| 1 |
{ RCSid $Id$ } |
| 2 |
{ |
| 3 |
Gaussian Integration |
| 4 |
|
| 5 |
10/20/87 |
| 6 |
|
| 7 |
order - order of integral (# of sample points) |
| 8 |
xmin - interval begin |
| 9 |
xmax - interval end |
| 10 |
f(x) - integrand |
| 11 |
integ - integral value |
| 12 |
} |
| 13 |
{ Defaults } |
| 14 |
order = 24 ; |
| 15 |
xmin = -1 ; |
| 16 |
xmax = 1 ; |
| 17 |
|
| 18 |
integ = q1(ceil(order/2-.25)) * (xmax-xmin)/2 ; |
| 19 |
|
| 20 |
q1(i) = if( i-1.5, wght(i)*(f1(absc(i))+f1(-absc(i))) + q1(i-1), |
| 21 |
wght(1)*(f1(absc(1))+if(even,f1(-absc(1)),0)) ) ; |
| 22 |
|
| 23 |
even = floor(order/2+.25) - order/2 + .25 ; |
| 24 |
|
| 25 |
f1(u) = f((xmax*(1+u)+xmin*(1-u))/2) ; |
| 26 |
|
| 27 |
{ Abscissas } |
| 28 |
absc(i) = select( order, |
| 29 |
0, |
| 30 |
.577350269189626, { n = 2 } |
| 31 |
select( i, { n = 3 } |
| 32 |
0, |
| 33 |
.774596669241483 |
| 34 |
), |
| 35 |
0, 0, |
| 36 |
select( i, { n = 6 } |
| 37 |
.238619186083197, |
| 38 |
.661209386466265, |
| 39 |
.932469514203152 |
| 40 |
), |
| 41 |
0, 0, 0, 0, 0, |
| 42 |
select( i, { n = 12 } |
| 43 |
.125233408511469, |
| 44 |
.367831498998180, |
| 45 |
.587317954286617, |
| 46 |
.769902674194305, |
| 47 |
.904117256370475, |
| 48 |
.981560634246719 |
| 49 |
), |
| 50 |
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, |
| 51 |
select( i, { n = 24 } |
| 52 |
.064056892862606, |
| 53 |
.191118867473616, |
| 54 |
.315042679696163, |
| 55 |
.433793507626045, |
| 56 |
.545421471388840, |
| 57 |
.648093651936976, |
| 58 |
.740124191578554, |
| 59 |
.820001985973903, |
| 60 |
.886415527004401, |
| 61 |
.938274552002733, |
| 62 |
.974728555971309, |
| 63 |
.995187219997021 |
| 64 |
) |
| 65 |
) ; |
| 66 |
|
| 67 |
{ Weights } |
| 68 |
wght(i) = select( order, |
| 69 |
0, |
| 70 |
1, { n = 2 } |
| 71 |
select( i, { n = 3 } |
| 72 |
.888888888888888, |
| 73 |
.555555555555555 |
| 74 |
), |
| 75 |
0, 0, |
| 76 |
select( i, { n = 6 } |
| 77 |
.467913934572691, |
| 78 |
.360761573048139, |
| 79 |
.171324492379170 |
| 80 |
), |
| 81 |
0, 0, 0, 0, 0, |
| 82 |
select( i, { n = 12 } |
| 83 |
.249147045813403, |
| 84 |
.233492536538355, |
| 85 |
.203167426723066, |
| 86 |
.160078328543346, |
| 87 |
.106939325995318, |
| 88 |
.047175336386512 |
| 89 |
), |
| 90 |
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, |
| 91 |
select( i, { n = 24 } |
| 92 |
.127938195346752, |
| 93 |
.125837456346828, |
| 94 |
.121670472927803, |
| 95 |
.115505668053726, |
| 96 |
.107444270115966, |
| 97 |
.097618652104114, |
| 98 |
.086190161531953, |
| 99 |
.073346481411080, |
| 100 |
.059298584915437, |
| 101 |
.044277438817420, |
| 102 |
.028531388628934, |
| 103 |
.012341229799987 |
| 104 |
) |
| 105 |
) ; |
