function [ xaprox ] = newton_raphson( f,df,x0,Eps,nmaxiter )
%newton_raphson 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) = 0 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\n');
        break;
    end
    if(abs(x(k)-x(k-1))/abs(x(k-1))) < Eps
        break
    end
end
xaprox = x(k);

end

------
function [ xaprox ] = met_secantei( f,a,b,Eps)

r= rand(1,2); %un vector cu 2 elemente random

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); %un vector cu 2 elemente random
          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 - 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;

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',8);

%A doua radacina
%In vecinatatea celui de-al doilea punct functia este convexa
%Alegem x0 ai f(x0) >0
x0=2.5;
r2 = newton_raphson(f,df,x0,Eps);
plot(r2,f(r2),'o' ,'MarkerFaceColor', 'r', 'MarkerSize',8);

%A treia radacina
x0=3.75;
r3 = newton_raphson(f,df,x0,Eps);
plot(r3,f(r3),'o' ,'MarkerFaceColor', 'r', 'MarkerSize',8);


%%
clear all
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);
dmu = matlabFunction(dmu,'vars',{x});
mu=matlabFunction(mu,'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','MarkerSize',10);

%%
%ex3
clear all
f = @(x) x.^3 - 18.*x -10;
fplot(f, [-5,5])

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