Monthly Archives: October 2011

The Bisection Method using Scilab

 

Recently, I have really been enjoying learning to use Scilab which is an open-source mathematics software that works alot like Matlab. Specifically we have run into some design problems in water and wastewater engineering that involved solving for variables that are described implicitly by an equation and cannot be solved through typical algebraic means. For example:

This cannot be solved directly for x. So how can we find the value of x that will satisfy the equation? Well this is where the numerical analysis technique of the bisection method comes in. The bisection method is a bounded or bracketed root-finding method. In other words, it will locate the root of an equation provided you give it the interval in which a root is located.

The search for the root is accomplished by the algorithm by dividing the interval in half and determining if the root is in one half or the other. Then it divides that interval in half and so on until it gets within a specified relative error. Below I have entered my code for this algorithm. If you save this file as Bisection_Method.sci in a scilab/modules/…/macros folder and execute it in the SciNotes editor, you can then use it by entering a command in the console like:

root = Bisection_Method(‘x-sinx-3’,0,6.5,0.5)

The first passed parameter is the function equation itself solved such that f(x) = 0. The second parameter is the lower end of the interval while the third is the upper end. The final parameter is the percent relative error that the algorithm much reach with the solution before it terminates and gives a result.

Please feel free to download Scilab for free here and copy this code and use it. Let me know if you have any troubles or if you find a problem with the code.

 

//The Bisection Method is a bracketed root locating method. Therefore
//the user must input the lower and upper bounds on the interval where
//the root is to be located. 
//func_expression - must be a string containing the function expression
//with the variable written as 'x'.
//es - is the relative percent error criterion that must be met for the
//method to terminate.
function [root]=Bisection_Method(func_expression, x_lower, x_upper, es)
    x = x_upper;
    fu = evstr(func_expression); //evaluate the function at x_upper
    x = x_lower;
    fl = evstr(func_expression); //evaluate the function at x_lower
    //***********************************************************
    //test to see if there is a root in the interval
    if (fu*fl >= 0) then
        root = 'no root in interval given';
    else
        //there is a root in the interval
        exact_solution = 'notfound';
        ea = 100;
        xr_new =(x_upper + x_lower)/2;
        //*******************************************************
        //iteratively progress toward to root until it is found or it
        //is approximated with a relative error less than required
        while ea > es & exact_solution == 'notfound',
            x = xr_new;
            fr = evstr(func_expression);
            x = x_lower;
            fl = evstr(func_expression);
            //***************************************************
            if(fl*fr < 0) then
                x_upper = xr_new;
            elseif(fl*fr > 0) then
                x_lower = xr_new;
            elseif(fl*fr == 0) then
                root = xr_new;
                exact_solution = 'found';
            end //of if statement
            //***************************************************
            //calculate the approximate relative error for the iteration
            xr_old = xr_new;
            xr_new =(x_upper + x_lower)/2;
            ea = abs((xr_new - xr_old)/xr_new)*100;
        end //of while loop 
        //*******************************************************
        //if error criterion has been met but the exact answer has not
        //been found, set the root to the adequate approximation.
        if (exact_solution == 'notfound') then
            root = xr_new;
        end //of if statement
        //*******************************************************      
    end //of if-else statement
    //***********************************************************
endfunction