rotations.h

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/*!
 * 
 *
 * \brief       Some operations for creating rotation matrices
 * 
 * 
 * 
 *
 * \author      O. Krause
 * \date        2011
 *
 *
 * \par Copyright 1995-2017 Shark Development Team
 * 
 * <BR><HR>
 * This file is part of Shark.
 * <https://shark-ml.github.io/Shark/>
 * 
 * Shark is free software: you can redistribute it and/or modify
 * it under the terms of the GNU Lesser General Public License as published 
 * by the Free Software Foundation, either version 3 of the License, or
 * (at your option) any later version.
 * 
 * Shark is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU Lesser General Public License for more details.
 * 
 * You should have received a copy of the GNU Lesser General Public License
 * along with Shark.  If not, see <http://www.gnu.org/licenses/>.
 *
 */
 
#ifndef SHARK_LINALG_ROTATIONS_H
#define SHARK_LINALG_ROTATIONS_H
 
#include <shark/LinAlg/Base.h>
#include <shark/Core/Random.h>
namespace shark{ namespace blas{
/**
 * \ingroup shark_globals
 * 
 * @{
 */
 
//! \brief Generates a Householder reflection from a vector to use with applyHouseholderLeft/Right
//!
//! Given a Vector x=(x0,x1,...,xn), finds a reflection with the property
//! (c, 0,0,...0) = (I-beta v v^t)x
//! and v = (x0-c,x1,x2,...,xn)
template<class X, class R>
typename X::value_type createHouseholderReflection(
    vector_expression<X, cpu_tag> const& x, 
    vector_expression<R, cpu_tag>& reflection
){
    SIZE_CHECK(x().size() != 0);
    SIZE_CHECK(x().size() == reflection().size());
    
    //special case for last iteration of QR etc
    //by convention diagonal elements are > 0
    if(x().size() == 1){
        reflection()(0) = 1;
        return 2;
    }
        
    
    typedef typename X::value_type Value;
    
    double norm = norm_2(x);
    if (x()(0) >= 0.0)
        norm *= -1.0;
    
    noalias(reflection()) = x;
    reflection()(0) -= norm;
    reflection() /= (x()(0) - norm);
    //if pivot is close to 0, this is one->numericaly stable
    //compared to the usual formula
    Value beta = (norm - x()(0)) / norm;
    return beta;
}
//\brief rotates a matrix using a householder reflection 
//
//calculates A*(1-beta*xx^T)
template<class Mat, class R, class T, class Device>
void applyHouseholderOnTheRight(
    matrix_expression<Mat, Device> & matrix,
    vector_expression<R, Device> const& reflection, 
    T beta
){
    SIZE_CHECK(matrix().size2() == reflection().size());
    SIZE_CHECK(reflection().size() != 0 );
    
    //special case for last iteration of QR etc
    if(reflection().size() == 1){
        matrix() *= 1-beta;
        return;
    }
    
    SIZE_CHECK(matrix().size2() == reflection().size());
    //Ax
    blas::vector<T> temp = prod(matrix,reflection);
    
    //A -=beta*(Ax)x^T
    noalias(matrix()) -= beta * outer_prod(temp,reflection);
}
 
 
/// \brief rotates a matrix using a householder reflection 
///
/// calculates (1-beta*xx^T)*A
template<class Mat, class R, class T, class Device>
void applyHouseholderOnTheLeft(
    matrix_expression<Mat, Device> & matrix,
    vector_expression<R, Device> const& reflection, 
    T const& beta
){
 
    SIZE_CHECK(matrix().size1() == reflection().size());
    SIZE_CHECK(reflection().size() != 0 );
    
    //special case for last iteration of QR etc
    if(reflection().size() == 1){
        matrix()*=1-beta;
        return;
    }
    //x^T A
    blas::vector<T> temp = prod(trans(matrix),reflection);
    
    //A -=beta*x(x^T A)
    noalias(matrix()) -= beta * outer_prod(reflection,temp);
}
 
/// \brief rotates a matrix using a householder reflection 
///
/// calculates (1-beta*xx^T)*A
template<class Mat, class R, class T, class Device>
void applyHouseholderOnTheLeft(
    matrix_expression<Mat, Device>&& matrix,
    vector_expression<R, Device> const& reflection, 
    T const& beta
){
    applyHouseholderOnTheLeft(matrix(),reflection,beta);
}
 
/// \brief Initializes a matrix such that it forms a random rotation matrix.
///
/// The matrix needs to be quadratic and have the proper size
/// (e.g. call matrix::resize before).
///
/// One common way to  do this is using Gram-Schmidt-Orthogonalisation 
/// on a matrix which is initialized with gaussian numbers. However, this is
/// highly unstable for big matrices. 
///
/// This algorithm is implemented from one of the algorithms presented in
/// Francesco Mezzadri "How to generate random matrices from the classical compact groups"
/// http://arxiv.org/abs/math-ph/0609050v2
///
/// He gives two algorithms: the first one uses QR decomposition on the random gaussian
/// matrix and ensures that the signs of Q are correct by multiplying every column of Q
/// with the sign of the diagonal of R. 
///
/// We use another algorithm implemented in the paper which works similarly, but 
/// reversed. We apply Householder rotations H_N H_{N-1}..H_1 where
/// H_1 is generated from a random vector on the n-dimensional unit sphere.
/// this requires less operations and is thus preferable. Also only half the
/// random numbers need to be generated
template< class MatrixT>
void randomRotationMatrix(random::rng_type& rng, matrix_container<MatrixT, cpu_tag>& matrixC){
    MatrixT& matrix = matrixC();
    SIZE_CHECK(matrix.size1() == matrix.size2());
    SIZE_CHECK(matrix.size1() > 0);
    size_t size = matrix.size1();
    diag(matrix) = repeat(1.0,size);
 
    RealVector v(size);
    //we skip the first dimension as the rotation of a 1d vector is just the identity
    for(std::size_t i = 2; i != size+1;++i){
        //create the random vector on the unit-sphere for the i-dimensional subproblem
        for(std::size_t j=0;j != i; ++j){
            v(j) = random::gauss(rng);
        }
        subrange(v,0,i) /=norm_2(subrange(v,0,i));//project on sphere
        
        double v0 = v(0);
        v(0) += v0/std::abs(v0);
        
        //compute new norm of v
        //~ double normvSqr = 1.0+(1+v0)*(1+v0)-v0*v0;
        double normvSqr = norm_sqr(subrange(v,0,i));
        
        //apply the householder rotation to the i-th submatrix
        applyHouseholderOnTheLeft(subrange(matrix,size-i,size,size-i,size),subrange(v,0,i),2.0/normvSqr);
        
    }
}
 
//! Creates a random rotation matrix with a certain size using the random number generator rng.
RealMatrix randomRotationMatrix(random::rng_type& rng, size_t size){
    RealMatrix mat(size,size);
    randomRotationMatrix(rng, mat);
    return mat;
}
 
 
/** @}*/
}}
#endif