/*!
 * \file
 * \brief Definitions of Singular Value Decompositions
 * \author Tony Ottosson
 *
 * -------------------------------------------------------------------------
 *
 * IT++ - C++ library of mathematical, signal processing, speech processing,
 *        and communications classes and functions
 *
 * Copyright (C) 1995-2007  (see AUTHORS file for a list of contributors)
 *
 * This program is free software; you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation; either version 2 of the License, or
 * (at your option) any later version.
 *
 * This program 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 General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License
 * along with this program; if not, write to the Free Software
 * Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
 *
 * -------------------------------------------------------------------------
 */

#ifndef SVD_H
#define SVD_H

#include <itpp/base/mat.h>


namespace itpp {

  /*!
   * \ingroup matrixdecomp
   * \brief Get singular values \c s of a real matrix \c A using SVD
   *
   * This function calculates singular values \f$s\f$ from the SVD
   * decomposition of a real matrix \f$A\f$. The SVD algorithm computes the
   * decomposition of a real \f$m \times n\f$ matrix \f$\mathbf{A}\f$ so
   * that
   * \f[
   * \mathrm{diag}(\mathbf{U}^T \mathbf{A} \mathbf{V}) = \mathbf{s}
   * = \sigma_1, \ldots, \sigma_p
   * \f]
   * where \f$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0\f$ are the
   * singular values of \f$\mathbf{A}\f$.
   * Or put differently:
   * \f[
   * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^T
   * \f]
   * where \f$ \mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$
   *
   * \note An external LAPACK library is required by this function.
   */
  bool svd(const mat &A, vec &s);

  /*!
   * \ingroup matrixdecomp
   * \brief Get singular values \c s of a complex matrix \c A using SVD
   *
   * This function calculates singular values \f$s\f$ from the SVD
   * decomposition of a complex matrix \f$A\f$. The SVD algorithm computes
   * the decomposition of a complex \f$m \times n\f$ matrix \f$\mathbf{A}\f$
   * so that
   * \f[
   * \mathrm{diag}(\mathbf{U}^H \mathbf{A} \mathbf{V}) = \mathbf{s}
   * = \sigma_1, \ldots, \sigma_p
   * \f]
   * where \f$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0\f$
   * are the singular values of \f$\mathbf{A}\f$.
   * Or put differently:
   * \f[
   * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H
   * \f]
   * where \f$ \mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$
   *
   * \note An external LAPACK library is required by this function.
   */
  bool svd(const cmat &A, vec &s);

/*!
   * \ingroup matrixdecomp
   * \brief Return singular values of a real matrix \c A using SVD
   *
   * This function returns singular values from the SVD decomposition
   * of a real matrix \f$A\f$. The SVD algorithm computes the decomposition
   * of a real \f$m \times n\f$ matrix \f$\mathbf{A}\f$ so that
   * \f[
   * \mathrm{diag}(\mathbf{U}^T \mathbf{A} \mathbf{V}) = \mathbf{s}
   * = \sigma_1, \ldots, \sigma_p
   * \f]
   * where \f$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0\f$ are the
   * singular values of \f$\mathbf{A}\f$.
   * Or put differently:
   * \f[
   * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^T
   * \f]
   * where \f$ \mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$
   *
   * \note An external LAPACK library is required by this function.
   */
  vec svd(const mat &A);

  /*!
   * \ingroup matrixdecomp
   * \brief Return singular values of a complex matrix \c A using SVD
   *
   * This function returns singular values from the SVD
   * decomposition of a complex matrix \f$A\f$. The SVD algorithm computes
   * the decomposition of a complex \f$m \times n\f$ matrix \f$\mathbf{A}\f$
   * so that
   * \f[
   * \mathrm{diag}(\mathbf{U}^H \mathbf{A} \mathbf{V}) = \mathbf{s}
   * = \sigma_1, \ldots, \sigma_p
   * \f]
   * where \f$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0\f$
   * are the singular values of \f$\mathbf{A}\f$.
   * Or put differently:
   * \f[
   * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H
   * \f]
   * where \f$ \mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$
   *
   * \note An external LAPACK library is required by this function.
   */
  vec svd(const cmat &A);

  /*!
   * \ingroup matrixdecomp
   * \brief Perform Singular Value Decomposition (SVD) of a real matrix \c A
   *
   * This function returns two orthonormal matrices \f$U\f$ and \f$V\f$
   * and a vector of singular values \f$s\f$.
   * The SVD algorithm computes the decomposition of a real \f$m \times n\f$
   * matrix \f$\mathbf{A}\f$ so that
   * \f[
   * \mathrm{diag}(\mathbf{U}^T \mathbf{A} \mathbf{V}) = \mathbf{s}
   * = \sigma_1, \ldots, \sigma_p
   * \f]
   * where the elements of \f$\mathbf{s}\f$, \f$\sigma_1 \geq \sigma_2 \geq
   * \ldots \sigma_p \geq 0\f$ are the singular values of \f$\mathbf{A}\f$.
   * Or put differently:
   * \f[
   * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^T
   * \f]
   * where \f$ \mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$
   *
   * \note An external LAPACK library is required by this function.
   */
  bool svd(const mat &A, mat &U, vec &s, mat &V);

  /*!
   * \ingroup matrixdecomp
   * \brief Perform Singular Value Decomposition (SVD) of a complex matrix \c A
   *
   * This function returns two orthonormal matrices \f$U\f$ and \f$V\f$
   * and a vector of singular values \f$s\f$.
   * The SVD algorithm computes the decomposition of a complex \f$m \times n\f$
   * matrix \f$\mathbf{A}\f$ so that
   * \f[
   * \mathrm{diag}(\mathbf{U}^H \mathbf{A} \mathbf{V}) = \mathbf{s}
   * = \sigma_1, \ldots, \sigma_p
   * \f]
   * where the elements of \f$\mathbf{s}\f$, \f$\sigma_1 \geq \sigma_2 \geq
   * \ldots \sigma_p \geq 0\f$ are the singular values of \f$\mathbf{A}\f$.
   * Or put differently:
   * \f[
   * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H
   * \f]
   * where \f$ \mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$
   *
   * \note An external LAPACK library is required by this function.
   */
  bool svd(const cmat &A, cmat &U, vec &s, cmat &V);


} // namespace itpp

#endif // #ifndef SVD_H