Muhammad两步三阶迭代格式
对于非线性方程\({F\left( x \right) = 0}\)，对方程函数在\({{x_n}}\)处泰勒展开得：
\[\begin{gathered}
F\left( x \right) = F\left( {{x_n}} \right) + \cdots + \frac{1}{{\left( {k - 1} \right)!}}{F^{\left( {k - 1} \right)}}\left( {{x_n}} \right){\left( {x - {x_n}} \right)^{\left( {k - 1} \right)}} + \hfill \\
\quad \quad \quad \int_0^1 {\frac{{{{\left( {1 - t} \right)}^{\left( {k - 1} \right)}}}}{{\left( {k - 1} \right)!}}{F^{\left( k \right)}}\left[ {{x_n} + t\left( {x - {x_n}} \right)} \right]} {\left( {x - {x_n}} \right)^k}dt \hfill \\
\end{gathered} \]
当\(k=1\)时，有
\[F\left( x \right) = F\left( {{x_n}} \right) + \int_0^1 {F'\left[ {{x_n} + t\left( {x - {x_n}} \right)} \right]} \left( {x - {x_n}} \right)dt\]
对上式采用不同的数值积分方法可以得到一系列迭代公式。
Muhammad Aslam Noor给出了如下两种的积分公式
\[\int_0^1 {F'\left[ {{x_n} + t\left( {x - {x_n}} \right)} \right]} \left( {x - {x_n}} \right)dt \approx \left[ {\frac{1}{4}F'\left( {{x_n}} \right) + \frac{3}{4}F'\left( {\frac{{{x_n} + 2x}}{3}} \right)} \right]\left( {x - {x_n}} \right)\]
\[\int_0^1 {F'\left[ {{x_n} + t\left( {x - {x_n}} \right)} \right]} \left( {x - {x_n}} \right)dt \approx \left[ {\frac{3}{4}F'\left( {\frac{{2{x_n} + x}}{3}} \right) + \frac{1}{4}F'\left( {{x_n}} \right)} \right]\left( {x - {x_n}} \right)\]
由牛顿法作为第一步，由此可以给出如下两种解算非线性方程的两步迭代格式：
\[\left\{ \begin{gathered}
{y_n} = {x_n} - F'{\left( {{x_n}} \right)^{ - 1}}F\left( {{x_n}} \right) \hfill \\
{x_{n + 1}} = {x_n} - \frac{{F\left( {{x_n}} \right)}}{{\frac{1}{4}F'\left( {{x_n}} \right) + \frac{3}{4}F'\left( {\frac{{{x_n} + 2{y_n}}}{3}} \right)}} \hfill \\
\end{gathered} \right.\quad \quad n = 1,2, \cdots \]
\[\left\{ \begin{gathered}
{y_n} = {x_n} - F'{\left( {{x_n}} \right)^{ - 1}}F\left( {{x_n}} \right) \hfill \\
{x_{n + 1}} = {x_n} - \frac{{F\left( {{x_n}} \right)}}{{\frac{3}{4}F'\left( {\frac{{2{x_n} + {y_n}}}{3}} \right) + \frac{1}{4}F'\left( {{y_n}} \right)}}\quad \hfill \\
\end{gathered} \right.n = 1,2, \cdots \]
以上格式为三阶收敛，且同样适用于非线性方程组的计算。
计算非线性方程
\[f\left( x \right) = 3{x^5} - 2{x^3} + 6x - 8\]
的根。
容易给出
\[\frac{{df\left( x \right)}}{{dx}} = 15{x^4} - 6{x^2} + 6\]
下面给出
go语言代码。
// muhammad
package main
import (
"fmt"
"math"
)
func main() {
/*------------------------------------------------------
! Author : Song Yezhi
! verison : 2021-10-04 19:02:15
! -----------------------------------------------------
! Input Parameters :
!
! Output Parameters :
!
------------------------------------------------------*/
var x0 float64 = 1.5
fmt.Printf("muhammad 1 method: \n")
_, _= muhammad1(x0)
fmt.Printf("------------------ \n")
fmt.Printf("muhammad 2 method: \n")
_, _= muhammad2(x0)
}
func muhammad1(x0 float64)(x1,fx float64){
/*------------------------------------------------------
Author : Song Yezhi
verison : 2021-10-4 18:51
go build -gcflags "-N -l"
-----------------------------------------------------
Input Parameters :
Output Parameters :
------------------------------------------------------*/
imax := 200
tol := 1e-8
x1 = x0
var y1,tmp,dx,x2 float64
for i:=0;i < imax ; i++ {
fx = funcX(x1)
y1 = x1 - fx/dfuncX(x1)
tmp = 0.25*dfuncX(x1) +0.75 * dfuncX((x1+2*y1)/3.0)
x2 = x1 - fx/tmp
dx = math.Abs(x2-x1)
if dx < tol {
break
}
x1 = x2
fmt.Printf("i= %4d x= %12.7f f(x)=%12.7f \n", i, x2, fx)
}
return x2,fx
}
func muhammad2(x0 float64)(x1,fx float64){
/*------------------------------------------------------
Author : Song Yezhi
verison : 2021.10.04
go build -gcflags "-N -l"
-----------------------------------------------------
Input Parameters :
Output Parameters :
------------------------------------------------------*/
imax := 200
tol := 1e-8
x1 = x0
var y1,tmp,dx,x2 float64
for i:=0;i < imax ; i++ {
fx = funcX(x1)
y1 = x1 - fx/dfuncX(x1)
tmp = 0.75*dfuncX((2.0*x1+y1)/3.0) +0.25 * dfuncX(y1)
x2 = x1 - fx/tmp
dx = math.Abs(x2-x1)
if dx < tol {
break
}
x1 = x2
fmt.Printf("i= %4d x= %12.7f f(x)=%12.7f \n", i, x2, fx)
}
return x2,fx
}
func funcX(x float64) float64 {
/*------------------------------------------------------
! Author : Song Yezhi
! verison : 2021.10.04
!
------------------------------------------------------*/
fx := 3.0*math.Pow(x,5)-2.0*x*x*x + 6*x -8.0
return fx
}
func dfuncX(x float64) float64 {
/*------------------------------------------------------
! Author : Song Yezhi
! verison : 2021.10.04
!
------------------------------------------------------*/
df := 15.0*math.Pow(x,4)-6.0*x*x + 6.0
return df
}