Question

Matlab Regula Falsi Method

A zero of f(x) = x^2 -4x-12.2 is known to exist on the interval [1.2 , 2.2 , 3.2,...9.2] and respective right endpoints x1 =[2.2 ,3.2, 4.2....10.2], find the sub-interval in which the zero exists. Set the left endpoint of this sub-interval =a and the right endpoint=b

With tolerance of 10^-9, use the Regular Falsi method to compute the root of (f) in [a,b]

User input is required at #####

CONTENTS Close Courses LMS Integration Documentation 1 %% INPUT: Define the tolerance 2 tol = 10^-9 %% found zero if f(c) <= tol 3 4 epsilon = 10^4; %% An arbitrarily large value to begin with 5 steps = 1; %% To keep track of the number of bisections 6 20 21 22 23 7 %% INPUT: Define the left endpoints of the coarse grid 8 X0 = 1.2 : 1 : 9.2; %% xo and x1 set the endpoints of a coarse grid 9 x1 = x@+1; %% for finding a zero. 10 11 y = f(x0); %% evaluate the function at the left endpoints 12 y1 = f(x1); %% evaluate the function at the right endpoints 13 product = yo .* y1; %% product of the endpoint values 14 index = find (product<0); %% find the position where product < 0 15 a = x(index); %% define the left and right boundaries for 16 b = x1(index); %% the finer grid. 17 18 while epsilon > tol 19 %% INPUT: insert the formula for computing C = a - f(a) * (b-a)/(f(b)-f(a)); epsilon = abs(f(c)); %% This is really f(c)-|| if epsilon <= tol %% Enter this if a root is found fprintf("With a tolerance of %.ee, after %d steps the root = %. 16F\n", tol, steps,c); 24 %% INPUT: store the final value of the root in fnroot 25 %% and store the total number of steps in numsteps 26 #### #### else 29 if f(b) f(c) < 0 30 %% INPUT: Pay close attention to the if test above and decide where b=c and a=c should go. a = c; else 33 %% INPUT: Pay close attention to the if test above and decide where b=c and a=C should go. 34 end steps = steps + 1; 37 end 38 end 39 40 function ans = f(x) 41 ans = X############ 42 end 27 28 31 32 b = c; 35 36 43 Run Scrint ?

Answer 1

The Matlab scripts regulaFalsiMethod.m and f.m are posted below. Run the script regulaFalsiMethod.m to generate the result in the Command Window. The screen shot of the result is attached at the end.

regulaFalsiMethod.m

%% INPUT: Define the tolerance
tol = 10^-9;     %% found zero if f(c) <= tol

epsilon = 10^4; %% an arbitrary large value to begin with
steps = 1;       %% to keep track of the number of bisections

%% INPUT: Define the left endpoints of the coarse grid
x0 = 1.2 : 1 : 9.2; %% x0 and x1 set the endpoints of a coarse grid
x1 = x0 + 1;         %% for finding a zero

y0 = f(x0);          %% evaluate the function at the left endpoints
y1 = f(x1);          %% evaluate the function at the right endpoints
product = y0 .* y1; %% product of the endpoint values
index = find( product < 0 ); %% find the position where product < 0
a = x0(index);                %% define the left and right boundaries for
b = x1(index);                %% the finer grid

while epsilon > tol
    %% INPUT: Insert the formula for computing
    c = a - f(a) * (b - a)/(f(b)-f(a));
    epsilon = abs(f(c)); %% this is really |f(c)-0|
    if epsilon <= tol %% enter this if root is found
        fprintf('With a tolerance of %.0e, ater %d steps the root = %.16f\n', tol, steps, c);
        %% INPUT: Store the final value of the root in fnroot
        %% and store the total number of steps in numsteps
        fnroot = c;
        numsteps = steps;
    else
        if f(b) * f(c) < 0
            %% INPUT: Pay close attention to the if test above and decide where b=c and a=c should go
            a = c;
        else
            %% INPUT: Pay close attention to the if test above and decide where b=c and a=c should go
            b = c;
        end
        steps = steps + 1;
    end
end

f.m

function an = f(x)
    an = x.^2 - 4 * x - 12.2;
end

28 regulaFalsiMethod.mx .m x + 24 fprintf(with a tolerance of $.ve, ater $d steps the root = 8.16f\n', tol, 25 %% INPUT: Store the final value of the root in fnroot 26 %% and store the total number of steps in numsteps 27 – fnroot = c; numsteps = steps; 29 else 30 – if f(b) + f(c) < 0 31 8% INPUT: Pay close attention to the if test above and decide where b= 32 - a = Ci 33 - else | | Command Window New to MATLAB? See resources for Getting Started. х >> regulaFalsiMethod 5 With a tolerance of le-09, ater 6 steps the root = 6.0249223594141226 fx >> script Ln 41 Col 1