Merge pull request #2071 from borglab/feature/cheb_wrapper
Wrap missing Chebyshev2 methods...release/4.3a0
Frank Dellaert
GitHub
commit
c03dba8579
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@ -40,10 +40,20 @@ class Chebyshev2 {
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static gtsam::Matrix WeightMatrix(size_t N, gtsam::Vector X);
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static gtsam::Matrix WeightMatrix(size_t N, gtsam::Vector X, double a, double b);
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static gtsam::Matrix CalculateWeights(size_t N, double x);
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static gtsam::Matrix DerivativeWeights(size_t N, double x);
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static gtsam::Matrix IntegrationMatrix(size_t N);
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static gtsam::Matrix DifferentiationMatrix(size_t N);
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static gtsam::Matrix IntegrationWeights(size_t N);
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static gtsam::Matrix DoubleIntegrationWeights(size_t N);
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static gtsam::Matrix CalculateWeights(size_t N, double x, double a, double b);
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static gtsam::Matrix DerivativeWeights(size_t N, double x, double a, double b);
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static gtsam::Matrix IntegrationWeights(size_t N, double a, double b);
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static gtsam::Matrix IntegrationMatrix(size_t N, double a, double b);
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static gtsam::Matrix DifferentiationMatrix(size_t N, double a, double b);
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static gtsam::Matrix IntegrationWeights(size_t N, double a, double b);
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static gtsam::Matrix DoubleIntegrationWeights(size_t N, double a, double b);
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};
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#include <gtsam/basis/BasisFactors.h>
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@ -561,8 +561,6 @@ TEST(Chebyshev2, DoubleIntegrationWeights) {
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// Let's integrate constant twice get a test case:
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Matrix P = Chebyshev2::IntegrationMatrix(N, a, b);
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auto ones = Vector::Ones(N);
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Vector ramp = P * ones;
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Vector quadratic = P * ramp;
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// Check the sum which should be 0.5*t^2 | [0,b] = b^2/2:
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Weights w = Chebyshev2::DoubleIntegrationWeights(N, a, b);
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@ -575,8 +573,6 @@ TEST(Chebyshev2, DoubleIntegrationWeights2) {
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// Let's integrate constant twice get a test case:
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Matrix P = Chebyshev2::IntegrationMatrix(N, a, b);
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auto ones = Vector::Ones(N);
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Vector ramp = P * ones;
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Vector quadratic = P * ramp;
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// Check the sum which should be 0.5*t^2 | [0,b] = b^2/2:
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Weights w = Chebyshev2::DoubleIntegrationWeights(N, a, b);
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@ -0,0 +1,279 @@
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"""
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GTSAM Copyright 2010-2019, Georgia Tech Research Corporation,
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Atlanta, Georgia 30332-0415
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All Rights Reserved
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See LICENSE for the license information
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Unit tests for Chebyshev2 Basis using the GTSAM Python wrapper.
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Converted from the C++ tests.
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"""
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import unittest
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import numpy as np
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from gtsam.utils.test_case import GtsamTestCase
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import gtsam
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from gtsam import Chebyshev2
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# Define test functions f and fprime:
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def f(x):
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return 3.0 * (x**3) - 2.0 * (x**2) + 5.0 * x - 11.0
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def fprime(x):
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return 9.0 * (x**2) - 4.0 * x + 5.0
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def Chebyshev2_vector(f, N, a=-1.0, b=1.0):
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points = Chebyshev2.Points(N, a, b)
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return np.array([f(x) for x in points])
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class TestChebyshev2(GtsamTestCase):
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def test_Point(self):
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"""Test that Chebyshev points are correctly calculated and symmetrical."""
