Python - 具有 2 个参数的频谱矩阵

Question

我对插入通用看涨期权很感兴趣，然后计算多项式关于两个参数的导数：西格玛和时间： 为了做到这一点，我正在使用 Chebyshev 节点上定义的重心插值。

现在我能够计算关于一个变量的多项式的一阶导数，但在这种情况下，我应该计算关于两个参数的函数（调用选项）的一阶导数，我不知道如何将谱矩阵视为一个有两个参数的函数，我将附上生成只有一个参数的光谱矩阵的代码

import sys
import warnings
import numpy as np
from numpy import linalg as LA
from numpy import fft as FFT
import math

def FirstPolynomialDerivativeMatrix(x):
    N = x.shape[0]
    w = BarycentricWeights(x)
    D = np.zeros((N,N))
    for i in range(0,N):
        for j in range(0,N):
            if (j != i):
                D[i,j] = w[j]/w[i] * 1./(x[i] - x[j])
                D[i,i] = D[i,i] - D[i,j]
    return D


def PolynomialDerivativeMatrix(x,k):
    N = x.shape[0]
    w = BarycentricWeights(x)
    D = np.zeros((N,N,k))
    D[:,:,0] = FirstPolynomialDerivativeMatrix(x)
    if ( k == 1 ): return D[:,:,k-1]
    for m in range(2,k+1):
        for i in range(0,N):
            for j in range(0,N):
                if ( j != i ):
                    D[i,j,m-1] = m/(x[i]-x[j]) * ( w[j]/w[i] * D[i,i,m-2] - D[i,j,m-2])
                    D[i,i,m-1] = D[i,i,m-1] - D[i,j,m-1]
    return D[:,:,k-1]


def BarycentricWeights(x):
    N = x.shape[0]
    w = np.zeros((N))
    for j in range(0,N):
        w[j] = 1.
    for j in range(1,N):
        for k in range(0,j):
            w[k] = w[k] * (x[k] - x[j])
            w[j] = w[j] * (x[j] - x[k])
    for j in range(0,N):
        w[j] = 1. / w[j]
    return w



x = np.array([0.55, 0.53535534, 0.5, 0.46464466, 0.45])
y = np.array([1.2,1.14142136, 1.,0.85857864, 0.8])

res1 = PolynomialDerivativeMatrix(x,2)
res2 = PolynomialDerivativeMatrix(y,2) #2 is the "second derivative"

所以在这个简短的例子中，我需要计算光谱矩阵作为两个参数的函数：x 和 y

感谢大家！