数学:非线性方程的求解---(笔记+代码）

code:

/**
*求解f(x) = x^4 - x - 2
*求解步骤
*1、对于给出的函数f(x),编写一个函数与之对应function()
*2、使用给定的算法进行求解f(x) = 0
*/
Double function(Double x){

return pow(x,4) - x - 2;
}

/**
*sign(x)符号函数，小于0返回0，否则返回1
*/
bool sign(Double x){

return x > 0;
}

Double dabs(Double x){
if(x < 0) x = -x;
return x;
}

/**
*二分法的代码实现
*参数是区间(a,b),而且
*f(a)f(b) < 0
*我们可以不指定a,b，使用试探法进行
*自动的a,b赋值,但是这种方法并不保险
*/

Double binary_calcutateX(Double a,Double b){

if(function(a) * function(b) > 0) return 0;//不满足条件
if(function(a) == 0) return a;
if(function(b) == 0) return b;
Double x;
while(dabs(b - a) > ERR){
x = a+(b-a)/2;
if(sign(function(x)) == sign(function(a))){
a = x;
}
else{
b = x;
}
}
return a + (b-a)/2;
}

code:

/**
*不动点迭代法
*参数x0
*需要用到f(x)的变形x = h(x) = x^4 - 2
*或者x = h(x) = (x+1)^(1/4)
*注意这个变换形式很重要
*/
Double h(Double x){
return pow(x,4) - 2;
}

Double h1(Double x){
return pow(x+2,1.0/4);
}
Double staticNode(Double x){//初始值不同时，会找到距离这个初始值最近的根
int k = 0;//记录迭代的次数
Double x1 = x;
while(dabs(function(x)) > ERR1 || dabs(x1 - x) > ERR2){
x = x1;
//x1 = h(x);
x1 = h1(x);
k += 1;
}
return x;
}

应用和问题：

一般可以用来求一个数n的平方根a；

1、无法保证全局收敛性

2、要求f(x)在其根值附近有连续的2阶导数

3、每一次计算都要计算f’(x),计算量大，一般用在f’(x)确定的情况下

Code:

/**
*牛顿迭代法
*参数：x0
*需要导函数ff
*/

//f(x) = x^4 - x - 2的导函数
Double ff(Double x){
return 4*pow(x,3) - 1;
}
Double niudun(Double x){
int k = 0;//记录迭代的次数
Double x1 = x;
while(dabs(function(x)) > ERR1 || dabs(x1 - x) > ERR2){
x = x1;
x1 = x - function(x)/ff(x);
k += 1;
}
return x;
}

Code:

/**
*平行弦法
*参数：x0
*需要导函数ff
*/
Double parallel(Double x){
int k = 0;//记录迭代的次数
Double xx = ff(x);
Double x1 = x;
while(dabs(function(x)) > ERR1 || dabs(x1 - x) > ERR2){
x = x1;
x1 = x - function(x)/xx;
k += 1;
}
return x;
}

Code:

/**
*割线法
*参数：x0，x1
*!!!!!!这个代码对这个样例出现的死循环
*/
Double gexian(Double x0,Double x1){
int k = 0;//记录迭代的次数
Double x2 = x1;
while(dabs(function(x1)) > ERR1 || dabs(x1 - x0) > ERR2){
cout<<x0<<" "<<function(x0)<<endl;
cout<<x1<<" "<<function(x1)<<endl;

x2 = x1 - function(x1)/(function(x1) - function(x0));//注意除0错误

x0 = x1;
x1 = x2;

k += 1;
}
return x1;
}

测试代码和头文件代码：

头文件head.h

#include<iostream>
#include<stdio.h>
#include<algorithm>
#include<math.h>
using namespace std;

typedef long double Double;

const Double ERR = 1e-4;
const Double ERR1 = 1e-4;
const Double ERR2 = 1e-4;

测试代码：

#include "head.h"
int main()
{
Double x5 = 0,x4 = 0,x3 = 0,x2 = 0,x1 = 0,a = 1.0,b = 1.5;

x1 = binary_calcutateX(a,b);
x2 = staticNode(b);
x3 = niudun(b);
x4 = parallel(b);
//x5 = gexian(a,b);

cout<<"x1 = "<<x1<<endl;
cout<<"f(x1) = "<<function(x1)<<endl;

cout<<"x2 = "<<x2<<endl;
cout<<"f(x2) = "<<function(x2)<<endl;

cout<<"x3 = "<<x3<<endl;
cout<<"f(x3) = "<<function(x3)<<endl;

cout<<"x4 = "<<x4<<endl;
cout<<"f(x4) = "<<function(x4)<<endl;

//cout<<"x5 = "<<x5<<endl;
//cout<<"f(x5) = "<<function(x5)<<endl;

return 0;
}