Evaluate a 3-D Chebyshev series on the Cartesian product of x, y and z with 2d array of coefficient in Python

To evaluate a 3-D Chebyshev series on the Cartesian product of x, y, z, use the polynomial.chebgrid3d() method in Python. This function evaluates a multidimensional Chebyshev series at points formed by the Cartesian product of the input arrays.

Understanding the Parameters

The chebgrid3d(x, y, z, c) method takes four parameters:

• x, y, z − The coordinates where the 3-D series is evaluated. If any parameter is a list or tuple, it's converted to an ndarray
• c − Array of coefficients ordered so that coefficients for terms of degree i,j are in c[i,j]. If c has fewer than three dimensions, ones are implicitly appended to make it 3-D

The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape.

Example

Let's create a 2D coefficient array and evaluate the 3-D Chebyshev series ?

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

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

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

# Check the Dimensions
print("\nDimensions of our Array...\n", c.ndim)

# Get the Datatype
print("\nDatatype of our Array object...\n", c.dtype)

# Get the Shape
print("\nShape of our Array object...\n", c.shape)

# Evaluate 3-D Chebyshev series on Cartesian product
print("\nResult...\n", C.chebgrid3d([1,2], [1,2], [1,2], c))

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

Dimensions of our Array...
2

Datatype of our Array object...
int64

Shape of our Array object...
(2, 2)

Result...
[[17. 28.]
  [28. 46.]]

How It Works

The function evaluates the Chebyshev polynomial at each point in the Cartesian product of the input coordinates. Since we provided a 2D coefficient array, the function automatically extends it to 3D by adding a dimension of size 1.

Different Coordinate Arrays

You can also use different coordinate arrays for each dimension ?

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

# Create coefficient array
c = np.array([[1, 2], [3, 4]])

# Use different coordinate arrays
x = [0, 1]
y = [0, 1, 2]  
z = [1]

result = C.chebgrid3d(x, y, z, c)
print("Result shape:", result.shape)
print("Result:\n", result)

Result shape: (2, 3, 1)
Result:
[[[10.]
  [10.]
  [18.]]]

Conclusion

The chebgrid3d() function provides an efficient way to evaluate 3-D Chebyshev series on Cartesian product grids. The function automatically handles dimension expansion and returns results with predictable shape based on input dimensions.