Python Sympy 任意近似于任意 Sympy 表达式？

Question

我发现自己想使用作为 mpmath 包的一部分提供的近似值，但对它们应该做什么感到困惑：

http://docs.sympy.org/dev/modules/mpmath/calculus/approximation.html

sympy 表达式和 sympy.mpmath 表达式之间到底有什么区别？

如果我想要一个符号表达式的泰勒近似而不理解 mpmath 包在做什么，我可以执行以下操作：

#Imports
import sympy
import sympy.parsing
import sympy.parsing.sympy_parser
import Library_TaylorApproximation

#Create a sympy expression to approximate
ExampleStringExpression = 'sin(x)'
ExampleSympyExpression = sympy.parsing.sympy_parser.parse_expr(ExampleStringExpression)


#Create a taylor expantion sympy expression around the point x=0
SympyTaylorApproximation = sympy.series( 
    ExampleSympyExpression,
    sympy.Symbol('x'),
    1, 
    4,
    ).removeO()

#Cast the sympy expressions to python functions which can be evaluated:
VariableNames = [str(var) for var in SympyTaylorApproximation.free_symbols]
PythonFunctionOriginal =  sympy.lambdify(VariableNames, ExampleSympyExpression)
PythonFunctionApproximation = sympy.lambdify(VariableNames, SympyTaylorApproximation)

#Evaluate the approximation and the original at a point:
print PythonFunctionOriginal(2)
print PythonFunctionApproximation(2)

#>>> 0.909297426826
#>>> 0.870987413961

但是，如果我尝试根据文档对 mpmath 执行相同的操作：

TaylorCoefficients = sympy.mpmath.taylor(ExampleSympyExpression, 1, 4 )
print 'TaylorCoefficients', TaylorCoefficients

#>>> TypeError: 'sin' object is not callable

我可以尝试将 python 函数塞入其中（可调用）：

TaylorCoefficients = sympy.mpmath.taylor(PythonFunctionOriginal, 1, 4 )
print 'TaylorCoefficients', TaylorCoefficients

#>>> TaylorCoefficients [mpf('0.8414709848078965'), mpf('0.0'), mpf('0.0'), mpf('0.0'), mpf('-8.3694689805155739e+57')]

但是以上没有任何意义，因为我知道不能对python函数进行导数。

我可以调用 mpmath 函数sin ：

TaylorCoefficients = sympy.mpmath.taylor(sympy.mpmath.sin, 1, 4 )
print 'TaylorCoefficients', TaylorCoefficients
#>>> TaylorCoefficients [mpf('0.8414709848078965'), mpf('0.54030230586813977'), mpf('-0.42073549240394825'), mpf('-0.090050384311356632'), mpf('0.035061291033662352')]

但是，我也无法按照我想要的方式对其进行操作 -> 喜欢如果我想要

SinTimesCos = sympy.mpmath.sin*sympy.mpmath.cos
TaylorCoefficients = sympy.mpmath.taylor(SinTimesCos, 1, 4 )
print 'TaylorCoefficients', TaylorCoefficients
#>>> TypeError: unsupported operand type(s) for *: 'function' and 'function'

究竟什么是 mpmath 函数？

它不是一个 sympy 表达式，也不是一个 python 函数。 如何对任意表达式进行操作？

看来我不能在文档中对任意 sympy 表达式进行近似处理。 http://docs.sympy.org/dev/modules/mpmath/calculus/approximation.html

如何对任意 sympy 表达式进行任意近似（Pade / Cheby Chev / Fourier）？

编辑：

所以我正在寻找的一个例子是以下近似值：

#Start with a sympy expression of (a, b, x)
expressionString = 'cos(a*x)*sin(b*x)*(x**2)'
expressionSympy = sympy.parsing.sympy_parser.parse_expr(expressionString)

#Do not want to decide on value of `a or b` in advance.
#Do want approximation with respect to x:

wantedSympyExpression = SympyChebyChev( expressionSympy, sympy.Symbol('x') ) 

结果可能或者是系数表达式是功能的列表a ，和b ：

wantedSympyExpressionCoefficients = [ Coef0Expression(a,b), Coef1Expression(a,b), ... , CoefNExpression(a,b)]

或者结果可能是整个 sympy 表达式本身（它本身是a ， b的函数）：

wantedSympyExpression = Coef0Expression(a,b) + Coef1Expression(a,b) *(x**2) + ... + CoefNExpression(a,b) (x**N)

请注意， a和b不是在执行近似之前选择的。

Answer 1

mpmath 函数是普通的 Python 函数。 他们只是在任意精度算术中进行数学运算。

但是以上没有任何意义，因为我知道不能对python函数进行导数。

您不能象征性地取导数，但您可以通过多次评估函数并使用数值微分技术来计算导数的近似值。 这就是sympy.mpmath.taylor所做的。 引用文档：

系数是使用高阶数值微分计算的。 该函数必须可以评估为任意精度。

Answer 2

如果您有一个 SymPy 表达式并希望将其计算为任意精度，请使用evalf ，例如

sympy.sin(1).evalf(100)

