cos(b) + sin(b)⋅cos(a)" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "expand(sin(a+b), trig=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "有关`expand`函数可以执行的各种扩展类型的详细说明，请参见`help(expand)`。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "与乘积展开相反的当然就是因式分解了。在SymPy中用`factor`函数来进行因式分解表达：" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\left(x + 1\\right) \\left(x + 2\\right) \\left(x + 3\\right)$$" ], "text/plain": [ "(x + 1)⋅(x + 2)⋅(x + 3)" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "factor(x**3 + 6 * x**2 + 11*x + 6)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 4.2 化简" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "`simplify`尝试使用不同的技巧，去将表达简化到易懂的形式。也有很多可以替代`simplify`的函数：`trigsimp`, `powsimp`, `logcombine`等等。\n", "\n", "这些函数的基本用法如下：" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\left(x + 1\\right) \\left(x + 2\\right) \\left(x + 3\\right)$$" ], "text/plain": [ "(x + 1)⋅(x + 2)⋅(x + 3)" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# simplify 展开乘积\n", "simplify((x+1)*(x+2)*(x+3))" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAJFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAzRAiu5mrdu/dZmiL4QAAAAAjSURBVAgdY2BgEGJgYDDZxMCgEgYkGNhJJVgzdmYB9TEwAACPpQrvlUCHcAAAAABJRU5ErkJggg==\n", "text/latex": [ "$$1$$" ], "text/plain": [ "1" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# simplify 使用三角恒等式\n", "simplify(sin(a)**2 + cos(a)**2)" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\frac{1}{\\tan{\\left (x \\right )}}$$" ], "text/plain": [ " 1 \n", "──────\n", "tan(x)" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "simplify(cos(x)/sin(x))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 4.3 apart 和 together" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "为了处理分数的符号表达式，我们可以用`apart`和`together`函数：" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [], "source": [ "f1 = 1/((a+1)*(a+2))" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAAvBAMAAADdrw/+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "text/latex": [ "$$\\frac{1}{\\left(a + 1\\right) \\left(a + 2\\right)}$$" ], "text/plain": [ " 1 \n", "───────────────\n", "(a + 1)⋅(a + 2)" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f1" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAI4AAAAsBAMAAABBB53eAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nUyuKBLIyAOpdmg617KB4AAAAAElFTkSuQmCC\n", "text/latex": [ "$$- \\frac{1}{a + 2} + \\frac{1}{a + 1}$$" ], "text/plain": [ " 1 1 \n", "- ───── + ─────\n", " a + 2 a + 1" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "apart(f1)" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [], "source": [ "f2 = 1/(a+2) + 1/(a+3)" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHwAAAAsBAMAAABVvsF6AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nnOfh0q8AAAAASUVORK5CYII=\n", "text/latex": [ "$$\\frac{1}{a + 3} + \\frac{1}{a + 2}$$" ], "text/plain": [ " 1 1 \n", "───── + ─────\n", "a + 3 a + 2" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f2" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAAvBAMAAADdrw/+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n", "text/latex": [ "$$\\frac{2 a + 5}{\\left(a + 2\\right) \\left(a + 3\\right)}$$" ], "text/plain": [ " 2⋅a + 5 \n", "───────────────\n", "(a + 2)⋅(a + 3)" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "together(f2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "简化通常结合分数但不分解：" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "ename": "NameError", "evalue": "name 'f2' is not defined", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0msimplify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", "\u001b[0;31mNameError\u001b[0m: name 'f2' is not defined" ] } ], "source": [ "simplify(f2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 5 微积分" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "除了代数操作之外，CAS的另一个主要用途是做微积分，比如代数表达式的导数和积分。