#ifndef Magnum_Math_Algorithms_GramSchmidt_h
#define Magnum_Math_Algorithms_GramSchmidt_h
/*
    This file is part of Magnum.

    Copyright © 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018
              Vladimír Vondruš <mosra@centrum.cz>

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    copy of this software and associated documentation files (the "Software"),
    to deal in the Software without restriction, including without limitation
    the rights to use, copy, modify, merge, publish, distribute, sublicense,
    and/or sell copies of the Software, and to permit persons to whom the
    Software is furnished to do so, subject to the following conditions:

    The above copyright notice and this permission notice shall be included
    in all copies or substantial portions of the Software.

    THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
    IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
    FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
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*/

/** @file
 * @brief Function @ref Magnum::Math::Algorithms::gramSchmidtOrthogonalizeInPlace(), @ref Magnum::Math::Algorithms::gramSchmidtOrthogonalize(), @ref Magnum::Math::Algorithms::gramSchmidtOrthonormalizeInPlace(), @ref Magnum::Math::Algorithms::gramSchmidtOrthonormalize()
 */

#include "Magnum/Math/RectangularMatrix.h"

namespace Magnum { namespace Math { namespace Algorithms {

/**
@brief In-place Gram-Schmidt matrix orthogonalization
@param[in,out] matrix   Matrix to perform orthogonalization on

Performs the [Gram-Schmidt process](https://en.wikipedia.org/wiki/Gram–Schmidt_process).
With a *projection operator* defined as
@f[
    \operatorname{proj}_{\boldsymbol{u}}\,(\boldsymbol{v}) = \frac{\boldsymbol{u} \cdot \boldsymbol{v}}{\boldsymbol{u} \cdot \boldsymbol{u}} \boldsymbol{u}
@f]

@m_class{m-noindent}

the process works as follows, with @f$ \boldsymbol{v}_k @f$ being columns of
@p matrix and @f$ \boldsymbol{u}_k @f$ columns of the output: @f[
    \boldsymbol{u}_k = \boldsymbol{v}_k - \sum_{j = 1}^{k - 1} \operatorname{proj}_{\boldsymbol{u}_j}\,(\boldsymbol{v}_k)
@f]

Note that the above is not performed directly due to numerical instability,
the stable [modified Gram-Schmidt](https://en.wikipedia.org/wiki/Gram–Schmidt_process#Numerical_stability)
algorithm is used instead.
@see @ref gramSchmidtOrthogonalize(), @ref gramSchmidtOrthonormalizeInPlace(),
    @ref Vector::projected()
*/
template<std::size_t cols, std::size_t rows, class T> void gramSchmidtOrthogonalizeInPlace(RectangularMatrix<cols, rows, T>& matrix) {
    static_assert(cols <= rows, "Unsupported matrix aspect ratio");
    for(std::size_t i = 0; i != cols; ++i) {
        for(std::size_t j = i+1; j != cols; ++j)
            matrix[j] -= matrix[j].projected(matrix[i]);
    }
}

/**
@brief Gram-Schmidt matrix orthogonalization

Unlike @ref gramSchmidtOrthogonalizeInPlace() returns the modified matrix
instead of performing the orthogonalization in-place.
*/
template<std::size_t cols, std::size_t rows, class T> RectangularMatrix<cols, rows, T> gramSchmidtOrthogonalize(RectangularMatrix<cols, rows, T> matrix) {
    gramSchmidtOrthogonalizeInPlace(matrix);
    return matrix;
}

/**
@brief In-place Gram-Schmidt matrix orthonormalization
@param[in,out] matrix   Matrix to perform orthonormalization on

Performs the [Gram-Schmidt process](https://en.wikipedia.org/wiki/Gram–Schmidt_process).
With a *projection operator* defined as
@f[
    \operatorname{proj}_{\boldsymbol{u}}\,(\boldsymbol{v}) = \frac{\boldsymbol{u} \cdot \boldsymbol{v}}{\boldsymbol{u} \cdot \boldsymbol{u}} \boldsymbol{u}
@f]

@m_class{m-noindent}

the Gram-Schmidt process works as follows, with @f$ \boldsymbol{v}_k @f$ being
columns of @p matrix and @f$ \boldsymbol{e}_k @f$ columns of the output: @f[
    \boldsymbol{u}_k = \boldsymbol{v}_k - \sum_{j = 1}^{k - 1} \operatorname{proj}_{\boldsymbol{u}_j}\,(\boldsymbol{v}_k), ~~~~~~ \boldsymbol{e}_k = \frac{\boldsymbol{u}_k}{|\boldsymbol{u}_k|}
@f]

In particular, this adds the normalization step on top of
@ref gramSchmidtOrthogonalizeInPlace(). Note that the above is not performed
directly due to numerical instability, the stable [modified Gram-Schmidt](https://en.wikipedia.org/wiki/Gram–Schmidt_process#Numerical_stability)
algorithm is used instead.
@see @ref gramSchmidtOrthonormalize(), @ref Vector::projectedOntoNormalized()
*/
template<std::size_t cols, std::size_t rows, class T> void gramSchmidtOrthonormalizeInPlace(RectangularMatrix<cols, rows, T>& matrix) {
    static_assert(cols <= rows, "Unsupported matrix aspect ratio");
    for(std::size_t i = 0; i != cols; ++i) {
        matrix[i] = matrix[i].normalized();
        for(std::size_t j = i+1; j != cols; ++j)
            matrix[j] -= matrix[j].projectedOntoNormalized(matrix[i]);
    }
}

/**
@brief Gram-Schmidt matrix orthonormalization

Unlike @ref gramSchmidtOrthonormalizeInPlace() returns the modified matrix
instead of performing the orthonormalization in-place.
*/
template<std::size_t cols, std::size_t rows, class T> RectangularMatrix<cols, rows, T> gramSchmidtOrthonormalize(RectangularMatrix<cols, rows, T> matrix) {
    gramSchmidtOrthonormalizeInPlace(matrix);
    return matrix;
}

}}}

#endif
