Generate a Vandermonde matrix of the Chebyshev polynomial in Python

To generate a Vandermonde matrix of the Chebyshev polynomial, use the chebyshev.chebvander() function in NumPy. The method returns the Vandermonde matrix where each column represents a different degree of the Chebyshev polynomial. The shape of the returned matrix is x.shape + (deg + 1,), where the last index corresponds to the degree of the Chebyshev polynomial.

The function takes two parameters: x is an array of points (converted to float64 or complex128), and deg is the degree of the resulting matrix.

Syntax

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

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. Must be a non-negative integer.

Example

Let's create a Vandermonde matrix of degree 2 for a set of points ?

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

# Create an array of points
x = np.array([0, 1, -1, 2])

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

# Check the dimensions and properties
print("\nDimensions of our Array:", x.ndim)
print("Datatype of our Array:", x.dtype)
print("Shape of our Array:", x.shape)

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

Our Array:
[ 0  1 -1  2]

Dimensions of our Array: 1
Datatype of our Array: int64
Shape of our Array: (4,)

Vandermonde matrix:
[[ 1.  0. -1.]
 [ 1.  1.  1.]
 [ 1. -1.  1.]
 [ 1.  2.  7.]]

How It Works

The Vandermonde matrix has three columns (degree + 1 = 3) representing Chebyshev polynomials of degrees 0, 1, and 2:

• Column 0: T?(x) = 1 for all x
• Column 1: T?(x) = x
• Column 2: T?(x) = 2x² - 1

Different Degrees

You can create matrices with different degrees ?

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

x = np.array([0, 1, -1])

# Degree 1 matrix
print("Degree 1 matrix:")
print(C.chebvander(x, 1))

# Degree 3 matrix
print("\nDegree 3 matrix:")
print(C.chebvander(x, 3))

Degree 1 matrix:
[[ 1.  0.]
 [ 1.  1.]
 [ 1. -1.]]

Degree 3 matrix:
[[ 1.  0. -1.  0.]
 [ 1.  1.  1.  1.]
 [ 1. -1.  1. -1.]]

Conclusion

The chebvander() function efficiently generates Vandermonde matrices for Chebyshev polynomials. Each row corresponds to a point in your input array, and each column represents a different polynomial degree, making it useful for polynomial fitting and numerical analysis.