Numeric Methods: notes

// minimos cuadrados
// A'A x = A' b
// si rang(A) = n, R no singular
// Rx = Q'b, se resuelve por sust. regresiva


function [Qu, R]= factorizacionQR(A)
    [m, n] = size(A)
    Qu = zeros(A)
    R = zeros(n,n)

    Qu(:, 1) = A(:, 1)
    R(1,1) = norm(A(:, 1), 2)
    Qu(:, 1) = Qu(:, 1) / R(1,1)


    for j = 2:n
        //Lleno la columna j de R
        for i = 1:j-1
            R(i,j) = A(:, j)' * Qu(:, i)
        end

        //Completo la columna j de Q.
        Qu(:, j)=  A(:, j) - Qu(:, 1:j-1) * R(1:j-1, j)

        //Completo la posición de la diagonal de R
        R(j,j) = norm(Qu(:, j), 2)
        Qu(:, j) = Qu(:, j) / R(j,j)
    end
endfunction

function sol = f1(x)
    sol = 1 + x(1)**2 - x(2)**2 + %e**x(1) * cos(x(2))
endfunction

function sol = f2(x)
    sol = 2*x(1)*x(2) + %e**x(1) * sin(x(2))
endfunction

function vec = F(x)
    vec(1) = f1(x)
    vec(2) = f2(x)
endfunction

function sol = Newton(f, vecini)
    eps = 10**-5
    iter = 0

    vec = vecini - inv(numderivative(f, vecini)) * f(vecini)

    while iter < 20 & (f(vecini) ~= eps)
        iter = iter + 1
        vec = vecini - inv(numderivative(f, vecini)) * f(vecini)
        vecini = vec
    end

    sol = vec
endfunction

function sol = f6(x)
    sol = x + (1/2.5)*(x**2 - 5)
endfunction

function sol = ptofijo6(g, x)
    Eps = 10**-5
    Iter = 0

    while (Iter < 100) & (abs(g(x)-Eps) > 0)
        x = g(x)
        Iter = Iter + 1
    end

    sol = x

endfunction

function sol = secante(f, x0, x1)
    Eps = 10**-6
    x2 = 0

    while abs(x1 - x0) > Eps
        x2 = x1 - (x1-x0)/(f(x1)-f(x0)) * f(x1)
        x0 = x1
        x1 = x2
    end
    sol = x2

endfunction

function sol = biseccion(f, a, b)
    Eps = 10**-3
    mid = (a-b)/2 + a

    while (b-a)/2 > Eps
        mid = (b-a)/2 + a
        if(f(a)*f(mid) <= 0) b=mid
        else a=mid
        end
    end

    sol = mid
endfunction