在评估之前sin(x).evalf(100, subs={x:1})您可以使用sin(x).evalf(100, subs={x:1})将x替换为1 。 evalf在evalf使用 mpmath，所以这会给你与 mpmath 相同的结果，但不必直接使用 mpmath。

Answer 3

编辑：重读我的答案 -> 我想我会填写一些缺失的部分作为服务，有一天我会真正使用它。 下面我标记了我如何命名我的库，以及需要什么导入。 我目前没有时间成为 sympy 的真正贡献者，但我觉得这个功能肯定会被其他数学/物理教授/学生使用。

请注意，出于空间原因，省略了以下两个库，我将在以后的某个日期抛出一个指向我的存储库的链接。

import Library_SympyExpressionToPythonFunction

在 sympy 表达式中创建一个具有相同参数（数字和名称）的自由变量的 python 可调用函数对象。

import Library_SympyExpressionToStringExpression

从字面上看只是 str(SympyExpression)

#-------------------------------------------------------------------------------

Library_GenerateChebyShevPolynomial:::

#-------------------------------------------------------------------------------


import pprint
import Library_SympyExpressionToPythonFunction
import Library_SympyExpressionToStringExpression
import sympy
import sympy.core

def Main(
    ApproximationSymbol = sympy.Symbol('x'),
    ResultType = 'sympy',
    Kind= None,
    Order= None,
    ReturnAll = False,
    CheckArguments = True,
    PrintExtra = False,
    ):

    Result = None

    if (CheckArguments):
        ArgumentErrorMessage = ""

        if (len(ArgumentErrorMessage) > 0 ):
            if(PrintExtra):
                print "ArgumentErrorMessage:\n", ArgumentErrorMessage
            raise Exception(ArgumentErrorMessage)

    ChebyChevPolynomials = []
    ChebyChevPolynomials.append(sympy.sympify(1.))
    ChebyChevPolynomials.append(ApproximationSymbol)

    #Generate the polynomial with sympy:
    for Term in range(Order + 1)[2:]:
        Tn = ChebyChevPolynomials[Term - 1]
        Tnminus1 = ChebyChevPolynomials[Term - 2]
        Tnplus1 = 2*ApproximationSymbol*Tn - Tnminus1

        ChebyChevPolynomials.append(Tnplus1.simplify().expand().trigsimp())

    if(PrintExtra): print 'ChebyChevPolynomials'
    if(PrintExtra): pprint.pprint(ChebyChevPolynomials)


    if (ReturnAll):
        Result = []
        for SympyChebyChevPolynomial in ChebyChevPolynomials:
            if (ResultType == 'python'):
                Result.append(Library_SympyExpressionToPythonFunction.Main(SympyChebyChevPolynomial))
            elif (ResultType == 'string'):
                Result.append(Library_SympyExpressionToStringExpression.Main(SympyChebyChevPolynomial))
            else:
                Result.append(SympyChebyChevPolynomial)

    else:
        SympyExpression = ChebyChevPolynomials[Order] #the last one

        #If the result type is something other than sympy, we can cast it into that type here:
        if (ResultType == 'python'):
            Result = Library_SympyExpressionToPythonFunction.Main(SympyExpression)
        elif (ResultType == 'string'):
            Result = Library_SympyExpressionToStringExpression.Main(SympyExpression)
        else:
            Result = SympyExpression



    return Result 


#-------------------------------------------------------------------------------

Library_SympyChebyShevApproximationOneDimension

#-------------------------------------------------------------------------------


import numpy
import sympy
import sympy.mpmath
import pprint
import Library_SympyExpressionToPythonFunction
import Library_GenerateChebyShevPolynomial

def Main(
    SympyExpression= None,
    DomainMinimumPoint= None,
    DomainMaximumPoint= None,
    ApproximationOrder= None,
    CheckArguments = True,
    PrintExtra = False,
    ):

    #Tsymb = sympy.Symbol('t')
    Xsymb = sympy.Symbol('x')
    DomainStart = DomainMinimumPoint[0]
    print 'DomainStart', DomainStart
    DomainEnd = DomainMaximumPoint[0]
    print 'DomainEnd', DomainEnd

    #Transform the coefficients and the result to be on arbitrary inverval instead of from 0 to 1
    DomainWidth = DomainEnd - DomainStart
    DomainCenter = (DomainEnd - DomainStart) / 2.
    t = (Xsymb*(DomainWidth) + DomainStart + DomainEnd) / 2.
    x = (2.*Xsymb - DomainStart - DomainEnd) / (DomainWidth)
    SympyExpression = SympyExpression.subs(Xsymb, t)

