非線性方程

f(x)=0 型別的方程是代數的或超越的。這些型別的方程式可以通過使用兩種方法來解決 -

1. 直接方法 ：此方法直接以有限的步數給出所有根的精確值。

2. 間接或迭代方法 ：迭代方法最適合計算機程式來求解方程。它基於逐次逼近的概念。在迭代方法中，有兩種方法可以解決方程 -

   • 包圍方法 ：我們採用兩個初始點，其中根位於它們之間。示例 - 二分法，假位置法。

   • 開放結束方法 ：我們採用一個或兩個初始值，其中根可以是任何地方。例子 - Newton-Raphson 方法，逐次逼近法，割線法。

在 C-中實施

/// Here define different functions to work with
#define f(x) ( ((x)*(x)*(x)) - (x) - 2 )
#define f2(x) ( (3*(x)*(x)) - 1 )
#define g(x) ( cbrt( (x) + 2 ) )

/**
* Takes two initial values and shortens the distance by both side.
**/
double BisectionMethod(){
    double root=0;

    double a=1, b=2;
    double c=0;

    int loopCounter=0;
    if(f(a)*f(b) < 0){
        while(1){
            loopCounter++;
            c=(a+b)/2;

            if(f(c)<0.00001 && f(c)>-0.00001){
                root=c;
                break;
            }

            if((f(a))*(f(c)) < 0){
                b=c;
            }else{
                a=c;
            }

        }
    }
    printf("It took %d loops.\n", loopCounter);

    return root;
}

/**
* Takes two initial values and shortens the distance by single side.
**/
double FalsePosition(){
    double root=0;

    double a=1, b=2;
    double c=0;

    int loopCounter=0;
    if(f(a)*f(b) < 0){
        while(1){
            loopCounter++;

            c=(a*f(b) - b*f(a)) / (f(b) - f(a));

            /*/printf("%lf\t %lf \n", c, f(c));/**////test
            if(f(c)<0.00001 && f(c)>-0.00001){
                root=c;
                break;
            }

            if((f(a))*(f(c)) < 0){
                b=c;
            }else{
                a=c;
            }
        }
    }
    printf("It took %d loops.\n", loopCounter);

    return root;
}

/**
* Uses one initial value and gradually takes that value near to the real one.
**/
double NewtonRaphson(){
    double root=0;

    double x1=1;
    double x2=0;

    int loopCounter=0;
    while(1){
        loopCounter++;

        x2 = x1 - (f(x1)/f2(x1));
        /*/printf("%lf \t %lf \n", x2, f(x2));/**////test

        if(f(x2)<0.00001 && f(x2)>-0.00001){
            root=x2;
            break;
        }

        x1=x2;
    }
    printf("It took %d loops.\n", loopCounter);

    return root;
}

/**
* Uses one initial value and gradually takes that value near to the real one.
**/
double FixedPoint(){
    double root=0;
    double x=1;

    int loopCounter=0;
    while(1){
        loopCounter++;

        if( (x-g(x)) <0.00001 && (x-g(x)) >-0.00001){
            root = x;
            break;
        }

        /*/printf("%lf \t %lf \n", g(x), x-(g(x)));/**////test

        x=g(x);
    }
    printf("It took %d loops.\n", loopCounter);

    return root;
}

/**
* uses two initial values & both value approaches to the root.
**/
double Secant(){
    double root=0;

    double x0=1;
    double x1=2;
    double x2=0;

    int loopCounter=0;
    while(1){
        loopCounter++;

        /*/printf("%lf \t %lf \t %lf \n", x0, x1, f(x1));/**////test

        if(f(x1)<0.00001 && f(x1)>-0.00001){
            root=x1;
            break;
        }

        x2 = ((x0*f(x1))-(x1*f(x0))) / (f(x1)-f(x0));

        x0=x1;
        x1=x2;
    }
    printf("It took %d loops.\n", loopCounter);

    return root;
}

int main(){
    double root;

    root = BisectionMethod();
    printf("Using Bisection Method the root is: %lf \n\n", root);
    
    root = FalsePosition();
    printf("Using False Position Method the root is: %lf \n\n", root);
    
    root = NewtonRaphson();
    printf("Using Newton-Raphson Method the root is: %lf \n\n", root);
    
    root = FixedPoint();
    printf("Using Fixed Point Method the root is: %lf \n\n", root);
    
    root = Secant();
    printf("Using Secant Method the root is: %lf \n\n", root);

    return 0;
}