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N = 5
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points = Chebyshev2.Points(N)
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expected = np.array([-1.0, -np.sqrt(2.0) / 2.0, 0.0, np.sqrt(2.0) / 2.0, 1.0])
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tol = 1e-15
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np.testing.assert_allclose(points, expected, rtol=0, atol=tol)
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# Check symmetry:
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p0 = Chebyshev2.Point(N, 0)
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p4 = Chebyshev2.Point(N, 4)
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p1 = Chebyshev2.Point(N, 1)
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p3 = Chebyshev2.Point(N, 3)
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self.assertAlmostEqual(p0, -p4, delta=tol)
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self.assertAlmostEqual(p1, -p3, delta=tol)
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def test_PointInInterval(self):
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"""Test that Chebyshev points map correctly to arbitrary intervals [a,b]."""
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N = 5
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points = Chebyshev2.Points(N, 0, 20)
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expected = (
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np.array(
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[0.0, 1.0 - np.sqrt(2.0) / 2.0, 1.0, 1.0 + np.sqrt(2.0) / 2.0, 2.0]
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)
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* 10.0
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)
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tol = 1e-15
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np.testing.assert_allclose(points, expected, rtol=0, atol=tol)
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# Also check all-at-once:
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actual = Chebyshev2.Points(N, 0, 20)
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np.testing.assert_allclose(actual, expected, rtol=0, atol=tol)
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def test_Decomposition(self):
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"""Test fitting a linear function with Chebyshev basis."""
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# Create a sequence: dictionary mapping x -> y.
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sequence = {}
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for i in range(16):
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x_val = (1.0 / 16) * i - 0.99
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sequence[x_val] = x_val
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fit = gtsam.FitBasisChebyshev2(sequence, None, 3)
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params = fit.parameters()
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expected = np.array([-1.0, 0.0, 1.0])
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np.testing.assert_allclose(params, expected, rtol=0, atol=1e-4)
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def test_DifferentiationMatrix3(self):
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"""Test the 3×3 differentiation matrix against known values."""
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N = 3
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# Expected differentiation matrix (from chebfun) then multiplied by -1.
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expected = np.array([[1.5, -2.0, 0.5], [0.5, -0.0, -0.5], [-0.5, 2.0, -1.5]])
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expected = -expected
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actual = Chebyshev2.DifferentiationMatrix(N)
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np.testing.assert_allclose(actual, expected, rtol=0, atol=1e-4)
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def test_DerivativeMatrix6(self):
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"""Test the 6×6 differentiation matrix against known values."""
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N = 6
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expected = np.array(
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[
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[8.5000, -10.4721, 2.8944, -1.5279, 1.1056, -0.5000],
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[2.6180, -1.1708, -2.0000, 0.8944, -0.6180, 0.2764],
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[-0.7236, 2.0000, -0.1708, -1.6180, 0.8944, -0.3820],
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[0.3820, -0.8944, 1.6180, 0.1708, -2.0000, 0.7236],
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[-0.2764, 0.6180, -0.8944, 2.0000, 1.1708, -2.6180],
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[0.5000, -1.1056, 1.5279, -2.8944, 10.4721, -8.5000],
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]
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)
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expected = -expected
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actual = Chebyshev2.DifferentiationMatrix(N)
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np.testing.assert_allclose(actual, expected, rtol=0, atol=1e-4)
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def test_CalculateWeights(self):
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"""Test interpolation weights for a cubic function at arbitrary points."""
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N = 32
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fvals = Chebyshev2_vector(f, N)
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x1, x2 = 0.7, -0.376
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w1 = Chebyshev2.CalculateWeights(N, x1)
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w2 = Chebyshev2.CalculateWeights(N, x2)
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self.assertAlmostEqual(w1.dot(fvals), f(x1), delta=1e-8)
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self.assertAlmostEqual(w2.dot(fvals), f(x2), delta=1e-8)
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def test_CalculateWeights2(self):
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"""Test interpolation weights in arbitrary interval [a,b]."""
