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IntegralGaussLagendre_test.go
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IntegralGaussLagendre_test.go
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// IntegralGaussLagendre_test
/*
------------------------------------------------------
作者 : Black Ghost
日期 : 2018-12-12
版本 : 0.0.0
------------------------------------------------------
不超过8次的Gauss-Lagendre求积分公式
理论：
对于积分
b
|f(x)dx
a
使用n+1次Lagendre多项式的零点作为高斯点，可获得代数
精度为2n+1的高斯型求积公式
其中区间[a, b]需要预先转换为区间[-1, 1]
参考 李信真, 车刚明, 欧阳洁, 等. 计算方法. 西北工业大学
出版社, 2000, pp 162-164.
------------------------------------------------------
输入 :
fun 被积分函数
a, b 积分区间
n 求积分公式次数
输出 :
sol 解
err 解出标志：false-未解出或达到步数上限；
true-全部解出
------------------------------------------------------
*/
package goNum_test
import (
"math"
"testing"
"github.com/chfenger/goNum"
)
// IntegralGaussLagendre 不超过8次的Gauss-Lagendre求积分公式
func IntegralGaussLagendre(fun func(float64) float64, a, b float64, n int) (float64, bool) {
/*
不超过8次的Gauss-Lagendre求积分公式
输入 :
fun 被积分函数
a, b 积分区间
n 求积分公式次数
输出 :
sol 解
err 解出标志：false-未解出或达到步数上限；
true-全部解出
*/
//判断n范围
if (n < 1) || (n > 8) {
panic("Error in goNum.IntegralGaussLagendre: n is a not correct input")
}
xi := [][]float64{
{0.0},
{-0.5773502692, 0.5773502692},
{-0.7745966692, 0.0, 0.7745966692},
{-0.8611363116, -0.3399810436, 0.3399810436, 0.8611363116},
{-0.9061798459, -0.5384693101, 0.0, 0.5384693101, 0.9061798459},
{-0.9324695142, -0.6612093865, -0.2386191861, 0.2386191861, 0.6612093865, 0.9324695142},
{-0.9491079123, -0.7415311856, -0.4058451514, 0.0, 0.4058451514, 0.7415311856, 0.9491079123},
{-0.9602898566, -0.7966664774, -0.5255324099, -0.1834346425, 0.1834346425, 0.5255324099, 0.7966664774, 0.9602898566},
}
Ai := [][]float64{
{2.0},
{1.0, 1.0},
{0.555555555555556, 0.888888888888889, 0.555555555555556},
{0.3478548451, 0.6521451549, 0.6521451549, 0.3478548451},
{0.2369268851, 0.4786286705, 0.568888889, 0.4786286705, 0.2369268851},
{0.1713244924, 0.3607615730, 0.4679139346, 0.4679139346, 0.3607615730, 0.1713244924},
{0.1294849662, 0.2797053915, 0.3818300505, 0.4179591837, 0.3818300505, 0.2797053915, 0.1294849662},
{0.1012285363, 0.2223810345, 0.3137066459, 0.3626837834, 0.3626837834, 0.3137066459, 0.2223810345, 0.1012285363},
}
//区间转换
c := (b - a) / 2.0
d := (a + b) / 2.0
switch n {
case 1:
sol := 0.0
for i := 0; i < len(xi[0]); i++ {
sol += Ai[0][i] * fun(d+c*xi[0][i])
}
return c * sol, true
case 2:
sol := 0.0
for i := 0; i < len(xi[1]); i++ {
sol += Ai[1][i] * fun(d+c*xi[1][i])
}
return c * sol, true
case 3:
sol := 0.0
for i := 0; i < len(xi[2]); i++ {
sol += Ai[2][i] * fun(d+c*xi[2][i])
}
return c * sol, true
case 4:
sol := 0.0
for i := 0; i < len(xi[3]); i++ {
sol += Ai[3][i] * fun(d+c*xi[3][i])
}
return c * sol, true
case 5:
sol := 0.0
for i := 0; i < len(xi[4]); i++ {
sol += Ai[4][i] * fun(d+c*xi[4][i])
}
return c * sol, true
case 6:
sol := 0.0
for i := 0; i < len(xi[5]); i++ {
sol += Ai[5][i] * fun(d+c*xi[5][i])
}
return c * sol, true
case 7:
sol := 0.0
for i := 0; i < len(xi[6]); i++ {
sol += Ai[6][i] * fun(d+c*xi[6][i])
}
return c * sol, true
case 8:
sol := 0.0
for i := 0; i < len(xi[7]); i++ {
sol += Ai[7][i] * fun(d+c*xi[7][i])
}
return c * sol, true
default:
return 0.0, false
}
}
func fun38(x float64) float64 {
return x * x * math.Cos(x)
}
func BenchmarkIntegralGaussLagendre(b *testing.B) {
for i := 0; i < b.N; i++ {
goNum.IntegralGaussLagendre(fun38, 0.0, math.Pi/2.0, 4) //0.467402...
}
}