用 sympy 对矩阵值函数进行泰勒展开

Question

我目前正在尝试象征性地表示矩阵指数函数的截断泰勒展开。为此，我将 Python 与 sympy 库一起使用。

我的代码看起来像

import sympy as sy

def taylor_exponential(x: sy.Symbol, order: int):
    res = sy.Integer(1)
    for k in range(1, order):
        res += (x ** k) / sy.factorial(k)
    return res
    # return sy.series(sy.exp(x), x, n=order)

n = sy.symbols("n")
lambd = sy.symbols("lambda")
H0 = sy.MatrixSymbol("H_0", n, n)

print(taylor_exponential(H0 * lambd, 2))

并且在代码结束前引发了以下异常：

Traceback (most recent call last):
  File "mwe.py", line 14, in <module>
    H0 = sy.MatrixSymbol("H_0", n, n)
  File "mwe.py", line 6, in taylor_exponential
    for k in range(1, order):
  File "[path]/lib64/python3.6/site-packages/sympy/core/numbers.py", line 2088, in __add__
    return Rational.__add__(self, other)
# [ More lines of errors ]
  File "[path]/lib64/python3.6/site-packages/sympy/matrices/expressions/matexpr.py", line 95, in __radd__
    return MatAdd(other, self).doit()
  File "[path]/lib64/python3.6/site-packages/sympy/matrices/expressions/matadd.py", line 36, in __new__
    validate(*args)
  File "[path]/lib64/python3.6/site-packages/sympy/matrices/expressions/matadd.py", line 67, in validate
    raise TypeError("Mix of Matrix and Scalar symbols")
TypeError: Mix of Matrix and Scalar symbols

经过一些研究，我在网上找到了一个代码，它按预期工作并且似乎还混合使用了矩阵和标量符号 (source):

from sympy import *

A = MatrixSymbol("A", 3, 3)
x = MatrixSymbol("x", 3, 1)
dt = symbols("dt")
k1 = A * x
k2 = A * (x + S(1) / 2 * k1 * dt)
k3 = A * (x + S(1) / 2 * k2 * dt)
k4 = A * (x + k3 * dt)
final = dt * S(1) / 6 * (k1 + k2 + k3 + k4)  # a matrix expression
ex = Matrix(final)  # an explicit representation of expressions, now in the

我的问题如下：为什么第二个代码片段在我的抛出异常时有效？有没有办法实现我想要做的事情（代表 lambda * H 的泰勒展开式，其中 lambda 是标量符号，H 是矩阵）？

Answer 1

您可以将矩阵乘以标量，但不能将矩阵与标量相加：

In [1]: M = MatrixSymbol('M', 2, 2)                                                                                               

In [2]: M                                                                                                                         
Out[2]: M

In [3]: 2*M                                                                                                                       
Out[3]: 2⋅M

In [4]: 2 + M                                                                                                                     
---------------------------------------------------------------------------
TypeError: Mix of Matrix and Scalar symbols

你的问题是 res 是一个整数。它应该是单位矩阵。我不知道如何创建符号维度的单位矩阵，所以我只使用符号 I:

import sympy as sy

def taylor_exponential(x: sy.Symbol, order: int):
    res = sy.MatrixSymbol('I', *x.shape)
    for k in range(1, order):
        res += (x ** k) / sy.factorial(k)
    return res
    # return sy.series(sy.exp(x), x, n=order)

n = sy.symbols("n")
lambd = sy.symbols("lambda")
H0 = sy.MatrixSymbol("H_0", n, n)

print(taylor_exponential(H0 * lambd, 4))

输出：

lambda*H_0 + (1/2)*(lambda*H_0)**2 + (1/6)*(lambda*H_0)**3 + I

用 sympy 对矩阵值函数进行泰勒展开

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