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N = 32
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a, b = 0.0, 10.0
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x1, x2 = 7.0, 4.12
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fvals = Chebyshev2_vector(f, N, a, b)
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w1 = Chebyshev2.CalculateWeights(N, x1, a, b)
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self.assertAlmostEqual(w1.dot(fvals), f(x1), delta=1e-8)
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w2 = Chebyshev2.CalculateWeights(N, x2, a, b)
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self.assertAlmostEqual(w2.dot(fvals), f(x2), delta=1e-8)
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def test_CalculateWeights_CoincidingPoint(self):
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"""Test that weights are correctly computed when x coincides with a Chebyshev point."""
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N = 5
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coincidingPoint = Chebyshev2.Point(N, 1)
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w = Chebyshev2.CalculateWeights(N, coincidingPoint)
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tol = 1e-9
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for j in range(N):
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expected = 1.0 if j == 1 else 0.0
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self.assertAlmostEqual(w[j], expected, delta=tol)
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def test_DerivativeWeights(self):
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"""Test derivative weights for polynomial function at arbitrary points."""
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N = 32
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fvals = Chebyshev2_vector(f, N)
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for x in [0.7, -0.376, 0.0]:
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dw = Chebyshev2.DerivativeWeights(N, x)
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self.assertAlmostEqual(dw.dot(fvals), fprime(x), delta=1e-9)
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x4 = Chebyshev2.Point(N, 3)
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dw4 = Chebyshev2.DerivativeWeights(N, x4)
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self.assertAlmostEqual(dw4.dot(fvals), fprime(x4), delta=1e-9)
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def test_DerivativeWeights2(self):
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"""Test derivative weights in arbitrary interval [a,b]."""
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N = 32
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a, b = 0.0, 10.0
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x1, x2 = 5.0, 4.12
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fvals = Chebyshev2_vector(f, N, a, b)
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dw1 = Chebyshev2.DerivativeWeights(N, x1, a, b)
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self.assertAlmostEqual(dw1.dot(fvals), fprime(x1), delta=1e-8)
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dw2 = Chebyshev2.DerivativeWeights(N, x2, a, b)
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self.assertAlmostEqual(dw2.dot(fvals), fprime(x2), delta=1e-8)
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x3 = Chebyshev2.Point(N, 3, a, b)
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dw3 = Chebyshev2.DerivativeWeights(N, x3, a, b)
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self.assertAlmostEqual(dw3.dot(fvals), fprime(x3), delta=1e-8)
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def test_DerivativeWeightsDifferentiationMatrix(self):
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"""Test that derivative weights match multiplication by differentiation matrix."""
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N6 = 6
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x1 = 0.311
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D6 = Chebyshev2.DifferentiationMatrix(N6)
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expected = Chebyshev2.CalculateWeights(N6, x1).dot(D6)
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actual = Chebyshev2.DerivativeWeights(N6, x1)
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np.testing.assert_allclose(actual, expected, rtol=0, atol=1e-12)
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a, b, x2 = -3.0, 8.0, 5.05
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D6_2 = Chebyshev2.DifferentiationMatrix(N6, a, b)
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expected1 = Chebyshev2.CalculateWeights(N6, x2, a, b).dot(D6_2)
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actual1 = Chebyshev2.DerivativeWeights(N6, x2, a, b)
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np.testing.assert_allclose(actual1, expected1, rtol=0, atol=1e-12)
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def test_DerivativeWeights6(self):
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"""Test that differentiating the identity function gives a constant."""
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N6 = 6
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D6 = Chebyshev2.DifferentiationMatrix(N6)
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x = Chebyshev2.Points(N6) # ramp with slope 1
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ones = np.ones(N6)
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np.testing.assert_allclose(D6.dot(x), ones, rtol=0, atol=1e-9)
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def test_DerivativeWeights7(self):
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"""Test that differentiating the identity function gives a constant (N=7)."""
