176 lines
5.8 KiB
Go
176 lines
5.8 KiB
Go
// InterpHermiteFunc
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/*
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------------------------------------------------------
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作者 : Black Ghost
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日期 : 2018-12-7
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版本 : 0.0.0
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------------------------------------------------------
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计算不高于2n+1次Hermite插值方程，拟合n+1个函数值数据
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点和对应的n+1个一阶导数点
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满阶插值，即阶数不高于2n+1
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理论：
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n
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H2n+1(x) = Sum (alphaj(x)*yj+betaj(x)*mj)
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j=0
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yj, mj分别为函数值和一阶导数值
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n 1
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alphaj(x) = (1-2(x-xj)*Sum -------)lj^2(x)
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k=0,k!=j xj-xk
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betaj(x) = (x-xj)lj^2(x)
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(x-x0)(x-x1)...(x-xn)
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lj(x) = --------------------------, (被减数不含xj项)
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(xj-x0)(xj-x1)...(xj-xn)
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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 111-113.
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------------------------------------------------------
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输入 :
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A 数据点矩阵，(n+1)x3，第一列xi；第二列yi；第三列y'i
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输出 :
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B 插值方程系数结果，从前到后对应从0到2n+1阶，(2n+2)x1
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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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//求解lj(x)
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func ljx_InterpHermiteFunc(A Matrix, j int) Matrix {
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Bljx := ZeroMatrix(A.Rows, 1) //出去j，加常数项总共n+1
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xj := A.GetFromMatrix(j, 0)
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// j!=0
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temp0 := (xj - A.GetFromMatrix(0, 0)) //x0
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Bljx.Data[0] = -1.0 * A.GetFromMatrix(0, 0)
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Bljx.Data[1] = 1.0
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// j==0
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if j == 0 {
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temp0 = (xj - A.GetFromMatrix(1, 0)) //x1
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Bljx.Data[0] = -1.0 * A.GetFromMatrix(1, 0)
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}
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//其它
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for i := 1; i < A.Rows; i++ {
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if (i != j) && (((j == 0) && (i > 1)) || (j > 0)) {
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if i < j {
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CA := ZeroMatrix(i+2, 1) //实际i+2行
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CB := ZeroMatrix(i+2, 1) //实际i+1行
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//先用x乘以之前每一项，相当于给每一项提升一阶,i+1
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//再用-xi乘以B的每一有效项,i+1
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CB.Data[0] = -1.0 * A.GetFromMatrix(i, 0) * Bljx.Data[0]
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for ii := 1; ii < i+1; ii++ {
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//单列可以这样，否则只能用SetMatrix和GetFromMatrix方法
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CA.Data[ii] = Bljx.Data[ii-1]
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CB.Data[ii] = -1.0 * A.GetFromMatrix(i, 0) * Bljx.Data[ii]
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}
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CA.Data[i+1] = Bljx.Data[i]
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//同阶相加赋予B
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for ii := 0; ii < i+2; ii++ {
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Bljx.Data[ii] = CA.Data[ii] + CB.Data[ii]
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}
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} else { //i>j
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CA := ZeroMatrix(i+1, 1) //实际i+1行
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CB := ZeroMatrix(i+1, 1) //实际i行
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//先用x乘以之前每一项，相当于给每一项提升一阶,i+1
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//再用-xi乘以B的每一有效项,i
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CB.Data[0] = -1.0 * A.GetFromMatrix(i, 0) * Bljx.Data[0]
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for ii := 1; ii < i; ii++ {
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//单列可以这样，否则只能用SetMatrix和GetFromMatrix方法
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CA.Data[ii] = Bljx.Data[ii-1]
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CB.Data[ii] = -1.0 * A.GetFromMatrix(i, 0) * Bljx.Data[ii]
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}
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CA.Data[i] = Bljx.Data[i-1]
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//同阶相加赋予B
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for ii := 0; ii < i+1; ii++ {
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Bljx.Data[ii] = CA.Data[ii] + CB.Data[ii]
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}
