Let’s understand the secant method in numerical analysis and learn how to implement secant method in C programming with an explanation, output, advantages, disadvantages and much more.
What is Secant Method?
The secant method is a root-finding method that uses a succession of the roots of secant lines to find a better approximation of root.
The secant method algorithm is a root bracketing method and is the most efficient method of finding the root of a function. It requires two initial guesses which are the start and end interval points. This method is very similar to the Regula Falsi method.
The secant method is a Quasi-Newton method and it is seen to be faster in execution in some scenarios as compared to Newton-Raphson method and the False Position Method well.
The secant algorithm does not ensure convergence. The rate of convergence of secant method algorithm is 1.618, which is really fast comparatively. The order of convergence of secant method is superlinear.
The equation used in the following secant method c programs are as follows:
- x3 – 5x + 3
- x3 – 3x – 8
Secant Method Formula
Secant Method Algorithm
1 2 3 4 5 6 7 8 | Step 1: Assume an equation f(x) = 0(which must be defined) Step 2: Input Start and Final values of the interval Step 3: Do xi+1 = xi - (f(xi) - (xi - xi-1)/(f(xi) - f(xi-1)) where i = 1, 2, 3, 4, ..... (while no convergence criteria is achieved) Step 4: Evaluated Result Step 5: Exit |
Convergence criteria for Secant Method
- Fixing apriori the total number of iterations (limit).
- Testing the condition (xi+1−xi), is less than some tolerance limit (epsilon).
Advantages
- It evaluates one function at every iteration as compared with Newton’s method which evaluates two.
- The convergence rate is very fast.
- The secant method rule does not use the derivatives of a function.
Disadvantages
- The convergence may not always happen.
- Newton’s method generalizes much efficiently to new methods for solving simultaneous systems of nonlinear equations as compared to the Secant method.
Note: This secant method in C programming is compiled with GNU GCC compiler using CodeLite IDE on Microsoft Windows 10 operating system.
Method 1: C Program For Secant Method using Do While Loop
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 | #include<stdio.h> double equation(float temp); int main() { float start_interval, end_interval, midpoint, allowed_error; int count, limit; printf("\nEnter Value for Start Interval Point:\t"); scanf("%f", &start_interval); printf("\nEnter Value for End Interval Point:\t"); scanf("%f", &end_interval); printf("\nEnter Value for Allowed Error:\t"); scanf("%f", &allowed_error); printf("\nEnter Limit for Iterations:\t"); scanf("%d", &limit); do { if(equation(start_interval) == equation(end_interval)) { printf("Please enter different start and end intervals\n"); return 0; } midpoint = (start_interval * equation(end_interval) - end_interval * equation(start_interval)) / (equation(end_interval) - equation(start_interval)); start_interval = end_interval; end_interval = midpoint; printf("\nIteration: %d\tRoot: %f\n", count, midpoint); count++; if(count == limit) { break; } }while(fabs(equation(midpoint)) > allowed_error); printf("\nRoot of the Equation:\t%f", midpoint); return 0; } double equation(float temp) { return(temp * temp * temp - 5 * temp + 3); } |
Output
Method 2: Implement Secant Method in C Programming using While Loop
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 | #include<stdio.h> #include<math.h> #define EQUATION(x) (x)*(x)*(x) - 3*(x) - 8 int main() { float interval_start, interval_end, midpoint; float a, b, temp, allowed_error; printf("\nEnter Value for Start Interval Point:\t"); scanf("%f", &interval_start); printf("\nEnter Value for End Interval Point:\t"); scanf("%f", &interval_end); printf("\nEnter value for Allowed Error:\t"); scanf("%f", &allowed_error); printf("\n"); printf("\nStart\t\tEnd\t\tMidPoint\tfunction(Start)\tfunction(End)"); printf("\n"); while(temp > allowed_error) { a = EQUATION(interval_start); b = EQUATION(interval_end); midpoint = interval_end - ((b * (interval_end - interval_start)) / (b - a)); printf("\n%f\t%f\t%f\t%f\t%f", interval_start, interval_end, midpoint, a, b); interval_start = interval_end; interval_end = midpoint; if(b < 0) { temp = fabs(b); } else { temp = b; } } printf("\n\nRoot of the Equation: %f\n", midpoint); } |
Output
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