Generate a Vandermonde matrix of the Chebyshev polynomial with float array of points in Python

To generate a Vandermonde matrix of the Chebyshev polynomial, use numpy.polynomial.chebyshev.chebvander(). This function returns a Vandermonde matrix where each column represents a different degree of the Chebyshev polynomial evaluated at the input points.

Syntax

numpy.polynomial.chebyshev.chebvander(x, deg)

Parameters

The function accepts the following parameters:

• x: Array of points. The dtype is converted to float64 or complex128 depending on whether any elements are complex. If x is scalar, it is converted to a 1-D array.
• deg: Degree of the resulting matrix. This determines the number of Chebyshev polynomial terms to include.

Return Value

Returns a Vandermonde matrix with shape x.shape + (deg + 1,), where the last index corresponds to the degree of the Chebyshev polynomial. The dtype matches the converted input array.

Example

Let's create a Vandermonde matrix for Chebyshev polynomials up to degree 2:

import numpy as np
from numpy.polynomial import chebyshev as C

# Create an array of points
x = np.array([0, 3.5, -1.4, 2.5])

# Display the array
print("Our Array:")
print(x)

# Check array properties
print("\nDimensions:", x.ndim)
print("Datatype:", x.dtype)
print("Shape:", x.shape)

# Generate Vandermonde matrix of degree 2
result = C.chebvander(x, 2)
print("\nVandermonde Matrix:")
print(result)

Our Array:
[ 0.   3.5 -1.4  2.5]

Dimensions: 1
Datatype: float64
Shape: (4,)

Vandermonde Matrix:
[[ 1.    0.   -1.  ]
 [ 1.    3.5  23.5 ]
 [ 1.   -1.4   2.92]
 [ 1.    2.5  11.5 ]]

How It Works

Each row in the matrix corresponds to one input point, and each column represents a Chebyshev polynomial of increasing degree:

• Column 0: T?(x) = 1 (always 1 for all points)
• Column 1: T?(x) = x (the input values themselves)
• Column 2: T?(x) = 2x² - 1 (second-degree Chebyshev polynomial)

Higher Degree Example

Here's an example with degree 3 to show more polynomial terms:

import numpy as np
from numpy.polynomial import chebyshev as C

# Create points and generate degree 3 Vandermonde matrix
x = np.array([0, 1, -1])
result = C.chebvander(x, 3)

print("Points:", x)
print("\nVandermonde Matrix (degree 3):")
print(result)

Points: [ 0  1 -1]

Vandermonde Matrix (degree 3):
[[ 1.  0. -1.  0.]
 [ 1.  1.  1.  1.]
 [ 1. -1.  1. -1.]]

Conclusion

The chebvander() function creates a Vandermonde matrix for Chebyshev polynomials, useful in polynomial fitting and numerical analysis. Each column represents a different degree polynomial evaluated at the input points.