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N7 = 7
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D7 = Chebyshev2.DifferentiationMatrix(N7)
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x = Chebyshev2.Points(N7)
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ones = np.ones(N7)
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np.testing.assert_allclose(D7.dot(x), ones, rtol=0, atol=1e-9)
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def test_IntegrationMatrix(self):
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"""Test integration matrix properties and accuracy on polynomial functions."""
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N = 10
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a, b = 0.0, 10.0
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P = Chebyshev2.IntegrationMatrix(N, a, b)
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F = P.dot(np.ones(N))
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self.assertAlmostEqual(F[0], 0.0, delta=1e-9)
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points = Chebyshev2.Points(N, a, b)
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ramp = points - a
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np.testing.assert_allclose(F, ramp, rtol=0, atol=1e-9)
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fp = Chebyshev2_vector(fprime, N, a, b)
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F_est = P.dot(fp)
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self.assertAlmostEqual(F_est[0], 0.0, delta=1e-9)
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F_est += f(a)
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F_true = Chebyshev2_vector(f, N, a, b)
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np.testing.assert_allclose(F_est, F_true, rtol=0, atol=1e-9)
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D = Chebyshev2.DifferentiationMatrix(N, a, b)
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ff_est = D.dot(F_est)
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np.testing.assert_allclose(ff_est, fp, rtol=0, atol=1e-9)
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def test_IntegrationWeights7(self):
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"""Test integration weights against known values for N=7."""
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N = 7
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actual = Chebyshev2.IntegrationWeights(N, -1, 1)
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expected = np.array(
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[
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0.0285714285714286,
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0.253968253968254,
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0.457142857142857,
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0.520634920634921,
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0.457142857142857,
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0.253968253968254,
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0.0285714285714286,
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]
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)
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np.testing.assert_allclose(actual, expected, rtol=0, atol=1e-9)
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self.assertAlmostEqual(np.sum(actual), 2.0, delta=1e-9)
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fp = Chebyshev2_vector(fprime, N)
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expectedF = f(1) - f(-1)
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self.assertAlmostEqual(actual.dot(fp), expectedF, delta=1e-9)
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P = Chebyshev2.IntegrationMatrix(N)
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p7 = P[-1, :]
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self.assertAlmostEqual(p7.dot(fp), expectedF, delta=1e-9)
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fvals = Chebyshev2_vector(f, N)
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self.assertAlmostEqual(p7.dot(fvals), actual.dot(fvals), delta=1e-9)
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def test_IntegrationWeights8(self):
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"""Test integration weights against known values for N=8."""
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N = 8
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actual = Chebyshev2.IntegrationWeights(N, -1, 1)
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expected = np.array(
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[
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0.0204081632653061,
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0.190141007218208,
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0.352242423718159,
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0.437208405798326,
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0.437208405798326,
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0.352242423718159,
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0.190141007218208,
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0.0204081632653061,
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]
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)
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np.testing.assert_allclose(actual, expected, rtol=0, atol=1e-9)
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self.assertAlmostEqual(np.sum(actual), 2.0, delta=1e-9)
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def test_DoubleIntegrationWeights(self):
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"""Test double integration weights for constant function (N=7)."""
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N = 7
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a, b = 0.0, 10.0
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P = Chebyshev2.IntegrationMatrix(N, a, b)
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ones = np.ones(N)
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w = Chebyshev2.DoubleIntegrationWeights(N, a, b)
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self.assertAlmostEqual(w.dot(ones), b * b / 2.0, delta=1e-9)
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def test_DoubleIntegrationWeights2(self):
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"""Test double integration weights for constant function (N=8)."""
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N = 8
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a, b = 0.0, 3.0
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P = Chebyshev2.IntegrationMatrix(N, a, b)
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ones = np.ones(N)
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w = Chebyshev2.DoubleIntegrationWeights(N, a, b)
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self.assertAlmostEqual(w.dot(ones), b * b / 2.0, delta=1e-9)
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if __name__ == "__main__":
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unittest.main()
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