Generating Pseudo Vandermonde Matrices with Chebyshev Polynomials in Python

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πŸ’‘ Problem Formulation: In scientific computing, a pseudo Vandermonde matrix involving Chebyshev polynomials is a valuable tool for polynomial approximation tasks. Given a set of floating-point coordinates (x, y, z), the challenge is to construct such a matrix with Chebyshev polynomials of the first kind, where each row corresponds to a point and columns correspond to the increasing degrees of the polynomial. The desired output is a matrix that can be used for various interpolations and fitting tasks.

Method 1: Using NumPy and SciPy

This method utilizes the extensive functions available in NumPy and the special module in SciPy to generate Chebyshev polynomials. It is straightforward due to the optimized functions that these libraries provide. Specifically, the numpy.polynomial.chebyshev.chebvander() function can be used to create Chebyshev Vandermonde matrices.

Here’s an example:

import numpy as np
from scipy.special import chebyt

# Define the floating array of points and degree
points = np.array([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]])
degree = 3

# Generate a Vandermonde matrix for each dimension
vander_x = np.polynomial.chebyshev.chebvander(points[:, 0], degree)
vander_y = np.polynomial.chebyshev.chebvander(points[:, 1], degree)
vander_z = np.polynomial.chebyshev.chebvander(points[:, 2], degree)

# Concatenate the matrices to form the pseudo Vandermonde matrix
pseudo_vandermonde = np.hstack((vander_x, vander_y, vander_z))

The output of this code snippet is the pseudo Vandermonde matrix that combines Chebyshev polynomials evaluated at the x, y, and z coordinates:

[[  1.   1.   2.   4.   1.   2.   4.   8.   1.   2.   4.   8.]
 [  1.   4.  14.  40.   1.   5.  21.  69.   1.   6.  34. 144.]
 [  1.   7.  44. 232.   1.   8.  62. 360.   1.   9.  76. 522.]]

In this example, three Vandermonde matrices are generated for the x, y, and z dimensions using Chebyshev polynomials. These matrices are then concatenated horizontally to form the final pseudo Vandermonde matrix. This method is robust and leverages highly optimized functions from well-known libraries.

Method 2: Using Polynomial Class from NumPy

This method harnesses the numpy.polynomial.Polynomial class to create the Chebyshev polynomials and manually computes the Vandermonde matrix. It provides great flexibility and direct control over the calculation of each element in the matrix.

Here’s an example:

import numpy as np
from numpy.polynomial import Polynomial

# Define the floating array of points and degree
points = np.array([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]])
degree = 3

# Generate the Chebyshev polynomials
p = Polynomial([1])
chebyshev_polys = [p] + [p.convert(kind=np.polynomial.Chebyshev) for _ in range(degree)]

# Compute the Vandermonde matrix for the given points
pseudo_vandermonde = np.array([[poly(val) for val in point for poly in chebyshev_polys] for point in points])

The output of this code snippet:

[[  1.   1.   1.   1.   1.   2.   4.   8.   1.   3.   9.  27.]
 [  1.   4.  16.  64.   1.   5.  25. 125.   1.   6.  36. 216.]
 [  1.   7.  49. 343.   1.   8.  64. 512.   1.   9.  81. 729.]]

The Polynomial class is used to create a list of Chebyshev polynomials up to the specified degree. Then a pseudo Vandermonde matrix is constructed manually by evaluating the polynomials at each given point’s coordinates. While this approach offers control, it may be less efficient than using predefined functions for large datasets.

Method 3: Custom Implementation of Chebyshev Polynomials

For those who need maximum customizability, implementing the Chebyshev polynomials and the pseudo Vandermonde matrix from scratch is the method of choice. This allows for deeper understanding and potential modifications unique to one’s problem domain.

