02-Mar-2019 15:25:58 quadrature_least_squares_test MATLAB version 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 2.2145 0.532304 LS 0 2.2145 0.532304 LS 1 2.2373 0.509502 LS 2 2.40531 0.341496 LS 3 2.36331 0.383487 LS 4 2.60629 0.140513 LS 5 2.72216 0.0246414 LS 6 2.53962 0.207184 LS 7 2.47627 0.270531 LS 8 2.62275 0.124049 LS 9 2.70048 0.0463255 LS10 2.54176 0.205045 LS11 2.35485 0.391947 LS12 2.73685 0.00995186 LS13 3.21145 0.464653 LS14 2.65108 0.0957229 LS15 1.634 1.1128 EXACT 2.7468 0 quadrature_least_squares_test Normal end of execution. 02-Mar-2019 15:25:58