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qls_prb_output.txt
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15 April 2014 10:18:45 AM
QLS_PRB
C version
Test the QUADRATURE_LEAST_SQUARES library.
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)
0 -1.0000 0.155556 0.155556
1 -0.5000 0.711111 0.711111
2 0.0000 0.266667 0.266667
3 0.5000 0.711111 0.711111
4 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)
0 -1.0000 0.0697707 0.0960373
1 -0.7500 0.415379 0.270085
2 -0.5000 -0.0654674 0.280963
3 -0.2500 0.740459 0.242113
4 0.0000 -0.320282 0.221601
5 0.2500 0.740459 0.242113
6 0.5000 -0.0654674 0.280963
7 0.7500 0.415379 0.270085
8 1.0000 0.0697707 0.0960373
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 0.00995189
LS13 3.21145 0.464653
LS14 2.65108 0.095724
LS15 1.634 1.1128
EXACT 2.7468 0
QLS_PRB
Normal end of execution.
15 April 2014 10:18:45 AM