08-May-2023 10:48:21 quadrature_least_squares_test(): MATLAB/Octave version 5.2.0 Test quadrature_least_squares(). quadrature_least_squares_test01(): weights_ls() computes the weights for a least squares quadrature rule. W1 = classical Newton Cotes weights, N = 5 W2 = least squares weights, D = 4, N = 5 I X(i) W1(i) W2(i) 1 -1.0000 0.155556 0.155556 2 -0.5000 0.711111 0.711111 3 0.0000 0.266667 0.266667 4 0.5000 0.711111 0.711111 5 1.0000 0.155556 0.155556 W1 = classical Newton Cotes weights, N = 9 W2 = least squares weights, D = 4, N = 9 I X(i) W1(i) W2(i) 1 -1.0000 0.0697707 0.0960373 2 -0.7500 0.415379 0.270085 3 -0.5000 -0.0654674 0.280963 4 -0.2500 0.740459 0.242113 5 0.0000 -0.320282 0.221601 6 0.2500 0.740459 0.242113 7 0.5000 -0.0654674 0.280963 8 0.7500 0.415379 0.270085 9 1.0000 0.0697707 0.0960373 quadrature_least_squares_test02(): weights_ls() computes the weights for a least squares quadrature rule. Pick 50 random values in [-1,+1]. Compare Monte Carlo (equal weight) integral estimate to least squares estimates of degree D = 0, 1, 2, 3, 4. For low values of D, the least squares estimate improves. As D increases, the estimate can deteriorate. Rule Estimate Error MC 2.57539 0.171414 LS 0 2.57539 0.171414 LS 1 2.55949 0.187312 LS 2 2.56589 0.180908 LS 3 2.5559 0.190901 LS 4 2.5838 0.162998 LS 5 2.59443 0.152373 LS 6 2.6547 0.0921002 LS 7 2.71541 0.0313926 LS 8 2.81832 0.0715188 LS 9 2.81847 0.0716675 LS10 2.78903 0.0422262 LS11 2.79529 0.0484921 LS12 2.85754 0.110737 LS13 2.85813 0.111325 LS14 2.69469 0.0521162 LS15 2.64374 0.103058 EXACT 2.7468 0 quadrature_least_squares_test(): Normal end of execution. 08-May-2023 10:48:21