正割法 Secant Method 解 equation = x^3 + x - 1

源自於
https://www.geeksforgeeks.org/program-to-find-root-of-an-equations-using-secant-method/

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Program to find root of an equations using secant method

The secant method is used to find the root of an equation f(x) = 0. It is started from two distinct estimates x1 and x2 for the root. It is an iterative procedure involving linear interpolation to a root. The iteration stops if the difference between two intermediate values is less than convergence factor.

Examples :

Input : equation = x3 + x - 1 
        x1 = 0, x2 = 1, E = 0.0001
Output : Root of the given equation = 0.682326
         No. of iteration=5

Algorithm

Initialize: x1, x2, E, n         // E = convergence indicator
calculate f(x1),f(x2)

if(f(x1) * f(x2) = E); //repeat the loop until the convergence
    print 'x0' //value of the root
    print 'n' //number of iteration
}
else
    print "can not found a root in the given interval"

// C++ Program to find root of an 

// equations using secant method 

#include <bits/stdc++.h> 

using namespace std; 

// function takes value of x and returns f(x) 

float f(float x) 

{ 

// we are taking equation as x^3+x-1 

float f = pow(x, 3) + x - 1; 

return f; 

} 

void secant(float x1, float x2, float E) 

{ 

float n = 0, xm, x0, c; 

if (f(x1) * f(x2) < 0) { 

do { 

// calculate the intermediate value 

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

// check if x0 is root of equation or not 

c = f(x1) * f(x0); 

// update the value of interval 

x1 = x2; 

x2 = x0; 

// update number of iteration 

n++; 

// if x0 is the root of equation then break the loop 

if (c == 0) 

break; 

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

} while (fabs(xm - x0) >= E); // repeat the loop 

// until the convergence 

cout << "Root of the given equation=" << x0 << endl; 

cout << "No. of iterations = " << n << endl; 

} else

cout << "Can not find a root in the given inteval"; 

} 

// Driver code 

int main() 

{ 

// initializing the values 

float x1 = 0, x2 = 1, E = 0.0001; 

secant(x1, x2, E); 

return 0; 

} 

Output :

Root of the given equation = 0.682326
No. of iterations = 5

Time Complexity = O(1)