如何使用 sympy 对未知函数 $f(x)$ 进行符号泰勒展开 [英] How to do a symbolic taylor expansion of an unknown function $f(x)$ using sympy

问题描述

在sage中，对未知函数进行泰勒展开是相当容易的f(x),

In sage it is fairly easy to do a Taylor expansion of an unknown function f(x),

x = var('x')
h = var('h')
f = function('f',x)
g1 = taylor(f,x,h,2)

如何在 sympy 中做到这一点?

How can this be done in sympy?

更新

asmeurer 指出，这是一个很快就会在 sympy 中从拉取请求中提供的功能 http://github.com/sympy/sympy/pull/1888.我使用 pip 安装了分支，

asmeurer points out that this is a feature which will be available soon in sympy from the pull request http://github.com/sympy/sympy/pull/1888. I installed the branch using pip,

pip install -e [e-mail]:renatocoutinho/sympy.git@897b#egg=sympy --upgrade

然而，当我尝试计算f(x)的级数时，

However, when I try to calculate the series of f(x),

x, h = symbols("x,h")
f = Function("f")
series(f,x,x+h)

我收到以下错误，

TypeError: unbound method series() must be called with f instance as第一个参数(取而代之的是 Symbol 实例)

TypeError: unbound method series() must be called with f instance as first argument (got Symbol instance instead)

推荐答案

正如@asmeurer 所描述的，现在可以使用

As @asmeurer described, this is now possible with

from sympy import init_printing, symbols, Function
init_printing()

x, h = symbols("x,h")
f = Function("f")

pprint(f(x).series(x, x0=h, n=3))

或

from sympy import series
pprint(series(f(x), x, x0=h, n=3))

两者都返回

                                              ⎛  2        ⎞│                          
                                            2 ⎜ d         ⎟│                          
                                    (-h + x) ⋅⎜────(f(ξ₁))⎟│                          
                                              ⎜   2       ⎟│                          
                ⎛ d        ⎞│                 ⎝dξ₁        ⎠│ξ₁=h    ⎛        3       ⎞
f(h) + (-h + x)⋅⎜───(f(ξ₁))⎟│     + ──────────────────────────── + O⎝(-h + x) ; x → h⎠
                ⎝dξ₁       ⎠│ξ₁=h                2                                    

如果你想要一个有限差分近似，你可以例如写

If you want a finite difference approximation, you can for example write

FW = f(x+h).series(x+h, x0=x0, n=3)
FW = FW.subs(x-x0,0)
pprint(FW)

得到前向近似，返回

                                  ⎛  2        ⎞│
                                2 ⎜ d         ⎟│
                               h ⋅⎜────(f(ξ₁))⎟│
                                  ⎜   2       ⎟│
          ⎛ d        ⎞│           ⎝dξ₁        ⎠│ξ₁=x₀    ⎛ 3    2        2    3                 ⎞
f(x₀) + h⋅⎜───(f(ξ₁))⎟│      + ────────────────────── + O⎝h  + h ⋅x + h⋅x  + x ; (h, x) → (0, 0)⎠
          ⎝dξ₁       ⎠│ξ₁=x₀             2

这篇关于如何使用 sympy 对未知函数 $f(x)$ 进行符号泰勒展开的文章就介绍到这了，希望我们推荐的答案对大家有所帮助，也希望大家多多支持IT屋！