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}
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temp0 = temp0 * (xj - A.GetFromMatrix(i, 0))
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}
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}
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for i := 0; i < Bljx.Rows; i++ {
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Bljx.Data[i] = Bljx.Data[i] / temp0
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}
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return Bljx
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}
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//求解ljx^2
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func ljx2_InterpHermiteFunc(A Matrix, j int) Matrix {
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ljx := ljx_InterpHermiteFunc(A, j) //n+1 rows
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BA := ZeroMatrix(2*ljx.Rows-1, 1) //2n+1 rows
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for i := ljx.Rows - 1; i >= 0; i-- {
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for j := ljx.Rows - 1; j >= 0; j-- {
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BA.Data[i+j] = BA.Data[i+j] + ljx.Data[i]*ljx.Data[j]
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}
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}
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return BA
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}
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//求解alphajx和betajx，合并是为了减少对ljx2_InterpHermiteFunc的调用
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func alphabetajx_InterpHermiteFunc(A Matrix, j int) (Matrix, Matrix) {
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var temp0 float64
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ljx2 := ljx2_InterpHermiteFunc(A, j) //2n+1 rows
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alphajx := ZeroMatrix(ljx2.Rows+1, 1) //2n+2 rows
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betajx := ZeroMatrix(ljx2.Rows+1, 1) //2n+2 rows
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xj := A.GetFromMatrix(j, 0)
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//计算alphajx中的求和
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for k := 0; k < A.Rows; k++ {
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if k != j {
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temp0 += 1.0 / (xj - A.GetFromMatrix(k, 0))
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}
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}
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temp0 = 2.0 * temp0
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//alphajx = (temp0*xj+1 - temp0*x) * ljx2
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//betajx = (x-xj) * ljx2
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//2n+1阶,2n+2行
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alphajx.Data[alphajx.Rows-1] = -1.0 * temp0 * ljx2.Data[ljx2.Rows-1]
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betajx.Data[betajx.Rows-1] = ljx2.Data[ljx2.Rows-1]
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//其它非零阶, alphajx.Rows-2 == betajx.Rows-2 == ljx2.Rows-1
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for i := alphajx.Rows - 2; i > 0; i-- {
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alphajx.Data[i] = (temp0*xj+1.0)*ljx2.Data[i] - 1.0*temp0*ljx2.Data[i-1]
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betajx.Data[i] = -1.0*xj*ljx2.Data[i] + ljx2.Data[i-1]
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}
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//零阶
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alphajx.Data[0] = (temp0*xj + 1.0) * ljx2.Data[0]
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betajx.Data[0] = -1.0 * xj * ljx2.Data[0]
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return alphajx, betajx
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}
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// InterpHermiteFunc 计算不高于2n+1次Hermite插值方程，拟合n+1个函数值数据点和对应的n+1个一阶导数点
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func InterpHermiteFunc(A Matrix) (Matrix, bool) {
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/*
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计算不高于2n+1次Hermite插值方程，拟合n+1个函数值数据点和对应的n+1个一阶导数点
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输入 :
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A 数据点矩阵，(n+1)x3，第一列xi；第二列yi；第三列y'i
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输出 :
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B 插值方程系数结果，从前到后对应从0到2n+1阶，(2n+2)x1
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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A列数是否为3
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if A.Columns != 3 {
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panic("Error in goNum.InterpHermite: give me xi, yi and y'i")
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}
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var err bool = false
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n := A.Rows - 1
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BA := ZeroMatrix(2*n+2, 1)
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for j := 0; j <= n; j++ {
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alphajx, betajx := alphabetajx_InterpHermiteFunc(A, j)
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for i := 0; i < alphajx.Rows; i++ {
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BA.Data[i] = BA.Data[i] + alphajx.Data[i]*A.GetFromMatrix(j, 1)
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BA.Data[i] = BA.Data[i] + betajx.Data[i]*A.GetFromMatrix(j, 2)
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}
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}
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err = true
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return BA, err
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}
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