131 lines
3.8 KiB
Go
131 lines
3.8 KiB
Go
// IntegralGaussLagendre
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/*
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------------------------------------------------------
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作者 : Black Ghost
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日期 : 2018-12-12
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版本 : 0.0.0
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------------------------------------------------------
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不超过8次的Gauss-Lagendre求积分公式
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理论：
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对于积分
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b
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|f(x)dx
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a
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使用n+1次Lagendre多项式的零点作为高斯点，可获得代数
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精度为2n+1的高斯型求积公式
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其中区间[a, b]需要预先转换为区间[-1, 1]
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参考 李信真, 车刚明, 欧阳洁, 等. 计算方法. 西北工业大学
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出版社, 2000, pp 162-164.
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------------------------------------------------------
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输入 :
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fun 被积分函数
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a, b 积分区间
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n 求积分公式次数
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输出 :
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sol 解
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err 解出标志：false-未解出或达到步数上限；
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true-全部解出
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------------------------------------------------------
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*/
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package goNum
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// IntegralGaussLagendre 不超过8次的Gauss-Lagendre求积分公式
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func IntegralGaussLagendre(fun func(float64) float64, a, b float64, n int) (float64, bool) {
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/*
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不超过8次的Gauss-Lagendre求积分公式
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输入 :
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fun 被积分函数
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a, b 积分区间
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n 求积分公式次数
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输出 :
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sol 解
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err 解出标志：false-未解出或达到步数上限；
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true-全部解出
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*/
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//判断n范围
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if (n < 1) || (n > 8) {
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panic("Error in goNum.IntegralGaussLagendre: n is a not correct input")
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}
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xi := [][]float64{
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{0.0},
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{-0.5773502692, 0.5773502692},
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{-0.7745966692, 0.0, 0.7745966692},
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{-0.8611363116, -0.3399810436, 0.3399810436, 0.8611363116},
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{-0.9061798459, -0.5384693101, 0.0, 0.5384693101, 0.9061798459},
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{-0.9324695142, -0.6612093865, -0.2386191861, 0.2386191861, 0.6612093865, 0.9324695142},
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{-0.9491079123, -0.7415311856, -0.4058451514, 0.0, 0.4058451514, 0.7415311856, 0.9491079123},
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{-0.9602898566, -0.7966664774, -0.5255324099, -0.1834346425, 0.1834346425, 0.5255324099, 0.7966664774, 0.9602898566},
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}
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Ai := [][]float64{
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{2.0},
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{1.0, 1.0},
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{0.555555555555556, 0.888888888888889, 0.555555555555556},
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{0.3478548451, 0.6521451549, 0.6521451549, 0.3478548451},
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{0.2369268851, 0.4786286705, 0.568888889, 0.4786286705, 0.2369268851},
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{0.1713244924, 0.3607615730, 0.4679139346, 0.4679139346, 0.3607615730, 0.1713244924},
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{0.1294849662, 0.2797053915, 0.3818300505, 0.4179591837, 0.3818300505, 0.2797053915, 0.1294849662},
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{0.1012285363, 0.2223810345, 0.3137066459, 0.3626837834, 0.3626837834, 0.3137066459, 0.2223810345, 0.1012285363},
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}
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//区间转换
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c := (b - a) / 2.0
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d := (a + b) / 2.0
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switch n {
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case 1:
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sol := 0.0
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for i := 0; i < len(xi[0]); i++ {
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sol += Ai[0][i] * fun(d+c*xi[0][i])
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}
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return c * sol, true
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case 2:
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sol := 0.0
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for i := 0; i < len(xi[1]); i++ {
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sol += Ai[1][i] * fun(d+c*xi[1][i])
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}
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return c * sol, true
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case 3:
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sol := 0.0
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for i := 0; i < len(xi[2]); i++ {
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sol += Ai[2][i] * fun(d+c*xi[2][i])
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}
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return c * sol, true
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case 4:
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sol := 0.0
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for i := 0; i < len(xi[3]); i++ {
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sol += Ai[3][i] * fun(d+c*xi[3][i])
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}
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return c * sol, true
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case 5:
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sol := 0.0
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for i := 0; i < len(xi[4]); i++ {
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sol += Ai[4][i] * fun(d+c*xi[4][i])
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}
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return c * sol, true
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case 6:
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sol := 0.0
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for i := 0; i < len(xi[5]); i++ {
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sol += Ai[5][i] * fun(d+c*xi[5][i])
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}
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return c * sol, true
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case 7:
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sol := 0.0
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for i := 0; i < len(xi[6]); i++ {
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sol += Ai[6][i] * fun(d+c*xi[6][i])
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}
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return c * sol, true
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case 8:
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sol := 0.0
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for i := 0; i < len(xi[7]); i++ {
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sol += Ai[7][i] * fun(d+c*xi[7][i])
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}
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return c * sol, true
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default:
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return 0.0, false
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}
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}
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