133 lines
3.7 KiB
Go
133 lines
3.7 KiB
Go
// IntegralRumberg
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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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Rumberg(龙贝格)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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b-a
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T1 = -----(f(a)+f(b))
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2
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1 b-a N b-a
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T2N = ---TN + -----Sum f(a+(2j-1)------)
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2 2N j=1 2N
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N=2^(k-1), k=1,2,3,...
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1 4T2N-TN
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SN = T2N + ---(T2N-TN) = --------
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3 4-1
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1 4^2S2N-SN
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CN = S2N + ----(S2N-SN) = -----------
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15 4^2-1
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1 4^3C2N-CN
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RN = C2N + ----(C2N-CN) = -----------
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63 4^3-1
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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 积分范围
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tol 控制误差
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Nn 最大循环步数
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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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import (
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"math"
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)
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// IntegralRumberg Rumberg(龙贝格)求积分公式
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func IntegralRumberg(fun func(float64) float64, a, b, tol float64, Nn int) (float64, bool) {
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/*
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Rumberg(龙贝格)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 积分范围
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tol 控制误差
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Nn 最大循环步数
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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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T := make([]float64, 0) //梯形序列
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S := make([]float64, 0) //辛浦生序列
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C := make([]float64, 0) //柯特斯序列
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R := make([]float64, 0) //龙贝格序列
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//第一步
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temp0 := (b - a) * (fun(a) + fun(b)) / 2.0
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T = append(T, temp0) //T[0]=T1
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//第二步, k=1
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temp0 = 0.0
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for j := 1; j < PowIInt(2, 0)+1; j++ {
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temp0 += fun(a + (2.0*float64(j)-1.0)*(b-a)/2.0)
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}
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temp0 = T[0]/2.0 + temp0*(b-a)/2.0
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T = append(T, temp0) //T[1]=T2
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temp1 := T[1] + (T[1]-T[0])/3.0
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S = append(S, temp1) //S[0]=S1
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//第三步, k=2
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temp0 = 0.0
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for j := 1; j < PowIInt(2, 1)+1; j++ {
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temp0 += fun(a + (2.0*float64(j)-1.0)*(b-a)/(2.0*2.0))
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}
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temp0 = T[1]/2.0 + temp0*(b-a)/(2.0*2.0)
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T = append(T, temp0) //T[2]=T4
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temp1 = T[2] + (T[2]-T[1])/3.0
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S = append(S, temp1) //S[1]=S2
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temp2 := S[1] + (S[1]-S[0])/15.0
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C = append(C, temp2) //C[0]=C1
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//第四步, k=3
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temp0 = 0.0
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for j := 1; j < PowIInt(2, 2)+1; j++ {
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temp0 += fun(a + (2.0*float64(j)-1.0)*(b-a)/(2.0*4.0))
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}
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temp0 = T[2]/2.0 + temp0*(b-a)/(2.0*4.0)
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T = append(T, temp0) //T[3]=T8
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temp1 = T[3] + (T[3]-T[2])/3.0
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S = append(S, temp1) //S[2]=S4
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temp2 = S[2] + (S[2]-S[1])/15.0
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C = append(C, temp2) //C[1]=C2
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temp3 := C[1] + (C[1]-C[0])/63.0
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R = append(R, temp3) //R[0]=R1
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//进入Rumberg循环
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for i := 1; i < Nn; i++ {
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temp0 = 0.0
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for j := 1; j < PowIInt(2, i+2)+1; j++ {
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temp0 += fun(a + (2.0*float64(j)-1.0)*(b-a)/(2.0*PowIIntF(2, i+2)))
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}
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temp0 = T[i+2]/2.0 + temp0*(b-a)/(2.0*PowIIntF(2, i+2))
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T = append(T, temp0) //T[i+3]
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temp1 = T[i+3] + (T[i+3]-T[i+2])/3.0
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S = append(S, temp1) //S[i+2]
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temp2 = S[i+2] + (S[i+2]-S[i+1])/15.0
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C = append(C, temp2) //C[i+1]
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temp3 = C[i+1] + (C[i+1]-C[i])/63.0
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R = append(R, temp3) //R[i]
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if math.Abs(R[i]-R[i-1]) < tol {
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return R[i], true
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}
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}
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return 0.0, false
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}
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