Question

Solve the following equations graphically and numerically using roots function on matlab

a) x ^ 5 - 3x ^ 3 + 2x ^ 2 - 1 = 0
b) exp(exp(-x)) -5 x^2=0

Answer 1

clc
clear
close all

f1 = @(x) x.^5 - 3.*x.^3 + 2.*x.^2 - 1;
f2 = @(x) exp(exp(-x)) - 5.*x.^2;

%% numerical method with roots() function
% on f1
c = [1 0 -3 2 0 -1]; %coefficients
r = roots(c); % all complex and real roots

% getting real roots for f1
j=1;
for i=1:numel(r)
if imag(r(i))==0
real_roots(j) = real(r(i));
j = j+1;
end
end

roots_f1 = real_roots
%% f2
x2 = linspace(-2,1,100);
ff2 = f2(x2);
pos = find(ff2>=0);
neg = find(ff2<=0);
root_id_f2 = ceil((pos(end)+neg(1))/2);
roots_f2 = x2(root_id_f2)

%% graphical method

% with f1
figure
x1 = linspace(-2,2,100);
plot(x1,f1(x1)) % plot f1(x)
hold on
plot(roots_f1,f1(roots_f1),'o r') % show roots
xlabel('x')
ylabel('f(x)')
legend('f(x)','roots','location','best')
title('f(x) = x^5 - 3*x^3 + 2*x^2 - 1')
grid on

% with f2
figure
plot(x2,f2(x2)) % plot f2(x)
hold on
plot(roots_f2,f1(roots_f2),'o r') % show roots
xlabel('x')
ylabel('f(x)')
legend('f(x)','roots','location','best')
title('f(x) = exp(exp(-x)) - 5*x^2')
grid on

%-----------------------------------------------------------------------

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