Sympy中的多元泰勒近似 [英] Multivariate Taylor approximation in sympy

问题描述

我旨在使用sympy编写多维泰勒逼近，其中

I aim to write a multidimensional Taylor approximation using sympy, which

• 使用尽可能多的内置代码，
• 计算两个变量的给定函数的截断泰勒逼近
• 不使用大零余项返回结果 ，例如在sin(x)=x - x**3/6 + O(x**4)中.

• uses as many builtin code as possible,
• computes the truncated Taylor approximation of a given function of two variables
• returns the result without the Big-O-remainder term, as e.g. in sin(x)=x - x**3/6 + O(x**4).

这是我到目前为止尝试过的:

Here is what I tryed so far:

方法1

天真的，对于每个变量，只需将series命令组合两次，不幸的是，该命令不起作用，如本例所示的功能sin(x*cos(y)):

Naively, one could just combine the series command twice for each variable, which unfortunately does not work, as this example shows for the function sin(x*cos(y)):

sp.sin(x*sp.cos(y)).series(x,x0=0,n=3).series(y,x0=0,n=3)
>>> NotImplementedError: not sure of order of O(y**3) + O(x**3)

方法2

基于这篇文章，我首先写了一维泰勒近似值:

Based on this post I first wrote a 1D taylor approximation:

def taylor_approximation(expr, x, max_order):
    taylor_series = expr.series(x=x, n=None)
    return sum([next(taylor_series) for i in range(max_order)])

使用一维示例进行检查很好

Checking it with 1D examples works fine

mport sympy as sp
x=sp.Symbol('x')
y=sp.Symbol('y')
taylor_approximation(sp.sin(x*sp.cos(y)),x,3)

>>> x**5*cos(y)**5/120 - x**3*cos(y)**3/6 + x*cos(y)

但是，如果我知道要同时在x和y中进行两个扩展，则链式调用会挂起

However, if I know do a chained call for doing both expansions in x and y, sympy hangs up

# this does not work
taylor_approximation(taylor_approximation(sp.sin(x*sp.cos(y)),x,3),y,3)

有人知道如何解决此问题或以其他方式实现它吗?

Does somebody know how to fix this or achieve it in an alternative way?

推荐答案

您可以使用expr.removeO()从表达式中删除大O.

You can use expr.removeO() to remove the big O from an expression.

Oneliner:expr.series(x, 0, 3).removeO().series(y, 0, 3).removeO()

Oneliner: expr.series(x, 0, 3).removeO().series(y, 0, 3).removeO()

这篇关于Sympy中的多元泰勒近似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！