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 5.1 微分" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "微分通常是简单的，用`diff`函数。第一个参数是求导的表达式，第二个参数是求导的符号:" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\left(x + \\pi\\right)^{2}$$" ], "text/plain": [ " 2\n", "(x + π) " ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "y" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$4 \\left(x + \\pi\\right)^{3}$$" ], "text/plain": [ " 3\n", "4⋅(x + π) " ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "diff(y**2, x)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "对于高阶导数我们可以这样：" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$12 \\left(x + \\pi\\right)^{2}$$" ], "text/plain": [ " 2\n", "12⋅(x + π) " ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "diff(y**2, x, x)" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$12 \\left(x + \\pi\\right)^{2}$$" ], "text/plain": [ " 2\n", "12⋅(x + π) " ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "diff(y**2, x, 2) # 和上面的一样" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "为了计算多变量表达的导数，我们可以这样做：" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [], "source": [ "x, y, z = symbols(\"x,y,z\")" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [], "source": [ "f = sin(x*y) + cos(y*z)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$\\frac{d^3f}{dxdy^2}$" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$- x \\left(x y \\cos{\\left (x y \\right )} + 2 \\sin{\\left (x y \\right )}\\right)$$" ], "text/plain": [ "-x⋅(x⋅y⋅cos(x⋅y) + 2⋅sin(x⋅y))" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "diff(f, x, 1, y, 2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 6. 积分" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "积分也是以同样的方式进行的:" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\sin{\\left (x y \\right )} + \\cos{\\left (y z \\right )}$$" ], "text/plain": [ "sin(x⋅y) + cos(y⋅z)" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$x \\cos{\\left (y z \\right )} + \\begin{cases} - \\frac{\\cos{\\left (x y \\right )}}{y} & \\text{for}\\: y \\neq 0 \\\\0 & \\text{otherwise} \\end{cases}$$" ], "text/plain": [ " ⎛⎧-cos(x⋅y) ⎞\n", " ⎜⎪────────── for y ≠ 0⎟\n", "x⋅cos(y⋅z) + ⎜⎨ y ⎟\n", " ⎜⎪ ⎟\n", " ⎝⎩ 0 otherwise⎠" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "integrate(f, x)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "通过提供积分变量的极限，我们可以计算定积分:" ] }, { "cell_type": "code", "execution_count": 57, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFAAAAAUBAMAAADo9qfkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQRf3RTl5VjX764CLv8JfPcJj8HGPsdwAAAAASUVORK5CYII=\n", "text/latex": [ "$$2 \\cos{\\left (y z \\right )}$$" ], "text/plain": [ "2⋅cos(y⋅z)" ] }, "execution_count": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "integrate(f, (x, -1, 1))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "还有反常积分" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB0AAAAVBAMAAABI7vhRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAInarRM2ZVBDdiWbv\nuzJCz3LGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAr0lEQVQYGWNggADG/2DwAcplYHaAsSC0KCqX\nIR2VzzEBlc9WgMrvROUyrEblg7TrAx3wDSrMtIGBa12RtKIAkO8CxJwMDM8ZFjI9ADKZjgIJJSBm\nuMDcACQl9B0YGEC28xkAVQFB/wUG7gVAmm0DfwOQYmD7yMAJYvQ38DsAKQbGbwy7QLQmA88CEM1g\n3zADRN1mYF4AohneL08A0zCC9WgDjAmm2WGuhIkGwhhAGgDwdic2xV4k0wAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\sqrt{\\pi}$$" ], "text/plain": [ " ___\n", "╲╱ π " ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(exp(-x**2), (x, -oo, oo))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "记住，`oo`在SymPy中代表无穷。