27-Jul-2021 20:35:17 chebyshev2_exactness_test: MATLAB/Octave version 9.9.0.1467703 (R2020b). Test chebyshev2_exactness. 27-Jul-2021 20:35:17 CHEBYSHEV2_EXACTNESS MATLAB/Octave version 9.9.0.1467703 (R2020b) Investigate the polynomial exactness of a Gauss-Chebyshev1 type 2 quadrature rule by integrating all monomials up to a given degree over the [-1,+1] interval. CHEBYSHEV2_EXACTNESS: User input: Quadrature rule X file = "cheby2_o4_x.txt". Quadrature rule W file = "cheby2_o4_w.txt". Quadrature rule R file = "cheby2_o4_r.txt". Maximum degree to check = 10 Spatial dimension = 1 Number of points = 4 The quadrature rule to be tested is a Gauss-Chebyshev type 2 rule ORDER = 4 Standard rule: Integral ( -1 <= x <= +1 ) f(x) * ( 1 - x^2 ) dx is to be approximated by sum ( 1 <= I <= ORDER ) w(i) * f(x(i)). Weights W: w(1) = 0.2170787134227061 w(2) = 0.5683194499747424 w(3) = 0.5683194499747423 w(4) = 0.2170787134227060 Abscissas X: x(1) = -0.8090169943749473 x(2) = -0.3090169943749473 x(3) = 0.3090169943749475 x(4) = 0.8090169943749475 Region R: r(1) = -1.000000e+00 r(2) = 1.000000e+00 A Gauss-Chebyshev type 2 rule would be able to exactly integrate monomials up to and including degree = 7 Error Degree 0.0000000000000001 0 0.0000000000000000 1 0.0000000000000000 2 0.0000000000000000 3 0.0000000000000001 4 0.0000000000000000 5 0.0000000000000001 6 0.0000000000000000 7 0.0714285714285715 8 0.0000000000000000 9 0.1904761904761904 10 CHEBYSHEV2_EXACTNESS: Normal end of execution. 27-Jul-2021 20:35:17 chebyshev2_exactness_test: Normal end of execution. 27-Jul-2021 20:35:17