householder_reflection.h

// Description: Householder reflection

#ifndef PASTELMATH_HOUSEHOLDER_REFLECTION_H
#define PASTELMATH_HOUSEHOLDER_REFLECTION_H

#include "pastel/sys/vector.h"
#include "pastel/sys/math/sign.h"

#include <cmath>

namespace Pastel
{

    //! Multiplies A in RR^{m x n} by a Householder subspace-reflection.
    /*!
   Preconditions:
   0 <= i < a.m()
   0 <= j < a.n()
   x.n() == a.m()

   Postconditions:
   A(1 : m, j : n) := (I - 2vv^T) A(1 : m, j : n),
   
   where

       v = u / |u|, and
       u = x + sgn(x_i) |x| e_i in RR^m.

   The algorithm acts as if x(0 : i) = 0. Therefore the first i 
   rows of A are always preserved. The first j rows of A are also
   preserved. This is useful in those situations where the skipped
   part of the matrix is known to be zero (e.g. QR-decomposition).
   */
    template <typename Real>
    void householder(
        Matrix<Real>& a, const Vector<Real>& x, 
        integer i, integer j);

}

#include "pastel/math/matrix/householder_reflection.hpp"

#endif