Nan Mihai - MPI - cu bonus (Metode numerice)

function [ lambda, x, OK ] = MPI(A, y, maxiter, epsilon)

    [n n] = size(A);
    k = 0;
    e = 2 * epsilon;
    OK = 1;
    u = y' * A * y;

    while e >= epsilon
        if k > maxiter
            OK = 0;
            error("Eroare!");
        endif
        %Rezolvarea sistemului folosind transformarea ortogonala
        B = A-u*eye(n);
        [Q, R] = Gram-Schmidt(B); %Algoritmul Gram-Schmidt
        y1 = Q'*y;
        x = SST(R, y1); %Rezolvarea unui sistem triunghiular superior prin subtitutii inverse
        %Sfarsit rezolvarea sistemului
        %x = (A - u*eye(n)) \ y;
        x = x / norm(x);
        lambda = x' * A * x;
        e = abs(1 - abs(y' * x));
        y = x;
        k = k + 1;
        u = lambda;
    endwhile
endfunction

function [Q, R] = Gram-Schmidt(A)
    [n , n] = size(A);
    Q = A;
    R = zeros(n);
    for k=1:n
        R(k, k) = norm(Q(:,k));
        Q(:. k) = Q(:, k) / R(k, k);
        for j = k+1:n
            R(k, j) = Q(:, k)'*Q(:, j);
            Q(:, j) = Q(:, j) - Q(:, k)*R(k, j):
        endfor
    endfor
endfunction

function x = SST (A,b)
    [n,n]=size(A);
    x=zeros(1,n);
    x(n)=b (n)/A(n,n);
    for i=n-1:-1:1
        x(i)=(b(i)-A(i,i+1:n)*x (i+1:n)')/A(i,i);
    endfor
endfunction