Generate a Vandermonde matrix of the Chebyshev polynomial in Python

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In this article, we will be looking at the approach using various functionalities of the Numpy packages to generate a Vandermonde matrix of the Chebyshev polynomial in Python.

NumPy.chebvander method

To generate a Vandermonde matrix of the Chebyshev polynomial, the user needs to call the np.chebvander() from the NumPy package in Python Numpy. And further, passing an array of integers to the function will return the Vandermonde matrix. The shape of the returned matrix is x.shape + (deg + 1), where the last index is the degree of the corresponding Chebyshev polynomial.

Syntax: np.chebvander(x, deg)

Parameters:

x: Array of points.

deg: Degree of the resulting matrix.

Return: Return the matrix having size.

Example 1:

In this example, we have created the array containing 10 data points from series 1 to 10 and with the use of the np.chebvander() function, we will pass the degree to be 2.

Python3

import numpy as np
from numpy.polynomial import chebyshev
 
gfg = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
 
# Print array
print("Array - ", gfg)
 
# Print dimension of the array
print("Dimension of Array:-", gfg.ndim)
 
# Print Datatype array
print("Datatype of Array:-", gfg.dtype)
 
# Print shape array
print("Shape of Array:-", gfg.shape)
 
# Generate a Vandermonde matrix of the 
# Chebyshev polynomial of degree 2
gfg_matrix = chebyshev.chebvander(gfg, 2)
print('\n', gfg_matrix)
 
# Print shape martrix
print("\nShape of Martix:-\n", gfg_matrix.shape)

Output:

Array -  [ 1  2  3  4  5  6  7  8  9 10]
Dimension of Array:- 1
Datatype of Array:- int64
Shape of Array:- (10,)

 [[  1.   1.   1.]
 [  1.   2.   7.]
 [  1.   3.  17.]
 [  1.   4.  31.]
 [  1.   5.  49.]
 [  1.   6.  71.]
 [  1.   7.  97.]
 [  1.   8. 127.]
 [  1.   9. 161.]
 [  1.  10. 199.]]

Shape of Matrix:-
 (10, 3)

Example 2:

In this example, we have created the array containing 5 data points and with the use of the np.chebvander() function passing the degree to be 5, and at last, we will check the shape of the generated matrix in the python.

Python3

import numpy as np
from numpy.polynomial import chebyshev
 
gfg = np.array([59,89,45,71,56])
 
# Print array
print("Array - ", gfg)
 
# Print dimension of the array
print("Dimension of Array:-",gfg.ndim)
 
# Print Datatype array
print("Datatype of Array:-",gfg.dtype)
 
# Print shape array
print("Shape of Array:-",gfg.shape)
 
# Generate a Vandermonde matrix of the
# Chebyshev polynomial of degree 5
gfg_matrix=chebyshev.chebvander(gfg, 5)
print('\n',gfg_matrix)
 
# Print shape martrix
print("\nShape of Martix:-\n",gfg_matrix.shape)

Output:

Array – [59 89 45 71 56]
Dimension of Array:- 1

Datatype of Array:- int64

Shape of Array:- (5,)

[[1.00000000e+00 5.90000000e+01 6.96100000e+03 8.21339000e+05

9.69110410e+07 1.14346815e+10]
[1.00000000e+00 8.90000000e+01 1.58410000e+04 2.81960900e+06

5.01874561e+08 8.93308522e+10]
[1.00000000e+00 4.50000000e+01 4.04900000e+03 3.64365000e+05

3.27888010e+07 2.95062772e+09]
[1.00000000e+00 7.10000000e+01 1.00810000e+04 1.43143100e+06

2.03253121e+08 2.88605118e+10]
[1.00000000e+00 5.60000000e+01 6.27100000e+03 7.02296000e+05

7.86508810e+07 8.80819638e+09]]
Shape of Martix:-

(5, 6)