lab3

syms x y
f = x^3-6*x^2
g=2*x^3+8*x^2
f+g
A=[x 2*y; 2*x 4*y]
A(1,1)
det(A)
inv(A)
diff(f,x)
int(f,x)
subs(f,{x},{2})
F=matlabFunction(f)
%F=matlabFunction(f, 'vars', {x})

%lab2 - 1a
function [ xaprox ] = newton_raphson(f, df, x0, Eps, nmaxiter)
%newton_rapshon rezolva ecuatii de forma f(x)=0
%synopsis newton_raphson(f, df, x0)
%          newton_raphson(f, df, x0, Eps)
%          newton_raphson(f, df, x0, Eps, nmaxiter)
%input f = functia
%       df = derivata
%       x0 = valoarea de pornire
%       Eps = eroarea, implicit Eps = eps * 10^6
%       nmaxiter = numarul maxim de iteratii, implicit nmaxiter = 100
%output xaprox = solutia aproximativa a ecuatiei f(x) = cu eroarea Eps
    if nargin < 5
        nmaxiter = 100;
    end
    if nargin < 4
        Eps = eps .* 10^6;
    end

    x(1) = x0
    k = 1;
    while true
        k = k+1;
        x(k) = x(k-1) - f(x(k-1))/df(x(k-1));
        if k > nmaxiter
            fprintf("S-a depasit numarul maxim de iteratii");
            break
        end

        if abs(x(k) - x(k-1))/abs(x(k-1)) < Eps
            break
        end
    end
    xaprox = x(k);


end

%lab2 - 1b
f = @(x) x.^3 - 7.*x.^2 + 14.*x - 6

%capetele intervalului
a = 0;
b = 4;

interval = linspace(a, b, 100);

plot(interval, f(interval), 'LineWidth', 3);

grid on

%prima radacina
x0 = 0.25;

syms x
df = diff(f(x), x)
df = matlabFunction(df, 'vars', {x});

Eps =  10^(-3);

r1 = newton_raphson(f, df, x0, Eps);
hold on
plot(r1, f(r1), 'o', 'MarkerFaceColor', 'r', 'MarkerSize', 10);


%a doua radacina
%in vecinatatea celui de-al doilea punct, functia este convexa
%alegem x0 astfel incat f(x0) > 0
x0 = 2.5;
r2 = newton_raphson(f, df, x0, Eps);
hold on
plot(r2, f(r2), 'o', 'MarkerFaceColor', 'g', 'MarkerSize', 10);

%a treia radacina
x0 = 3.75;
r3 = newton_raphson(f, df, x0, Eps);
hold on
plot(r3, f(r3), 'o', 'MarkerFaceColor', 'y', 'MarkerSize', 10);

%lab2 - 2
f = @(x) (x-1.5).^2 .* (x-4);
fplot(f,[1,2],'LineWidth', 3);

syms x

mu = (f(x)/diff(f(x),x));
mu = simplify(mu);
dmu = diff(mu, x);
dmu = simplify(dmu);
mu = matlabFunction(mu, 'vars', {x});
dmu = matlabFunction(dmu, 'vars', {x});

hold on;
fplot(mu, [1,2], ...
    'LineWidth', 3);
grid on;

x0=2
r1 = newton_raphson(mu,dmu,x0);
plot(r1,mu(r1),'o')

%lab2 - 3
function [ xaprox ] = metoda_secantei(f, a, b, Eps)

r = rand(1,2);
alt_r = a +(b-a)*r;

x(1) = alt_r(1);
x(2) = alt_r(2);
k = 2;

while abs(x(k) - x(k-1))/abs(x(k-1)) >= Eps
    k = k+1;
    x(k) = (x(k-2) * f(x(k-1)) - x(k-1) * f(x(k-2)))/(f(x(k-1)) - f(x(k-2)));
    if x(k) < a | x(k) > b
        r = rand(1,2);
        alt_r = a +(b-a)*r;
        clear x
        x(1) = alt_r(1);
        x(2) = alt_r(2);
        k=2;
    end
end
xaprox = x(k)



end


f = @(x) x.^3 - 18.*x -10
fplot(f, [-5, 5])


Eps = 10^(-5)
a = -5;
b = -3;
tic
r1 = metoda_secantei(f,a,b,Eps)
toc
hold on
plot(r1, f(r1), 'o')