/* multifit/multilinear.c

 * 

 * Copyright (C) 2000, 2007, 2010 Brian Gough

 * Copyright (C) 2013, 2015 Patrick Alken

 * 

 * 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 3 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 Street, Fifth Floor, Boston, MA 02110-1301, USA.

 */

#include <config.h>

#include <gsl/gsl_errno.h>

#include <gsl/gsl_multifit.h>

#include <gsl/gsl_blas.h>

#include <gsl/gsl_vector.h>

#include <gsl/gsl_matrix.h>

#include <gsl/gsl_linalg.h>

#include "linear_common.c"

static int multifit_linear_svd (const gsl_matrix * X,

                                const int balance,

                                gsl_multifit_linear_workspace * work);

int

gsl_multifit_linear (const gsl_matrix * X,

                     const gsl_vector * y,

                     gsl_vector * c,

                     gsl_matrix * cov,

                     double *chisq, gsl_multifit_linear_workspace * work)

{

  size_t rank;

  int status = gsl_multifit_linear_tsvd(X, y, GSL_DBL_EPSILON, c, cov, chisq, &rank, work);

  return status;

}

/*

gsl_multifit_linear_tsvd()

  Solve linear least squares system with truncated SVD

Inputs: X     - least squares matrix, n-by-p

        y     - right hand side vector, n-by-1

        tol   - tolerance for singular value truncation; if

                s_j <= tol * s_0

                then it is discarded from series expansion

        c     - (output) solution vector, p-by-1

        cov   - (output) covariance matrix, p-by-p

        chisq - (output) cost function chi^2

        rank  - (output) effective rank (number of singular values

                used in solution)

        work  - workspace

*/

int

gsl_multifit_linear_tsvd (const gsl_matrix * X,

                          const gsl_vector * y,

                          const double tol,

                          gsl_vector * c,

                          gsl_matrix * cov,

                          double * chisq,

                          size_t * rank,

                          gsl_multifit_linear_workspace * work)

{

  const size_t n = X->size1;

  const size_t p = X->size2;

  if (y->size != n)

    {

      GSL_ERROR("number of observations in y does not match matrix",

                GSL_EBADLEN);

    }

  else if (p != c->size)

    {

      GSL_ERROR ("number of parameters c does not match matrix",

                 GSL_EBADLEN);

    }

  else if (tol <= 0)

    {

      GSL_ERROR ("tolerance must be positive", GSL_EINVAL);

    }

  else

    {

      int status;

      double rnorm = 0.0, snorm;

      /* compute balanced SVD */

      status = gsl_multifit_linear_bsvd (X, work);

      if (status)

        return status;

      status = multifit_linear_solve (X, y, tol, -1.0, rank,

                                      c, &rnorm, &snorm, work);

      *chisq = rnorm * rnorm;

      /* variance-covariance matrix cov = s2 * (Q S^-1) (Q S^-1)^T */

      {

        double r2 = rnorm * rnorm;

        double s2 = r2 / (double)(n - *rank);

        size_t i, j;

        gsl_matrix_view QSI = gsl_matrix_submatrix(work->QSI, 0, 0, p, p);

        gsl_vector_view D = gsl_vector_subvector(work->D, 0, p);

        for (i = 0; i < p; i++)

          {

            gsl_vector_view row_i = gsl_matrix_row (&QSI.matrix, i);

            double d_i = gsl_vector_get (&D.vector, i);

            for (j = i; j < p; j++)

              {

                gsl_vector_view row_j = gsl_matrix_row (&QSI.matrix, j);

                double d_j = gsl_vector_get (&D.vector, j);

                double s;

                gsl_blas_ddot (&row_i.vector, &row_j.vector, &s);

                gsl_matrix_set (cov, i, j, s * s2 / (d_i * d_j));

                gsl_matrix_set (cov, j, i, s * s2 / (d_i * d_j));

              }

          }

      }

      return status;

    }

}

/*

gsl_multifit_linear_svd()

  Perform SVD decomposition of the matrix X and store in work without

balancing

*/

int

gsl_multifit_linear_svd (const gsl_matrix * X,

                         gsl_multifit_linear_workspace * work)

{

  /* do not balance by default */

  int status = multifit_linear_svd(X, 0, work);

  return status;

}

/*

gsl_multifit_linear_bsvd()

  Perform SVD decomposition of the matrix X and store in work with

balancing

*/

int

gsl_multifit_linear_bsvd (const gsl_matrix * X,

                          gsl_multifit_linear_workspace * work)

{

  int status = multifit_linear_svd(X, 1, work);

  return status;

}

size_t

gsl_multifit_linear_rank(const double tol, const gsl_multifit_linear_workspace * work)

{

  double s0 = gsl_vector_get (work->S, 0);

  size_t rank = 0;

  size_t j;

  for (j = 0; j < work->p; j++)

    {

      double sj = gsl_vector_get (work->S, j);

      if (sj > tol * s0)

        ++rank;

    }

  return rank;

