牛顿迭代法

求平方根

数学迭代式：x[n+1] = (x[n] + a/x[n])/2.

初始化：x[0] = a/2.

保持：x[n+1] = (x[n] + a/x[n])/2.

终止条件：|x[n+1]-x[n]| < 0.00001时，迭代终止。

代码：

#include "stdio.h"
#include "math.h"

#define MAXSIZE 100
void main() {

    float x[MAXSIZE] = {0};
    float a = 2;
    x[0] = a/2;
    x[1] = (x[0] + a/x[0])/2;
    int count = 0;
    for(int i = 1; fabs(x[i] - x[i-1]) > 0.0000001; i++) {
        x[i+1] = (x[i] + a/x[i])/2;
        count = i;
    }
    printf("x[%d] = %f", count, x[count]);

    
}

输出结果：

求方程的根

数学迭代式：

f(x) = 2*x*x*x – 4*x*x + 3*x – 6;

f(x)的导数g(x)=6*x*x – 8*x + 3;

迭代式：x[n+1] = x[n] – f(x[n])/g(x[n]);

初始化：x[0] = 1.5.

保持：x[n+1] = x[n] – f(x)/g(x).

终止条件：|x[n+1]-x[n]| < 0.00001时，迭代终止。

代码：

#include "stdio.h"
#include "math.h"

#define MAXSIZE 100
void main() {

    float x[MAXSIZE] = {0};
    x[0] = 1.5;
    x[1] = x[0] - (2*x[0]*x[0]*x[0] -4*x[0]*x[0] + 3*x[0] - 6)/(6*x[0]*x[0] - 8*x[0] + 3);
    int count = 0;
    for(int i = 1; fabs(x[i] - x[i-1]) > 0.0000001; i++) {
        x[i+1] = x[i] - (2*x[i]*x[i]*x[i] -4*x[i]*x[i] + 3*x[i] - 6)/(6*x[i]*x[i] - 8*x[i] + 3);
        count = i;
    }
    printf("x[%d] = %f", count, x[count]);

    
}

输出结果：

二分法

数学迭代式：

a = -10, b = 10.

x = (a + b) /2;

if  f(x)*f(a) > 0  b = x;

else a = x.

初始化：a = -10, b = 10; x =(a+b)/2.

保持：if  f(x)*f(a) > 0  b = x; else a = x.

终止条件：|f(x)| < 0.00001时，迭代终止。

代码：

#include "stdio.h"
#include "math.h"

float f(float x);
float binary_get_root(float a, float b);
float root = 0;
void main() {

    printf("The binary root is: %f", binary_get_root(-10, 10));
}

float f(float x) {
    return 2*x*x*x -4*x*x + 3*x - 6;
}

float binary_get_root(float a, float b) {
    if(fabs(f(a)) < 0.000001) {
        root = a;
        return a;
    } else if(fabs(f(b)) < 0.000001) {
        root = b;
        return root;
    }
    float x = (a + b) /2;
    if(f(x)*f(a) > 0) binary_get_root(x, b);
    else binary_get_root(a, x);
    return root;
}

输出结果：