Chebyshev2 - GTSAM Docs

Overview¶

Chebyshev2 implements a pseudo-spectral parameterization on Chebyshev points of the second kind. Instead of coefficients, the parameters are function values at Chebyshev points, and evaluation uses barycentric interpolation. The class also provides differentiation and integration matrices for spectral calculus.

Authors: Frank Dellaert, et al. (see THANKS for the full author list)

See LICENSE for the license information

# Install GTSAM from pip if running in Google Colab
try:
    import google.colab
    %pip install --quiet gtsam-develop
except ImportError:
    pass  # Not in Colab

Key Functionality / API¶

Point(N, j[, a, b]) and Points(N[, a, b]) return Chebyshev points.
CalculateWeights(N, x[, a, b]) returns barycentric interpolation weights.
DerivativeWeights(N, x[, a, b]) returns derivative weights.
DifferentiationMatrix(N[, a, b]) and IntegrationMatrix(N[, a, b]) provide spectral operators.
IntegrationWeights(N[, a, b]) and DoubleIntegrationWeights(N[, a, b]) provide quadrature weights.

Usage Example¶

This example inspects the Chebyshev points, interpolation weights, and the differentiation matrix.

import numpy as np
import gtsam

np.set_printoptions(precision=3, suppress=True)

N = 8
points = gtsam.Chebyshev2.Points(N)
print("Chebyshev points:", np.asarray(points).ravel())

x = 0.2
weights = gtsam.Chebyshev2.CalculateWeights(N, x)
print("Interpolation weights at x=0.2:", np.asarray(weights).ravel())

D = gtsam.Chebyshev2.DifferentiationMatrix(N)
print("D shape:", np.asarray(D).shape)
print("First row of D:", np.asarray(D)[0])

Chebyshev points: [-1.    -0.901 -0.623 -0.223  0.223  0.623  0.901  1.   ]
Interpolation weights at x=0.2: [-0.009  0.02  -0.027  0.053  0.998 -0.053  0.032 -0.014]
D shape: (8, 8)
First row of D: [-16.5    20.196  -5.312   2.572  -1.636   1.232  -1.052   0.5  ]

Source¶

Chebyshev2.h