}

/* Estimation of values for given x */

int

gsl_multifit_linear_est (const gsl_vector * x,

                         const gsl_vector * c,

                         const gsl_matrix * cov, double *y, double *y_err)

{

  if (x->size != c->size)

    {

      GSL_ERROR ("number of parameters c does not match number of observations x",

         GSL_EBADLEN);

    }

  else if (cov->size1 != cov->size2)

    {

      GSL_ERROR ("covariance matrix is not square", GSL_ENOTSQR);

    }

  else if (c->size != cov->size1)

    {

      GSL_ERROR ("number of parameters c does not match size of covariance matrix cov",

         GSL_EBADLEN);

    }

  else

    {

      size_t i, j;

      double var = 0;

      gsl_blas_ddot(x, c, y);       /* y = x.c */

      /* var = x' cov x */

      for (i = 0; i < x->size; i++)

        {

          const double xi = gsl_vector_get (x, i);

          var += xi * xi * gsl_matrix_get (cov, i, i);

          for (j = 0; j < i; j++)

            {

              const double xj = gsl_vector_get (x, j);

              var += 2 * xi * xj * gsl_matrix_get (cov, i, j);

            }

        }

      *y_err = sqrt (var);

      return GSL_SUCCESS;

    }

}

/*

gsl_multifit_linear_rcond()

  Return reciprocal condition number of LS matrix;

gsl_multifit_linear_svd() must first be called to

compute the SVD of X and its reciprocal condition number

*/

double

gsl_multifit_linear_rcond (const gsl_multifit_linear_workspace * w)

{

  return w->rcond;

}

/*

gsl_multifit_linear_residuals()

  Compute vector of residuals from fit

Inputs: X - design matrix

        y - rhs vector

        c - fit coefficients

        r - (output) where to store residuals

*/

int

gsl_multifit_linear_residuals (const gsl_matrix *X, const gsl_vector *y,

                               const gsl_vector *c, gsl_vector *r)

{

  if (X->size1 != y->size)

    {

      GSL_ERROR

        ("number of observations in y does not match rows of matrix X",

         GSL_EBADLEN);

    }

  else if (X->size2 != c->size)

    {

      GSL_ERROR ("number of parameters c does not match columns of matrix X",

                 GSL_EBADLEN);

    }

  else if (y->size != r->size)

    {

      GSL_ERROR ("number of observations in y does not match number of residuals",

                 GSL_EBADLEN);

    }

  else

    {

      /* r = y - X c */

      gsl_vector_memcpy(r, y);

      gsl_blas_dgemv(CblasNoTrans, -1.0, X, c, 1.0, r);

      return GSL_SUCCESS;

    }

} /* gsl_multifit_linear_residuals() */

/* Perform a SVD decomposition on the least squares matrix X = U S Q^T

 *

 * Inputs: X       - least squares matrix

 *         balance - 1 to perform column balancing

 *         work    - workspace

 *

 * Notes:

 * 1) On output,

 *    work->A contains the matrix U

 *    work->Q contains the matrix Q

 *    work->S contains the vector of singular values

 * 2) The matrix X may have smaller dimensions than the workspace

 *    in the case of stdform2() - but the dimensions cannot be larger

 * 3) On output, work->n and work->p are set to the dimensions of X

 * 4) On output, work->rcond is set to the reciprocal condition number of X

 */

static int

multifit_linear_svd (const gsl_matrix * X,

                     const int balance,

                     gsl_multifit_linear_workspace * work)

{

  const size_t n = X->size1;

  const size_t p = X->size2;

  if (n > work->nmax || p > work->pmax)

    {

      GSL_ERROR("observation matrix larger than workspace", GSL_EBADLEN);

    }

  else

    {

      gsl_matrix_view A = gsl_matrix_submatrix(work->A, 0, 0, n, p);

      gsl_matrix_view Q = gsl_matrix_submatrix(work->Q, 0, 0, p, p);

      gsl_matrix_view QSI = gsl_matrix_submatrix(work->QSI, 0, 0, p, p);

      gsl_vector_view S = gsl_vector_subvector(work->S, 0, p);

      gsl_vector_view xt = gsl_vector_subvector(work->xt, 0, p);

      gsl_vector_view D = gsl_vector_subvector(work->D, 0, p);

      /* Copy X to workspace,  A <= X */

      gsl_matrix_memcpy (&A.matrix, X);

      /* Balance the columns of the matrix A if requested */

      if (balance) 

        {

          gsl_linalg_balance_columns (&A.matrix, &D.vector);

        }

      else

        {

          gsl_vector_set_all (&D.vector, 1.0);

        }

      /* decompose A into U S Q^T */

      gsl_linalg_SV_decomp_mod (&A.matrix, &QSI.matrix, &Q.matrix,

                                &S.vector, &xt.vector);

      /* compute reciprocal condition number rcond = smin / smax */

      {

        double smin, smax;

        gsl_vector_minmax(&S.vector, &smin, &smax;;

        work->rcond = smin / smax;

      }

      work->n = n;

      work->p = p;

      return GSL_SUCCESS;

    }

}