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 6.1 加数和乘积" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "我们可以计算加数和乘积使用函数：'Sum'" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [], "source": [ "n = Symbol(\"n\")" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAADsAAAA9BAMAAADhUgydAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMruZq91U7yJ2iWYQRM2bSrnQAAACDklEQVQ4Ec2VO0gcQRjH/8ednne7eyexsvHAyspXUiWCNiKH4KPSgHqr+CgvKRMidyQQCETu4DQBqyuTJllQDKRa0thY2EggjdeKonfgG3Uy+5idzT6mzlfs/r//j+9jZ2ZnBjDiEZD4WjVlwKNtGPiJxQBiWUvAJNJ6GF+CfI24Go4jd2jSxLgswLS5AMsNpPfDqzEr+PIVoOIZ9wlv9S6XRfLpADeAt3fuzKt/PRFiJP4HfEZ43DgjcD4tT1pNc29tu3DlxzEywUwppzLpVCN3zjxEa0xynCEaM9HHFMcSuWYmEraKjlyMMrPngqnA9y7pDfRtUyGXIozvV7qIp0hNhOXCrQgjT/ZFPEreiDByru5pvoJ1u6g0I6z+XBRhpeGnyf4BZmZUpvj7C8bsRH7FXUdNI1O0ktSQY3KxipI9Gd3cdKu8bmZJPqpmN35uJT+KjrnjKCBp/QWuUSkvXHjL0iWVefKgWfCn68N6GUpVaaVAfs3o+3FrR/R2lukx17G8oFOSejg1Y4quhPlPRtQ5tDRQIMSoG+QLRMw9Jsv3kOoGCo7mOmK1YGS4dCulNT2Ux6vI69VQTOf6m2QMyhv0VqBxQI+1T15Ec+NWEAW9FUThx9Kz9opTEYAPtXgxZkzrMeDHHzdcB7kfI4vfguZ4iccRs/lRUPOWOmbnWTm9FTyhDGFTsz3jVvg3/gK+T7KE+MaJIAAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\sum_{n=1}^{10} \\frac{1}{n^{2}}$$" ], "text/plain": [ " 10 \n", " ____ \n", " ╲ \n", " ╲ 1 \n", " ╲ ──\n", " ╱ 2\n", " ╱ n \n", " ╱ \n", " ‾‾‾‾ \n", "n = 1 " ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Sum(1/n**2, (n, 1, 10))" ] }, { "cell_type": "code", "execution_count": 61, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAPBAMAAAAIUwCQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nt0ciqgAAAABJRU5ErkJggg==\n", "text/latex": [ "$$1.54976773116654$$" ], "text/plain": [ "1.54976773116654" ] }, "execution_count": 61, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Sum(1/n**2, (n,1, 10)).evalf()" ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAPBAMAAAAIUwCQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWbdiTJE\nuyIU2bFIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACpklEQVQ4Ea1TQWsTQRh9223TdLPZ5FBPHmqt\nIEIrS62V9tJQo1Q8ZNEGoShEoQp6aKQeDV0U1JPVgwiCGMRKrZTsQS9ejIIHT1bP1u5FUA9KGmna\nYhvffNMUf4ALeftm3puXmfm+BRqP9fQ5aczTYI49CYEfcx+V/BI4OKtU8Qj0Xzy9pQhzsosBnLn5\nJCeNLqXAvIpTfLUtaOj1nf1wPHT4HD8ELmMg1B5t9NCsFlMR1gfrD77CWAX6RqsUgLiHRb7OqUDC\nMHAEkQoSd4BID6K7YOW1R4yGDycvimaHgd2YSGKNqyM68ENIDucmAxXUgBLsbgxx2bcC4ikYVYhH\nwFqAyb+iotk+IBO8LZtc1gj8pPJgtTKQYG4AEwEn1JGLBSRSsCsQj4C94Vpa0WzKZSCvj0duBNZe\njZeB2ypQwV7ukGPzARANCxjyYK9APNqYWZ3VCoQx55ELDBRJ9JHNmo9OmDlmCZSAKd7Ts5NAPwr4\nVURsXTzaCKs+qRXNAHudTXGPoY3AuosTrgUGCrTmjFKS6uMyiirQU4HKc0CMaH+/rBXN2B4puq3u\n7UBVn+nyThUogBuXSgHVpi4jYKA+sngEoilMaEUYjVn+WGpucavK9xl4vchAU4ESM64TIl4ZBANZ\nFKMC5fEFmkLEtCKMKR7wE1hKbgey8abfpNOZPWcV5Bh4BYkqA9vT6eWj8QVEq6o5p8sCQ9RHRBEG\n7IAT8jL+CXzN+6GrSW2OcC2Mbap32ybH0tgteYhHgPvCF1E04xcbCXtYas6rI7N9m4tmJx0JFUi4\n4B7yEfXRkeOYTXkXvYH2iNGeRFQrmn2fmzmGWzBWgJbh2oiUaGY0YJmWfvsCVlZ9/O+yn4nz9REM\nnj9OJh6BF2NKV4qwqXp9DXb2TJmT//n5CzSjDuDy+pXbAAAAAElFTkSuQmCC\n", "text/latex": [ "$$1.64493406684823$$" ], "text/plain": [ "1.64493406684823" ] }, "execution_count": 62, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Sum(1/n**2, (n, 1, oo)).evalf()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "乘积也是以类似的方式进行的:" ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC8AAAA9BAMAAADPFy0PAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "text/latex": [ "$$\\prod_{n=1}^{10} n$$" ], "text/plain": [ " 10 \n", "┬───┬ \n", "│ │ n\n", "│ │ \n", "n = 1 " ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Product(n, (n, 1, 10)) # 10!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 7. 极限" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "极限可以用`limit`函数来计算。