    #GET THE COEFFICIENTS:
    Coefficients = []
    for CoefficientNumber in range(ApproximationOrder):
        if(PrintExtra): print 'CoefficientNumber', CoefficientNumber

        Coefficient = 0.0
        for k in range(1, ApproximationOrder + 1):
            if(PrintExtra): print '  k', k

            CoefficientFunctionPart = SympyExpression.subs(Xsymb, sympy.cos( sympy.pi*( float(k) - .5 )/ float(ApproximationOrder) )  )
            if(PrintExtra): print '  CoefficientFunctionPart', CoefficientFunctionPart

            CeofficientCosArg = float(CoefficientNumber)*( float(k) - .5 )/ float( ApproximationOrder)
            if(PrintExtra): print '  ',CoefficientNumber,'*','(',k,'-.5)/(', ApproximationOrder ,') == ', CeofficientCosArg

            CoefficientCosPart      =   sympy.cos( sympy.pi*CeofficientCosArg )
            if(PrintExtra): print '  CoefficientCosPart', CoefficientCosPart

            Coefficient += CoefficientFunctionPart*CoefficientCosPart

        if(PrintExtra): print 'Coefficient==', Coefficient

        Coefficient = (2./ApproximationOrder)*Coefficient.evalf(10)

        if(PrintExtra): print 'Coefficient==', Coefficient

        Coefficients.append(Coefficient)

    print '\n\nCoefficients'
    pprint.pprint( Coefficients )


    #GET THE POLYNOMIALS:
    ChebyShevPolynomials = Library_GenerateChebyShevPolynomial.Main(
        ResultType = 'sympy',
        Kind= 1,
        Order= ApproximationOrder-1,
        ReturnAll = True,
        )

    print '\nChebyShevPolynomials'
    pprint.pprint( ChebyShevPolynomials )


    Result = 0.0 -.5*(Coefficients[0])
    for Coefficient, ChebyShevPolynomial in zip(Coefficients, ChebyShevPolynomials):
        Result += Coefficient*ChebyShevPolynomial

    #Transform the coefficients and the result to be on arbitrary inverval instead of from 0 to 1
    Result = Result.subs(Xsymb, x)

    return Result

-------------------------------------------------- -----------------------------

Example_SympyChebyShevApproximationOneDimension：

#------------------------------------------------------------------------------
import sympy
import sympy.mpmath
import matplotlib.pyplot as plt
import json
import pprint




import Library_GenerateBesselFunction
import Library_SympyChebyShevApproximationOneDimension
import Library_SympyExpressionToPythonFunction
import Library_GraphOneDimensionalFunction


ApproximationOrder = 10

#CREATE THE EXAMPLE EXRESSION:
Kind = 1
Order = 2
ExampleSympyExpression = sympy.sin(sympy.Symbol('x'))

"""
Library_GenerateBesselFunction.Main(
    ResultType =  'sympy',
    Kind =  Kind,
    Order =  Order,
    VariableNames = ['x'],
    ) 
"""
PythonOriginalFunction = Library_SympyExpressionToPythonFunction.Main( 
    ExampleSympyExpression ,
    FloatPrecision = 100,
    )

#CREATE THE NATIVE CHEBY APPROXIMATION

ChebyDomainMin = 5.
ChebyDomainMax = 10.
ChebyDomain = [ChebyDomainMin, ChebyDomainMax]
ChebyExpandedPolynomialCoefficients, ChebyError = sympy.mpmath.chebyfit(
    PythonOriginalFunction, 
    ChebyDomain, 
    ApproximationOrder, 
    error=True
    )
print 'ChebyExpandedPolynomialCoefficients'
pprint.pprint( ChebyExpandedPolynomialCoefficients )
def PythonChebyChevApproximation(Point):
    Result = sympy.mpmath.polyval(ChebyExpandedPolynomialCoefficients, Point)
    return Result


#CREATE THE GENERIC ONE DIMENSIONAL CHEBY APPROXIMATION:
SympyChebyApproximation = Library_SympyChebyShevApproximationOneDimension.Main(
    SympyExpression = ExampleSympyExpression*sympy.cos( sympy.Symbol('a') ),
    ApproximationSymbol = sympy.Symbol('x'),
    DomainMinimumPoint = [ChebyDomainMin],
    DomainMaximumPoint = [ChebyDomainMax],
    ApproximationOrder = ApproximationOrder
    )


print 'SympyChebyApproximation', SympyChebyApproximation

SympyChebyApproximation = SympyChebyApproximation.subs(sympy.Symbol('a'), 0.0)

print 'SympyChebyApproximation', SympyChebyApproximation

PythonCastedChebyChevApproximationGeneric = Library_SympyExpressionToPythonFunction.Main( 
    SympyChebyApproximation ,
    FloatPrecision = 100,
    )

print 'PythonCastedChebyChevApproximationGeneric(1)', PythonCastedChebyChevApproximationGeneric(1.)