modifying regula falsi method to the secant method - c

Question

I have implemented the regula falsi method. I am trying to modify it so it becomes the secant method. A pdf I read mentioned that it is essentially the same with just one change.
Future guesses for my 'm' value should have a slightly different formula, instead of:
m = a - f(a) * ( (b-a)/( f(b)-f(a) ) );
it should be:
m = a - f(a) * ( (m-a)/( f(m)-f(a) ) );
but unfortunately it doesn't work (It never finds the root). What should I fix to get this into the secant method?
source code as follows:
#include <stdio.h>
#include <math.h>
void
secant(double a, double b, double e, double (*f)(double), int maxiter ) {
double m, fm, fa, fb;
int i;
fa=(*f)(a);
fb=(*f)(b);
m = a - fa * ( (b-a)/( fb - fa ) );
fm=(*f)(m);
for(i=0; i<maxiter; i++) {
if ( fabs(fm) <= e ) {
printf("f(%f) = %f\n", m, fm);
return;
} else if ((fa*fm) < 0) {
b=m;
fb=fm;
} else {
a=m;
fa=fm;
}
// the guess below works for regula falsi method:
// m = a - fa * ( (b-a)/(fb - fa));
//this was supposed to be the change to turn this into the secant method
m = a - fa * ( (m-a)/(fm - fa) );
fm=(*f)(m);
}
}
int main(){
secant(1,4,0.0001,sin,500);
return 0;
}
Thanks in advance
EDIT: Ok after playing around with pen and paper I finally got it it wasnt a simple change as I initially thought:
void secant(double a, double b, double e, double (*f)(double), int maxiter ) {
double m, fm, fa, fb;
int i;
fa=(*f)(a);
fb=(*f)(b);
for(i=0; i<maxiter; i++) {
m = a - fa * ( (b-a)/(fb - fa) );
fm=(*f)(m);
if ( fabs(fm) <= e ) {
printf("f(%f)=%f, iter: %d\n", m,fm,i);
return;
}
a=b;
b=m;
fa=fb;
fb=fm;
}
}

Answer 1

It's easier for the secant method to not find the root. Are you sure it should find it?
For testing, here is an example: http://www.mathcs.emory.edu/ccs/ccs315/ccs315/node18.html
(example 4.7) You'd want to run that example ( f(x)=x^6-x-1 , x0=1 x1=2, root x=1.347)

Answer 2

Either the PDF is wrong or you misunderstood it. Without reading the PDF it is impossible to say which or explain further. When I explain both methods, I say that the difference between regula falsi and secant method is the rule for updating a and b.
Compute the first two iterations of the secant method by hand. Then modify your program to print the values of a, b and m at every iteration (or use a debugger). That should give you a hint of what is happening.
On your example, the secant method should converge in a few iterations.

Answer 3

Secant method is an open-bracket method, whereas Regula-Falsi is closed-bracket type. So, your PDF is correct. To modify Regula-Falsi method to secant method you have to change the closed-bracket to open type.
See this example for Regula-Falsi method.
Here, the coding is such that one interval always remains constant.
Now, here is a sample program for Secant method.
Here, like the Regula method, two initial guesses are taken, but both the intervals keep on changing until the correct root is obtained. Obtain the initial two guesses manually.
And, about the rate of convergence, for Secant method it is super-linear, whereas for Regula it is linear. So, secant method converges way faster.