Mathematical Methods for Physics and Engineering : A Comprehensive Guide

NUMERICAL METHODS

Gauss–Laguerre and Gauss–Hermite integration

∫∞

0

e−xf(x)dx=

∑n

i=1

wif(xi)

∫∞

−∞

e−x

2

f(x)dx=

∑n

i=1

wif(xi)

xi wi ±xi wi

n=2 n=2

0.58578 64376 0.85355 33906 0.70710 67812 0.88622 69255

3.41421 35624 0.14644 66094

n=3

n=3 0.00000 00000 1.18163 59006

0.41577 45568 0.71109 30099 1.22474 48714 0.29540 89752

2.29428 03603 0.27851 77336

6.28994 50829 0.01038 92565 n=4

0.52464 76233 0.80491 40900

n=4 1.65068 01239 0.08131 28354

0.32254 76896 0.60315 41043

1.74576 11012 0.35741 86924 n=5

4.53662 02969 0.03888 79085 0.00000 00000 0.94530 87205

9.39507 09123 0.00053 92947 0.95857 24646 0.39361 93232

2.02018 28705 0.01995 32421

n=5

0.26356 03197 0.52175 56106 n=6

1.41340 30591 0.39866 68111 0.43607 74119 0.72462 95952

3.59642 57710 0.07594 24497 1.33584 90740 0.15706 73203

7.08581 00059 0.00361 17587 2.35060 49737 0.00453 00099

12.6408 00844 0.00002 33700

n=7

n=6 0.00000 00000 0.81026 46176

0.22284 66042 0.45896 46740 0.81628 78829 0.42560 72526

1.18893 21017 0.41700 08308 1.67355 16288 0.05451 55828

2.99273 63261 0.11337 33821 2.65196 13568 0.00097 17812

5.77514 35691 0.01039 91975

9.83746 74184 0.00026 10172 n=8

15.9828 73981 0.00000 08985 0.38118 69902 0.66114 70126

1.15719 37124 0.20780 23258

n=7 1.98165 67567 0.01707 79830

0.19304 36766 0.40931 89517 2.93063 74203 0.00019 96041

1.02666 48953 0.42183 12779

2.56787 67450 0.14712 63487 n=9

4.90035 30845 0.02063 35145 0.00000 00000 0.72023 52156

8.18215 34446 0.00107 40101 0.72355 10188 0.43265 15590

12.7341 80292 0.00001 58655 1.46855 32892 0.08847 45274

19.3957 27862 0.00000 00317 2.26658 05845 0.00494 36243

3.19099 32018 0.00003 96070

Table 27.9 The integration points and weights for a number ofn-point
Gauss–Laguerre and Gauss–Hermite integration formulae. Where the points
are given as±xi, the contributions from both +xiand−ximust be included.
However, the contribution from any pointxi= 0 must be counted only once.