例如：" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAJFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAzRAiu5mrdu/dZmiL4QAAAAAjSURBVAgdY2BgEGJgYDDZxMCgEgYkGNhJJVgzdmYB9TEwAACPpQrvlUCHcAAAAABJRU5ErkJggg==\n", "text/latex": [ "$$1$$" ], "text/plain": [ "1" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "limit(sin(x)/x, x, 0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "我们可以用`limit`去检查`diff`函数的推导结果：" ] }, { "cell_type": "code", "execution_count": 65, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKAAAAAUBAMAAAD4uit9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nXd6gLq79C/gHf4qzfkFAyyAAAAAASUVORK5CYII=\n", "text/latex": [ "$$\\sin{\\left (x y \\right )} + \\cos{\\left (y z \\right )}$$" ], "text/plain": [ "sin(x⋅y) + cos(y⋅z)" ] }, "execution_count": 65, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f" ] }, { "cell_type": "code", "execution_count": 66, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFMAAAAUBAMAAAADwRznAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "text/latex": [ "$$y \\cos{\\left (x y \\right )}$$" ], "text/plain": [ "y⋅cos(x⋅y)" ] }, "execution_count": 66, "metadata": {}, "output_type": "execute_result" } ], "source": [ "diff(f, x)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$\\displaystyle \\frac{\\mathrm{d}f(x,y)}{\\mathrm{d}x} = \\frac{f(x+h,y)-f(x,y)}{h}$" ] }, { "cell_type": "code", "execution_count": 67, "metadata": {}, "outputs": [], "source": [ "h = Symbol(\"h\")" ] }, { "cell_type": "code", "execution_count": 68, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFMAAAAUBAMAAAADwRznAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHaZIu+JVM27RDKr\nZt2dj8xZAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABlUlEQVQoFXWTPUvDUBSG31jbfDUxFBQEh9BK\nXYRmEkXEiBRRUDo46RJaEXEKgrMZHDo6uAiKXXS0/gMFN0EIRZyLCK5qQWg76IlJ9V6jBxLOc86T\n+8G9Af6Lfp/t3LPwO1/kCmmbQx72ORQ8DjnQLA5R5pEl3WYJGOSRpRUWKJdqEHImigYgTFRcDGW3\noY41PGrl6MmML+XNSEhYSIklFKhcdNVV9QmyOQw0iU/oa0d71L1I0DwsJDwcUasKsas7ULtnBkzi\nZyBlpF5FOxLEJoykiTcaoUXtKxd419ujlGKNapAO0BPUF6BuqF1AIR17NaAtbH1YoQoMmJSFQqBO\noq/EjNqag9YhgRZA01zSKxSUJnAM+ZoKVUBMWlA6O8AdMW1LNAroMyKBtoUp1H1qFX1sqOuQnF0D\nh8QVmtk9xXJPSDjASHnWppZwk7cxv3mL88ZDjfiCOtnMDI0SClKwFlrMH8EebCDQ6aU9IdhFLL6v\nSyTQddF92Yx5QSFYcBChEFxCJTv9VYm9elc7FGQ7JvwUYj/MJ3CTXM0O35peAAAAAElFTkSuQmCC\n", "text/latex": [ "$$y \\cos{\\left (x y \\right )}$$" ], "text/plain": [ "y⋅cos(x⋅y)" ] }, "execution_count": 68, "metadata": {}, "output_type": "execute_result" } ], "source": [ "limit((f.subs(x, x+h) - f)/h, h, 0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "OK!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "我们可以改变方向，从我们接近极限点使用`dir`关键字参数:" ] }, { "cell_type": "code", "execution_count": 69, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAALBAMAAABv+6sJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKvvZom7mXYyzVQi\n3UQ6SGZXAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAaklEQVQIHWNgYBBgAAIQwaj82YGBIayogYGB\nbQLHLwapDQxTGRg8GRj2J6xkYGA5wACUYP0LJBgcQEyGfBDRAGYm/wNqd2BwZGDgiDE+wMBxgIGd\ngSGcYb4dgytQolxtAwNjvXEAUDncNgBJUBUwaYAbUgAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\infty$$" ], "text/plain": [ "∞" ] }, "execution_count": 69, "metadata": {}, "output_type": "execute_result" } ], "source": [ "limit(1/x, x, 0, dir=\"+\")" ] }, { "cell_type": "code", "execution_count": 70, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAALBAMAAAAHCCkxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMqvvZom7mXZU\nIkRJD0iWAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfklEQVQIHWNggAMBEAtMQIUYw74VMDB0Lt0A\nV8LA6cD9iUHoAIMHQqiEgeH8BBUGBvYLDELGIKDCANTA8RmkC6gdCkC8+SAChCEAxJr2j4GBsQAm\nwlDIwMDdm3aBgfsCXIiLgaGLwT+PoQIuwsC4KvIAA+P6tAaEENThAgwMAMSLGqu/gFQwAAAAAElF\nTkSuQmCC\n", "text/latex": [ "$$-\\infty$$" ], "text/plain": [ "-∞" ] }, "execution_count": 70, "metadata": {}, "output_type": "execute_result" } ], "source": [ "limit(1/x, x, 0, dir=\"-\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 8. 序列" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "系列扩展也是CAS最有用的特性之一。