Question

Determine the second highest and the second lowest real root at an interval of 10 81 for the following function y-(ap +9)sin()-(P -6+34)cos(2 + m) (1) Apply Secant method to determine both roots with stopping error of &a = 0.1% For all calculation, use at least 5 slignificant figures for better accuracy

Answer 1

The Matlab Scripts secant.m and secantSolver.m are posted below. Run the Matlab script secantSolver.m to display the second lowest and highest root in the interval [0, 8] in the command window.

Also note that the estimate of the interval within the interval [0, 8] within which the second lowest and second highest root lies are obtained by plotting the graph of y vs t which will also be displayed when you run the script.

secant.m

function [root, fx, iter]= secant(func, xl, xu, es, maxIter)

% [root, fx, iter] = secant(func, xl, xu, es, maxIter):

% input:
% func = name of function
% xl, xu = lower and upper guesses
% es = desired relative error
% maxIter = maximum allowable iterations

% output:
% root = real root
% fx = function value at root
% iter = number of iterations

for iter = 1: maxIter
  
    xi = xu - func(xu) * (xl - xu) /(func(xl) - func(xu)) ;
    if abs((xi - xu) / xu) < es
        root = xi;
        fx = func(root);
        break
    end
    xl = xu;
    xu = xi;
end
if iter > maxIter
    fprintf('Solution was not obtained in %i iterations. \n' , maxIter)
    root = fprintf('No answer') ;
end

secantSolver.m

clear, clc, close all;

tt = linspace(0, 8);
yy = (4*tt.^2+9).*sin(tt) - (tt.^3-6*tt+34).*cos(2*tt+pi);

figure
plot( tt, yy )
xlabel( 't' )
ylabel( 'y' )
grid on

y = @(t) (4*t^2+9)*sin(t) - (t^3-6*t+34)*cos(2*t+pi);
sec_low = secant(y, 1, 2, 1e-3, 1000);
sec_high = secant(y, 5, 6, 1e-3, 1000);

fprintf('The second lowest real root in interval [0, 8] is %0.5f.\n', sec_low);
fprintf('The second highest real root in interval [0, 8] is %0.5f.\n', sec_high);