同情的多元泰勒近似

Question

我的目标是使用sympy哪个来编写多维泰勒近似

• 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近似
• 返回没有 Big-O-remainder项的结果,例如in sin(x)=x - x**3/6 + O(x**4).

这是我到目前为止尝试的内容:

方法1

天真地,人们可以只series为每个变量组合两次命令,遗憾的是这不起作用,如本例所示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,我首先写了一个1D泰勒近似:

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)])

用1D示例检查它可以正常工作

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)

不过,如果我知道这样做既扩张做一个链接调用x和y,sympy挂断

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

有人知道如何解决这个问题或以另一种方式实现它吗？

Answer 1

您可以使用expr.removeO()从表达式中删除大O.

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

Answer 2

这是与 Sympy 一起使用的多元泰勒级数展开式：

def Taylor_polynomial_sympy(function_expression, variable_list, evaluation_point, degree):
    """
    Mathematical formulation reference:
    https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Multivariable_Calculus/3%3A_Topics_in_Partial_Derivatives/Taylor__Polynomials_of_Functions_of_Two_Variables
    :param function_expression: Sympy expression of the function
    :param variable_list: list. All variables to be approximated (to be "Taylorized")
    :param evaluation_point: list. Coordinates, where the function will be expressed
    :param degree: int. Total degree of the Taylor polynomial
    :return: Returns a Sympy expression of the Taylor series up to a given degree, of a given multivariate expression, approximated as a multivariate polynomial evaluated at the evaluation_point
    """
    from sympy import factorial, Matrix, prod
    import itertools

    n_var = len(variable_list)
    point_coordinates = [(i, j) for i, j in (zip(variable_list, evaluation_point))]  # list of tuples with variables and their evaluation_point coordinates, to later perform substitution

    deriv_orders = list(itertools.product(range(degree + 1), repeat=n_var))  # list with exponentials of the partial derivatives
    deriv_orders = [deriv_orders[i] for i in range(len(deriv_orders)) if sum(deriv_orders[i]) <= degree]  # Discarding some higher-order terms
    n_terms = len(deriv_orders)
    deriv_orders_as_input = [list(sum(list(zip(variable_list, deriv_orders[i])), ())) for i in range(n_terms)]  # Individual degree of each partial derivative, of each term

    polynomial = 0
    for i in range(n_terms):
        partial_derivatives_at_point = function_expression.diff(*deriv_orders_as_input[i]).subs(point_coordinates)  # e.g. df/(dx*dy**2)
        denominator = prod([factorial(j) for j in deriv_orders[i]])  # e.g. (1! * 2!)
        distances_powered = prod([(Matrix(variable_list) - Matrix(evaluation_point))[j] ** deriv_orders[i][j] for j in range(n_var)])  # e.g. (x-x0)*(y-y0)**2
        polynomial += partial_derivatives_at_point / denominator * distances_powered
    return polynomial

以下是二变量问题的验证，遵循以下练习和答案：https ://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Multivariable_Calculus/3%3A_Topics_in_Partial_Derivatives/Taylor__Polynomials_of_Functions_of_Two_Variables

# Solving the exercises in section 13.7 of https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Multivariable_Calculus/3%3A_Topics_in_Partial_Derivatives/Taylor__Polynomials_of_Functions_of_Two_Variables
from sympy import symbols, sqrt, atan, ln

# Exercise 1
x = symbols('x')
y = symbols('y')
function_expression = x*sqrt(y)
variable_list = [x,y]
evaluation_point = [1,4]
degree=1
print(Taylor_polynomial_sympy(function_expression, variable_list, evaluation_point, degree))
degree=2
print(Taylor_polynomial_sympy(function_expression, variable_list, evaluation_point, degree))

# Exercise 3
x = symbols('x')
y = symbols('y')
function_expression = atan(x+2*y)
variable_list = [x,y]
evaluation_point = [1,0]
degree=1
print(Taylor_polynomial_sympy(function_expression, variable_list, evaluation_point, degree))
degree=2
print(Taylor_polynomial_sympy(function_expression, variable_list, evaluation_point, degree))

# Exercise 5
x = symbols('x')
y = symbols('y')
function_expression = x**2*y + y**2
variable_list = [x,y]
evaluation_point = [1,3]
degree=1
print(Taylor_polynomial_sympy(function_expression, variable_list, evaluation_point, degree))
degree=2
print(Taylor_polynomial_sympy(function_expression, variable_list, evaluation_point, degree))

# Exercise 7
x = symbols('x')
y = symbols('y')
function_expression = ln(x**2+y**2+1)
variable_list = [x,y]
evaluation_point = [0,0]
degree=1
print(Taylor_polynomial_sympy(function_expression, variable_list, evaluation_point, degree))
degree=2
print(Taylor_polynomial_sympy(function_expression, variable_list, evaluation_point, degree))

simplify()对结果进行执行可能很有用。