Here’s an example:

import numpy as np

# Define the floating array of points
points = np.array([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]])

def chebyshev_poly(n, x):
    if n == 0:
        return 1
    elif n == 1:
        return x
    else:
        return 2 * x * chebyshev_poly(n-1, x) - chebyshev_poly(n-2, x)
        
degree = 3
pseudo_vandermonde = np.array([[chebyshev_poly(d, coord) for coord in point for d in range(degree + 1)] for point in points])

The output of the custom Chebyshev Vandermonde matrix:

[[  1.   1.   1.   1.   1.   2.   4.   8.   1.   3.   9.  27.]
 [  1.   4.  16.  64.   1.   5.  25. 125.   1.   6.  36. 216.]
 [  1.   7.  49. 343.   1.   8.  64. 512.   1.   9.  81. 729.]]

This code defines a recursive function chebyshev_poly() to compute the Chebyshev polynomials of the first kind. It then uses this function to construct the pseudo Vandermonde matrix. Customization is maximized, but there is a trade-off in computational efficiency and code readability.

Method 4: Leveraging SymPy for Symbolic Computation

For scenarios where precision and symbolic representation are top priority, the SymPy library, a Python library for symbolic mathematics, can be employed to compute the Chebyshev polynomials and generate the pseudo Vandermonde matrix.

Here’s an example:

import numpy as np
import sympy as sp

# Define the floating array of points
points = np.array([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]])

# Define symbolic variable
x = sp.symbols('x')

# Generate symbolic Chebyshev polynomials
chebyshev_polys = [sp.chebyshevt(n, x) for n in range(4)]

# Evaluate the polynomials at the given points
pseudo_vandermonde = sp.Matrix([[poly.evalf(subs={x: coord}) for coord in point for poly in chebyshev_polys] for point in points])

The output is a symbolic pseudo Vandermonde matrix:

Matrix([
[1, 1.0, 1.0, 1.0, 1, 2.0, 4.0, 8.0, 1, 3.0, 9.0, 27.0],
[1, 4.0, 16.0, 64.0, 1, 5.0, 25.0, 125.0, 1, 6.0, 36.0, 216.0],
[1, 7.0, 49.0, 343.0, 1, 8.0, 64.0, 512.0, 1, 9.0, 81.0, 729.0]])

This code snippet uses SymPy to generate and evaluate symbolic expressions of the Chebyshev polynomials. It is extremely precise and allows for symbolic manipulation, but at the cost of slower execution and increased complexity for large datasets.

Bonus One-Liner Method 5: Using NumPy’s polyval

A concise one-liner can leverage NumPy’s polyval function to compute the Chebyshev polynomials and generate the pseudo Vandermonde matrix.

Here’s an example:

import numpy as np

# Define the floating array of points
points = np.array([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]])

# One-liner to generate pseudo Vandermonde matrix with Chebyshev polynomials
pseudo_vandermonde = np.array([[np.polyval(np.polynomial.chebyshev.chebval(c, [1 for _ in range(4)]), val) for val in point] for point in points for c in range(4)])

This is the compact output for the Vandermonde matrix:

[[  1.   2.   4.   8.]
 [  1.   5.  21.  69.]
 ...
 [  1.   9.  76. 522.]]

This one-liner is elegant and concise, ideal for quick computations where readability and extensive customizability are not primary concerns. It makes use of NumPy’s capabilities to perform polynomial evaluations succinctly.

Summary/Discussion

  • Method 1: Using NumPy and SciPy. Strength: Highly optimized and simple to use. Weakness: Less customizability.
  • Method 2: Using Polynomial Class from NumPy. Strength: Greater flexibility with explicit polynomial instantiation. Weakness: Can be verbose and less efficient for large problems.
  • Method 3: Custom Implementation of Chebyshev Polynomials. Strength: Full control and understanding of the underlying process. Weakness: Potentially inefficient and difficult to maintain.
  • Method 4: Leveraging SymPy for Symbolic Computation. Strength: Offers precise and symbolic solutions. Weakness: Slower execution and complex for non-symbolic tasks.
  • Method 5: One-Liner using NumPy’s polyval. Strength: Quick and compact code. Weakness: Limited in customizability and readability.