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 matrices and exact rectangular 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]) differentiates values at N nodes.
IntegrationMatrix(N[, a, b]) maps N derivative values to N+1 exact antiderivative values with F(a)=0.
IntegrationWeights(N[, a, b]) and DoubleIntegrationWeights(N[, a, b]) provide quadrature weights.

Usage Example¶

This example inspects the Chebyshev points, interpolation weights, differentiation matrix, and exact integration 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])

P = gtsam.Chebyshev2.IntegrationMatrix(N)
integration_weights = gtsam.Chebyshev2.IntegrationWeights(N)
print("P shape:", np.asarray(P).shape)
print("P final row equals weights:", np.allclose(np.asarray(P)[-1], integration_weights))

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  ]
P shape: (9, 8)
P final row equals weights: True

Source¶

Chebyshev2.h