15-May-2025 20:28:34 quadrature_least_squares_test(): MATLAB/Octave version 6.4.0 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 3.21624 0.469436 LS 0 3.21624 0.469436 LS 1 3.24901 0.502204 LS 2 2.54177 0.205036 LS 3 2.48485 0.261955 LS 4 3.02449 0.277684 LS 5 3.07607 0.329266 LS 6 2.56689 0.179907 LS 7 2.45536 0.291443 LS 8 2.95253 0.205723 LS 9 3.31748 0.570682 LS10 2.21984 0.526961 LS11 1.43838 1.30843 LS12 3.57805 0.831253 LS13 7.08247 4.33567 LS14 1.72478 1.02202 LS15 -10.3575 13.1043 EXACT 2.7468 0 quadrature_least_squares_test(): Normal end of execution. 15-May-2025 20:28:34