在SymPy中，我们可以用`series` 函数对表达式进行级数展开:" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + O\\left(x^{6}\\right)$$" ], "text/plain": [ " 2 3 4 5 \n", " x x x x ⎛ 6⎞\n", "1 + x + ── + ── + ── + ─── + O⎝x ⎠\n", " 2 6 24 120 " ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "series(exp(x), x)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "默认情况下，它在$x=0$附近扩展表达式，但是我们可以通过在函数调用中显式包含一个值来扩展$x$的任何值:" ] }, { "cell_type": "code", "execution_count": 72, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAvwAAAAoBAMAAABwRjOsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "text/latex": [ "$$e + e \\left(x - 1\\right) + \\frac{e}{2} \\left(x - 1\\right)^{2} + \\frac{e}{6} \\left(x - 1\\right)^{3} + \\frac{e}{24} \\left(x - 1\\right)^{4} + \\frac{e}{120} \\left(x - 1\\right)^{5} + \\mathcal{O}\\left(\\left(x - 1\\right)^{6}; x\\rightarrow1\\right)$$" ], "text/plain": [ " 2 3 4 5 \n", " ℯ⋅(x - 1) ℯ⋅(x - 1) ℯ⋅(x - 1) ℯ⋅(x - 1) ⎛ 6\n", "ℯ + ℯ⋅(x - 1) + ────────── + ────────── + ────────── + ────────── + O⎝(x - 1) \n", " 2 6 24 120 \n", "\n", " \n", " ⎞\n", "; x → 1⎠\n", " " ] }, "execution_count": 72, "metadata": {}, "output_type": "execute_result" } ], "source": [ "series(exp(x), x, 1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "可以明确定义级数展开的顺序为:" ] }, { "cell_type": "code", "execution_count": 73, "metadata": {}, "outputs": [ { "data": { "image/png": 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2R1/8g1/EK8Me1mh+EX/2nm6NjnGI22FzPdUaRfTfhG8KIK2KKq1RKeCUBnJuh7RU+Iyo\nMXYIBvF5B7LA3VveaXCVng+JRCk03VZftrf1zjKdguJt/PVgIwUOfd3VyxqdTn8q44PDYN091aYQ\nf/EX+FxPAaQ8oRabuEbH5MVGAVcX6klJhfoLN47iPwyGTw6RpbwIG2/kKj2flxL1eYY31ENjYKkH\nsIAO1FNQ7MA12rLHoS8HnzK27ZHiO1bSincru+n8ZltuIMAkfg2I9/UpLrgeA3y2Sr53cQDpVKwA\nXKMsIDsDI6gC9aQMiAnwNwzlY8xSmMHvPrlKyRdI558CvR47V8z0SABwB2C4tuXhdMdR9PWlPXv2\n3lqz9Se/tSat+FzW924tMZGDqwqKdfjiSBvK+IzAttwOcHGFfH+IA0in4u49e955gAWMHqI1Snro\nQXOKgJhgaI6Oo8RCVzKPcpWSL5CPk5qBbk8d23+4IsJZqQ4D9mt0Aa9HI8i6DElfhXYXu6jd9OyJ\n8ftBvuKKsjKPlfA8TS8izyjPQZqHIXWAHfR4nF6vYQApVQDwM/zBJaTIv8l6UlIxQQmvR+eYBQ+h\n8ApXKfkwLFWy8k2eGknfGmmlx0rk5eNrf2NPMlR1brFHSV+jbXson+sZn5/Hf/3EvhRxjWavG980\niz8bWtW0x18K5e+C9I0BpFQB8D6wgHILdteEnnRUTAC/hlXzzJJ7Bso1rtJL8yXly752j80P9ohH\n+EudDt4AWZe1m5rWGOkr8/rRljWWv5hjrc6aDSlc0w8TYRhf2s/i/zXiEWv3uL6nHqyA8E0BpFQB\nn+g8KgRcM/2q1JOSigggM/2IZLlv6jwArlLypcjJccjxDPz/ZOB/nQjAqqJ2ecoAAAAASUVORK5C\nYII=\n", "text/latex": [ "$$e + e \\left(x - 1\\right) + \\frac{e}{2} \\left(x - 1\\right)^{2} + \\frac{e}{6} \\left(x - 1\\right)^{3} + \\frac{e}{24} \\left(x - 1\\right)^{4} + \\frac{e}{120} \\left(x - 1\\right)^{5} + \\frac{e}{720} \\left(x - 1\\right)^{6} + \\frac{e}{5040} \\left(x - 1\\right)^{7} + \\frac{e}{40320} \\left(x - 1\\right)^{8} + \\frac{e}{362880} \\left(x - 1\\right)^{9} + \\mathcal{O}\\left(\\left(x - 1\\right)^{10}; x\\rightarrow1\\right)$$" ], "text/plain": [ " 2 3 4 5 6\n", " ℯ⋅(x - 1) ℯ⋅(x - 1) ℯ⋅(x - 1) ℯ⋅(x - 1) ℯ⋅(x - 1) \n", "ℯ + ℯ⋅(x - 1) + ────────── + ────────── + ────────── + ────────── + ──────────\n", " 2 6 24 120 720 \n", "\n", " 7 8 9 \n", " ℯ⋅(x - 1) ℯ⋅(x - 1) ℯ⋅(x - 1) ⎛ 10 ⎞\n", " + ────────── + ────────── + ────────── + O⎝(x - 1) ; x → 1⎠\n", " 5040 40320 362880 " ] }, "execution_count": 73, "metadata": {}, "output_type": "execute_result" } ], "source": [ "series(exp(x), x, 1, 10)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "级数展开式包含了近似的阶数，当我们对不同阶数的级数展开式进行计算时，这对于跟踪有效性的阶数非常有用:" ] }, { "cell_type": "code", "execution_count": 74, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMQAAAAwBAMAAAC8i8hXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n", "text/latex": [ "$$1 - \\frac{x^{2}}{2} + \\frac{x^{4}}{24} + \\mathcal{O}\\left(x^{5}\\right)$$" ], "text/plain": [ " 2 4 \n", " x x ⎛ 5⎞\n", "1 - ── + ── + O⎝x ⎠\n", " 2 24 " ] }, "execution_count": 74, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s1 = cos(x).series(x, 0, 5)\n", "s1" ] }, { "cell_type": "code", "execution_count": 75, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF8AAAAcBAMAAAD1rn4EAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n", "text/latex": [ "$$x + \\mathcal{O}\\left(x^{2}\\right)$$" ], "text/plain": [ " ⎛ 2⎞\n", "x + O⎝x ⎠" ] }, "execution_count": 75, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s2 = sin(x).series(x, 0, 2)\n", "s2" ] }, { "cell_type": "code", "execution_count": 76, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF8AAAAcBAMAAAD1rn4EAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACCUlEQVQ4EY1Sv2tTURT+Xsx7SV5+9FKh0MnX\nCm61obhYB99SHBv8A0wUwbFO6qJDUcShtCiCwcEIuuhglg7i0Afi4CBJ7SDSwS4OTiYiCCLEc867\n971cSfFdOOd833fOx7l5N0DG4+3vZpzUY5dxOpvhnR67jV6Yy2CpdPXQHaw23Pr/HSWVzLxSuJKQ\nf4C7O7P5ULSPaWcdKKb2VGe04G5/GDUI5IdJw+sDOYpJ50hwQWFxi1o8pc9NqvmBYVL9QNM95ysw\nxc1aR0twW24E3DVUqjG4w2oXKPFtespMvH/xkvC+oZah2PeJl3nDrWTg+2hE+CTF9PGVE4HoZkMz\nYlo+oPSZwpk/df4SVT5NBadVWC9uCTOGL3KPWp/ENxQzuBHckwG6YgRPeUM/FG4MO8J6DSoPKC5i\nTT0TCfTacFDqMvPa7Uff2u06wSeKhdecNigUHjOUs9qhMhUIBsyGsxEJlZ+s8gbgh2ROtEHuFQvG\nsMZrZrkVGyp/4j7lXghfLSInd0g2HPtFb7otQ2coP80NMKct9JWanR2ci6nZUBgtr+yFotF3d3+X\nB/mr2kDvcnR++pOmxoC3mxtRPMHffWlu4bqeP+SlTZdqTX5JKtxPISEntCiTasuSnPTfbukpqRyk\nmFChbtFJZNkSi5FFJ5FZS7xmsYnE74/J+e4YOQw+H2tU1RjJCv8Cl4Bl3Hr5MZEAAAAASUVORK5C\nYII=\n", "text/latex": [ "$$x + \\mathcal{O}\\left(x^{2}\\right)$$" ], "text/plain": [ " ⎛ 2⎞\n", "x + O⎝x ⎠" ] }, "execution_count": 76, "metadata": {}, "output_type": "execute_result" } ], "source": [ "expand(s1 * s2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "如果我们想摆脱阶数信息，我们可以使用`removeO`方法:" ] }, { "cell_type": "code", "execution_count": 77, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAG8AAAAwBAMAAADtMzlxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACRUlEQVRIDc2Vv2vUYBjHv8klubt4XsMVOqdV\nCg6HtS3oIJhBORfx8B/o6XCLSG/TSQ8VsTjo2MOhFR0VuiuYyU0aFNqhFBwVRe78hagYn8vv5p6n\noA76Dnmf5/t9PnnfhDxvgGBUjjY6YZS/GvVDeSmb6/7HbJqJN7GVyUZC/aE1ooXCBq4ITiDrTdnd\nfcUXdWlJ5Z18T0BrFtcE/4AtGJFc+ST5u261YCmfJXB9VXJIrzragLXpCU93WCcUDcvwePs6Lguv\nrbbvxLSN9sYIp0zNnzmHZ9PzI04gKK3i1dIdzpvAJXuJM0KNNjkwHc4/i0XrAWeEmoLyGu9auMsb\nsTpmx1F+/pAXdubr7s48ySo/kpAJTOsgVPaF31P7mGSISFpYfYoGZ+vf9vS1DueE2vhUbZO1ldnJ\n+kWZ+78d0/7D/f07MOyn39h2tFW5n6R7RWDUT346CFCWb9JYcikcS40+pUavt/ym15uhGrGfqIwd\nyVuV+4nlkIBiP/FcDMr9JHAxKPaTxMVg3E+ntp4MS59z9dr2YzfVk2cMJGUGi02KbqUFaTQOI/Nn\nUJzUAQoWqq8A9W1WjGP6i7+P4/xc7aJAh1KDPQpfA8fcPBHl5UEAtlhwxZJB4ksDFB0WJPO2JaxI\n8kILNenU1r/IHK4BLQks0Vctjb0eTFcCtyWK9DowAQFUPRk0PRy/Pzf3c5YreQTN4fShRt/afprY\nD6DiQXXI5IZ2o/2yS8Z3zjzZvnCE04damU6ILnDeP8xUrPj+V0b+K+kXPYuKl+Zu0xMAAAAASUVO\nRK5CYII=\n", "text/latex": [ "$$\\frac{x^{5}}{24} - \\frac{x^{3}}{2} + x$$" ], "text/plain": [ " 5 3 \n", "x x \n", "── - ── + x\n", "24 2 " ] }, "execution_count": 77, "metadata": {}, "output_type": "execute_result" } ], "source": [ "expand(s1.removeO() * s2.removeO())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "但请注意，这不是$\\cos(x)\\sin(x)$到$5$的正确扩展:" ] }, { "cell_type": "code", "execution_count": 78, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAANsAAAAwBAMAAABqLhIyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "text/latex": [ "$$x - \\frac{2 x^{3}}{3} + \\frac{2 x^{5}}{15} + \\mathcal{O}\\left(x^{6}\\right)$$" ], "text/plain": [ " 3 5 \n", " 2⋅x 2⋅x ⎛ 6⎞\n", "x - ──── + ──── + O⎝x ⎠\n", " 3 15 " ] }, "execution_count": 78, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(cos(x)*sin(x)).series(x, 0, 6)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 9. 