07-Jan-2022 19:16:28 equidistribution_test(): MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 Test equidistribution(). chebyshev1_test(): Examine the equidistribution property of the Chebyshev1 points X. The points X are defined on [-1,+1] They have density 1/pi * 1/sqrt(1-x^2) They can be bracketed by points X2 Let H(I) be the width of the interval containing X(I). Let R(I) be the density function at X(I). We seek an equidistribution property that relates H and R, probably of the form: H proportional to 1 / R^J, with J = 1, 2 or 3. So compute S(J) = H * R^J for each point. T = ( 2.0 * n + 1 - 2 * ( 1 : n ) ) * pi / ( 2 * n ) X = cos ( T ) T2 = ( 2.0 * n - 2 * ( 0 : n ) ) * pi / ( 2 * n ) X2 = cos ( T2 ) H = X2(2:n+1) - X2(1:n) R = 1/sqrt(1-x^2) S(J) = H / R^J, J = 1, 2, 3 I X H Rho H*Rho^1 H*Rho^2 H*Rho^3 1 -0.9957 0.01703 10.84 0.1845 2 21.68 2 -0.9618 0.0505 3.654 0.1845 0.6743 2.464 3 -0.8952 0.08226 2.243 0.1845 0.414 0.9288 4 -0.798 0.1112 1.659 0.1845 0.3062 0.5081 5 -0.6737 0.1364 1.353 0.1845 0.2497 0.3379 6 -0.5264 0.1569 1.176 0.1845 0.217 0.2553 7 -0.3612 0.1721 1.072 0.1845 0.1979 0.2122 8 -0.1837 0.1814 1.017 0.1845 0.1877 0.191 9 6.123e-17 0.1845 1 0.1845 0.1845 0.1845 10 0.1837 0.1814 1.017 0.1845 0.1877 0.191 11 0.3612 0.1721 1.072 0.1845 0.1979 0.2122 12 0.5264 0.1569 1.176 0.1845 0.217 0.2553 13 0.6737 0.1364 1.353 0.1845 0.2497 0.3379 14 0.798 0.1112 1.659 0.1845 0.3062 0.5081 15 0.8952 0.08226 2.243 0.1845 0.414 0.9288 16 0.9618 0.0505 3.654 0.1845 0.6743 2.464 17 0.9957 0.01703 10.84 0.1845 2 21.68 chebyshev1_test() Normal end of execution. equidistribution_test(): Normal end of execution. 07-Jan-2022 19:16:28