Evaluate a Chebyshev series at points x when coefficients are multi-dimensional in Python

To evaluate a Chebyshev series at points x with multi-dimensional coefficients, use the chebyshev.chebval() method in NumPy. This method handles coefficient arrays where each column represents a different polynomial series.

Syntax

numpy.polynomial.chebyshev.chebval(x, c, tensor=True)

Parameters

x: Points at which to evaluate the series. Can be scalar, list, or array.

c: Array of coefficients. For multi-dimensional arrays, each column represents a separate polynomial.

tensor: If True (default), evaluates every column of coefficients for every element of x. If False, broadcasts x over the columns.

Example

Let's create a 2D coefficient array and evaluate Chebyshev series at multiple points ?

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

# Create a multidimensional array of coefficients
c = np.arange(4).reshape(2,2)

# Display the array
print("Our Array...")
print(c)

# Check the dimensions and shape
print("\nDimensions:", c.ndim)
print("Datatype:", c.dtype)
print("Shape:", c.shape)

# Evaluate Chebyshev series at points [1, 2]
result = C.chebval([1, 2], c, tensor=True)
print("\nResult (chebval)...")
print(result)

Our Array...
[[0 1]
 [2 3]]

Dimensions: 2
Datatype: int64
Shape: (2, 2)

Result (chebval)...
[[2. 4.]
 [4. 7.]]

How It Works

The coefficient array has shape (2, 2), representing two polynomials:

− First polynomial: coefficients [0, 2]

− Second polynomial: coefficients [1, 3]

For each point x, the Chebyshev series is evaluated as: c[0] + c[1]*T?(x), where T?(x) is the first Chebyshev polynomial.

Different Tensor Values

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

c = np.arange(6).reshape(3,2)
x = [1, 2]

print("Coefficients:")
print(c)

print("\nWith tensor=True:")
print(C.chebval(x, c, tensor=True))

print("\nWith tensor=False:")  
print(C.chebval(x, c, tensor=False))

Coefficients:
[[0 1]
 [2 3]
 [4 5]]

With tensor=True:
[[ 6. 10.]
 [12. 18.]]

With tensor=False:
[ 6. 18.]

Conclusion

Use chebval() with multi-dimensional coefficients to evaluate multiple Chebyshev series simultaneously. The tensor parameter controls how the evaluation is broadcast across coefficient columns.