线性代数" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 9.1 矩阵" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "使用`Matrix`类可以定义矩阵：" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [], "source": [ "m11, m12, m21, m22 = symbols(\"m11, m12, m21, m22\")\n", "b1, b2 = symbols(\"b1, b2\")" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\left[\\begin{matrix}m_{11} & m_{12}\\\\m_{21} & m_{22}\\end{matrix}\\right]$$" ], "text/plain": [ "⎡m₁₁ m₁₂⎤\n", "⎢ ⎥\n", "⎣m₂₁ m₂₂⎦" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A = Matrix([[m11, m12],[m21, m22]])\n", "A" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\left[\\begin{matrix}b_{1}\\\\b_{2}\\end{matrix}\\right]$$" ], "text/plain": [ "⎡b₁⎤\n", "⎢ ⎥\n", "⎣b₂⎦" ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" } ], "source": [ "b = Matrix([[b1], [b2]])\n", "b" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "有`Matrix`类实例的基础上我们可以做常规的矩阵代数运算：" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\left[\\begin{matrix}m_{11}^{2} + m_{12} m_{21} & m_{11} m_{12} + m_{12} m_{22}\\\\m_{11} m_{21} + m_{21} m_{22} & m_{12} m_{21} + m_{22}^{2}\\end{matrix}\\right]$$" ], "text/plain": [ "⎡ 2 ⎤\n", "⎢ m₁₁ + m₁₂⋅m₂₁ m₁₁⋅m₁₂ + m₁₂⋅m₂₂⎥\n", "⎢ ⎥\n", "⎢ 2 ⎥\n", "⎣m₁₁⋅m₂₁ + m₂₁⋅m₂₂ m₁₂⋅m₂₁ + m₂₂ ⎦" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A**2" ] }, { "cell_type": "code", "execution_count": 83, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJEAAAAyBAMAAACufiRQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nDbOYk3PHxJxH96vsyH4p/gHeHTH4ApAfnQAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\left[\\begin{matrix}b_{1} m_{11} + b_{2} m_{12}\\\\b_{1} m_{21} + b_{2} m_{22}\\end{matrix}\\right]$$" ], "text/plain": [ "⎡b₁⋅m₁₁ + b₂⋅m₁₂⎤\n", "⎢ ⎥\n", "⎣b₁⋅m₂₁ + b₂⋅m₂₂⎦" ] }, "execution_count": 83, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A * b" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "计算行列式和逆，诸如此类:" ] }, { "cell_type": "code", "execution_count": 84, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKYAAAAMBAMAAAAaIdvMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n9AyNtKqM6qgiUQAAAABJRU5ErkJggg==\n", "text/latex": [ "$$m_{11} m_{22} - m_{12} m_{21}$$" ], "text/plain": [ "m₁₁⋅m₂₂ - m₁₂⋅m₂₁" ] }, "execution_count": 84, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A.det()" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\left[\\begin{matrix}\\frac{m_{22}}{m_{11} m_{22} - m_{12} m_{21}} & - \\frac{m_{12}}{m_{11} m_{22} - m_{12} m_{21}}\\\\- \\frac{m_{21}}{m_{11} m_{22} - m_{12} m_{21}} & \\frac{m_{11}}{m_{11} m_{22} - m_{12} m_{21}}\\end{matrix}\\right]$$" ], "text/plain": [ "⎡ m₂₂ -m₁₂ ⎤\n", "⎢───────────────── ─────────────────⎥\n", "⎢m₁₁⋅m₂₂ - m₁₂⋅m₂₁ m₁₁⋅m₂₂ - m₁₂⋅m₂₁⎥\n", "⎢ ⎥\n", "⎢ -m₂₁ m₁₁ ⎥\n", "⎢───────────────── ─────────────────⎥\n", "⎣m₁₁⋅m₂₂ - m₁₂⋅m₂₁ m₁₁⋅m₂₂ - m₁₂⋅m₂₁⎦" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A.inv()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 10. 求解方程" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "对于求解方程和方程组我们可以用`solve`函数：" ] }, { "cell_type": "code", "execution_count": 86, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEoAAAAUBAMAAADYerbFAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAdt3NMolEEFTvq5lmIsfa\npuIAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAB3SURBVCgVYxAyYSAEDqsxhBFSA5QPQ1bFMR2Ljs4C\nVFVVu55jqGJfvQ5NFQMjpioGBrlBrqpYCQSAgTuYXV9ngBb+6KHKvfLxGgYmdVRVu+ZZHUCNbbB8\nL6oqMA8lTYBFAohRxS5AjCoeLIqA7hJSwSaOIiakBgD0ZimSClhTrgAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\left [ -1, \\quad 1\\right ]$$" ], "text/plain": [ "[-1, 1]" ] }, "execution_count": 86, "metadata": {}, "output_type": "execute_result" } ], "source": [ "solve(x**2 - 1, x)" ] }, { "cell_type": "code", "execution_count": 87, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAjEAAABLCAMAAAC2lyZIAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nUA9GUf8DYe6QJALMc1YAAAAASUVORK5CYII=\n", "text/latex": [ "$$\\left [ - i \\sqrt{- \\frac{1}{2} + \\frac{\\sqrt{5}}{2}}, \\quad i \\sqrt{- \\frac{1}{2} + \\frac{\\sqrt{5}}{2}}, \\quad - \\sqrt{\\frac{1}{2} + \\frac{\\sqrt{5}}{2}}, \\quad \\sqrt{\\frac{1}{2} + \\frac{\\sqrt{5}}{2}}\\right ]$$" ], "text/plain": [ "⎡ _____________ _____________ ___________ ________\n", "⎢ ╱ ___ ╱ ___ ╱ ___ ╱ _\n", "⎢ ╱ 1 ╲╱ 5 ╱ 1 ╲╱ 5 ╱ 1 ╲╱ 5 ╱ 1 ╲╱ \n", "⎢-ⅈ⋅ ╱ - ─ + ───── , ⅈ⋅ ╱ - ─ + ───── , - ╱ ─ + ───── , ╱ ─ + ───\n", "⎣ ╲╱ 2 2 ╲╱ 2 2 ╲╱ 2 2 ╲╱ 2 2\n", "\n", "___⎤\n", "__ ⎥\n", "5 ⎥\n", "── ⎥\n", " ⎦" ] }, "execution_count": 87, "metadata": {}, "output_type": "execute_result" } ], "source": [ "solve(x**4 - x**2 - 1, x)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "方程组：" ] }, { "cell_type": "code", "execution_count": 88, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH4AAAAVBAMAAAByPkciAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/1sDfgDNr2XH9LJoSQAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\left \\{ x : 1, \\quad y : 0\\right \\}$$" ], "text/plain": [ "{x: 1, y: 0}" ] }, "execution_count": 88, "metadata": {}, "output_type": "execute_result" } ], "source": [ "solve([x + y - 1, x - y - 1], [x,y])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "在其他符号表达方面:" ] }, { "cell_type": "code", "execution_count": 89, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOAAAAAmBAMAAAAvsop7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n4r9Q1gIGAlZHdQAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\left \\{ x : \\frac{a}{2} + \\frac{c}{2}, \\quad y : \\frac{a}{2} - \\frac{c}{2}\\right \\}$$" ], "text/plain": [ "⎧ a c a c⎫\n", "⎨x: ─ + ─, y: ─ - ─⎬\n", "⎩ 2 2 2 2⎭" ] }, "execution_count": 89, "metadata": {}, "output_type": "execute_result" } ], "source": [ "solve([x + y - a, x - y - c], [x,y])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 进一步的阅读" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* http://sympy.org/en/index.html - The SymPy projects web page.\n", "* https://github.com/sympy/sympy - The source code of SymPy.\n", "* http://live.sympy.org - Online version of SymPy for testing and demonstrations." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 版本" ] }, { "cell_type": "code", "execution_count": 90, "metadata": {}, "outputs": [ { "data": { "application/json": { "Software versions": [ { "module": "Python", "version": "2.7.10 64bit [GCC 4.2.1 (Apple Inc. build 5577)]" }, { "module": "IPython", "version": "3.2.1" }, { "module": "OS", "version": "Darwin 14.1.0 x86_64 i386 64bit" }, { "module": "numpy", "version": "1.9.2" }, { "module": "matplotlib", "version": "1.4.3" }, { "module": "sympy", "version": "0.7.6" } ] }, "text/html": [ "
SoftwareVersion
Python2.7.10 64bit [GCC 4.2.1 (Apple Inc. build 5577)]
IPython3.2.1
OSDarwin 14.1.0 x86_64 i386 64bit
numpy1.9.2
matplotlib1.4.3
sympy0.7.6
Sat Aug 15 11:37:37 2015 JST
" ], "text/latex": [ "\\begin{tabular}{|l|l|}\\hline\n", "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n", "Python & 2.7.10 64bit [GCC 4.2.1 (Apple Inc. build 5577)] \\\\ \\hline\n", "IPython & 3.2.1 \\\\ \\hline\n", "OS & Darwin 14.1.0 x86\\_64 i386 64bit \\\\ \\hline\n", "numpy & 1.9.2 \\\\ \\hline\n", "matplotlib & 1.4.3 \\\\ \\hline\n", "sympy & 0.7.6 \\\\ \\hline\n", "\\hline \\multicolumn{2}{|l|}{Sat Aug 15 11:37:37 2015 JST} \\\\ \\hline\n", "\\end{tabular}\n" ], "text/plain": [ "Software versions\n", "Python 2.7.10 64bit [GCC 4.2.1 (Apple Inc. build 5577)]\n", "IPython 3.2.1\n", "OS Darwin 14.1.0 x86_64 i386 64bit\n", "numpy 1.9.2\n", "matplotlib 1.4.3\n", "sympy 0.7.6\n", "Sat Aug 15 11:37:37 2015 JST" ] }, "execution_count": 90, "metadata": {}, "output_type": "execute_result" } ], "source": [ "%reload_ext version_information\n", "\n", "%version_information numpy, matplotlib, sympy" ] } ], "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": 1 }