Question

Practice for Matlab. You can create own f(x).

a. Write code to find and plot a root using the modified secant method

b. Find the roots using the roots command - identify and plot root of interest

c. Use polyder and roots to find and plot the minimum value

d. use fminbnd to find and plot minimum value

Thank you!

Answer 1

The matlab code is given below:

Matlab code:

close all; clear all; clc
F = @(x) x^2-10.56*x+5.6; % Define the function
p = [1 -10.56 5.6]; % Define same polynomial function with coefficients

[x,e] = Modified_Secant_method(F,0.1,0,10,0.1) % Used Modified Secant method
figure % New figure
title('Using roots')
scatter(real(x),imag(x),'filled','red') % Plot solution from modified secant method

r = roots(p); % Use roots method
figure % New figure
title('Using roots')
xlabel('x value')
ylabel('Minimum value')
scatter(real(r),imag(r),'filled','red') % Plot solution from roots

dF = polyder(p); % Polyder to find differential of equation
rd = roots(dF); % Find the roots of differentiated polynomial
ddF = polyder(dF);
for i = 1:length(rd)
if(polyval(ddF,rd(1)) > 0)
minr = rd;
end
end
figure % New figure
title('Using polyder')
plot(minr,F(minr),'*r')% Plot minimum value
xlabel('x value')
ylabel('Minimum value')

figure
title('Using fminbnd')
xmin = fminbnd(F,0,10); % Find value of x for which the function is minimum
plot(xmin,F(xmin),'*r')% Plot minimum value
xlabel('x value')
ylabel('Minimum value')

The script that calculates roots using modified secant method is given below:

Modified_Secant_method.m:

function [x,e] = Modified_Secant_method(F,del,x_0,N,err)

x(1) = x_0;
i = 1;
e = 10;
while((i<N) & (e>err))
x(i+1) = x(i)-(F(x(i))*del)/(F(x(i)+del)-F(x(i)));
if (i ~= 1)
e(i) = (abs(x(i+1)-x(i))/x(i+1))*100;
end
i = i+1;
end
end