π‘ Problem Formulation: Evaluating a 3D Chebyshev series entails computing the values of the series at specific points on a three-dimensional grid formed by the Cartesian product of x, y, and z coordinates. Given coefficients of a Chebyshev series and points (x, y, z), we seek the series values at these points. For example, with input coefficients c
and point grid arrays x
, y
, z
, the output would be an array of evaluated series values at each point.
Method 1: Using NumPy’s polynomial.chebyshev.chebgrid3d
This method involves NumPy’s polynomial.chebyshev
module, which provides the chebgrid3d
function to evaluate a 3D Chebyshev series at points in the space defined by the arrays x, y, and z. This function is convenient for handling multidimensional array operations and is well-optimized for numerical computations in Python.
Here’s an example:
import numpy as np from numpy.polynomial.chebyshev import chebgrid3d c = np.random.rand(4, 4, 4) # Coefficients of the Chebyshev series x = np.linspace(-1, 1, 10) y = np.linspace(-1, 1, 10) z = np.linspace(-1, 1, 10) vals = chebgrid3d(x, y, z, c)
The output will be a 10x10x10 array where each element represents the evaluated series value at corresponding points.
This code snippet initializes a random 3D array c
of Chebyshev coefficients and generates arrays x
, y
, and z
using np.linspace
to represent the Cartesian grid. The function chebgrid3d
is called to evaluate the series at every point on the grid. The result is a 3D array of values.
Method 2: Using scipy.interpolate.Chebyshev3D
The scipy.interpolate
module’s Chebyshev3D
class provides an interpolation object that can be used to evaluate a 3D Chebyshev series. This method is useful for when the series coefficients correspond to an approximation of some function over a 3D grid.
Here’s an example:
from scipy.interpolate import Chebyshev3D c = np.random.rand(4, 4, 4) # Coefficients of the Chebyshev series cheb_interp = Chebyshev3D(c) x = np.linspace(-1, 1, 10) y = np.linspace(-1, 1, 10) z = np.linspace(-1, 1, 10) xx, yy, zz = np.meshgrid(x, y, z) vals = cheb_interp(xx, yy, zz)
The output would be a 10x10x10 array containing the evaluated series values at the meshgrid points.
After setting up the Chebyshev coefficient array c
, this snippet creates a Chebyshev3D
interpolation object. Arrays x
, y
, and z
are created and combined into a meshgrid. The cheb_interp
object is then called as a function, passing the meshgrid points to obtain the evaluations at each point.
Method 3: Brute Force Using Multidimensional Arrays and Loops
One can manually compute the Chebyshev series evaluation using nested loops over the coordinates x, y, and z. This method, while simple, may not be as efficient as using specialized functions but is good for understanding the series evaluation at a fundamental level.
Here’s an example:
import numpy.polynomial.chebyshev as cheb c = np.random.rand(4, 4, 4) # Chebyshev series coefficients x = np.linspace(-1, 1, 10) y = np.linspace(-1, 1, 10) z = np.linspace(-1, 1, 10) vals = np.empty((10, 10, 10)) for i, xi in enumerate(x): for j, yj in enumerate(y): for k, zk in enumerate(z): vals[i, j, k] = cheb.chebval3d(xi, yj, zk, c)
The output will be the same 10x10x10 array of values as before, containing the series evaluations at the Cartesian product of x
, y
, and z
.
Here, the nested for-loops iterate over every possible combination of points in the 3D Cartesian grid created by x
, y
, and z
. At each iteration, the chebval3d
function from NumPy’s Chebyshev module evaluates the series at a single point.
Method 4: Using Tensor Products and Matrix Multiplication
Matrix multiplication can be used to evaluate the Chebyshev series by exploiting the tensor product structure of 3D Chebyshev polynomials. This method involves broadcasting and reshaping operations for efficient computation.
Here’s an example:
from numpy.polynomial.chebyshev import chebvander3d c = np.random.rand(4, 4, 4) # Chebyshev series coefficients x = np.linspace(-1, 1, 10) y = np.linspace(-1, 1, 10) z = np.linspace(-1, 1, 10) Txyz = chebvander3d(x, y, z, [3, 3, 3]) vals = np.tensordot(c, Txyz, axes=([0, 1, 2], [3, 4, 5]))
The output will be the familiar 10x10x10 array.
The chebvander3d
function generates a Vandermonde-like tensor for the 3D Chebyshev polynomials evaluated at the grid points. The resulting tensor Txyz
is then combined with the coefficients c
using the np.tensordot
function, performing the necessary summation to evaluate the series.
Bonus One-Liner Method 5: Using NumPy’s apply_along_axis
You can utilize apply_along_axis
from NumPy to apply a function, such as the Chebyshev series evaluator, along one dimension of an array. This method provides a quick one-liner solution, although it may not be as efficient for large-scale problems.
Here’s an example:
from numpy.polynomial.chebyshev import chebval3d c = np.random.rand(4, 4, 4) xyz = np.meshgrid(x, y, z, indexing='ij', sparse=True) vals = np.apply_along_axis(lambda coords: chebval3d(*coords, c), 0, np.stack(xyz))
Output will be a 10x10x10 array of series values.
This one-liner constructs a sparsely-represented meshgrid and then stacks it into a single array whose axes correspond to x, y, and z coordinates. The apply_along_axis
method applies the chebval3d
function to each coordinate triple, unpacked as arguments via the lambda function.
Summary/Discussion
- Method 1: NumPy’s chebgrid3d. Pros: Well-optimized and concise. Cons: Less control over the evaluation process.
- Method 2: SciPy’s Chebyshev3D. Pros: Flexible and part of a powerful interpolation framework. Cons: Uses more memory to create interpolation objects.
- Method 3: Manual Evaluation with Loops. Pros: Easy to understand and implement. Cons: Inefficient and slow for larger arrays.
- Method 4: Tensor Products with Matrix Multiplication. Pros: Mathematically elegant and efficient for large computations. Cons: Requires a deeper understanding of tensor operations.
- Method 5: NumPy’s apply_along_axis. Pros: Compact one-liner solution. Cons: Potentially slower for large data sets due to not being parallelized.