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cholesky.c

/* Cholesky Decomposition
 *
 * Copyright (C) 2000  Thomas Walter
 *
 * 03 May 2000: Modified for GSL by Brian Gough
 * 29 Jul 2005: Additions by Gerard Jungman
 * Copyright (C) 2000,2001, 2002, 2003, 2005, 2007 Brian Gough, Gerard Jungman
 *
 * This 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, or (at your option) any
 * later version.
 *
 * This source 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.
 */

/*
 * Cholesky decomposition of a symmetrix positive definite matrix.
 * This is useful to solve the matrix arising in
 *    periodic cubic splines
 *    approximating splines
 *
 * This algorithm does:
 *   A = L * L'
 * with
 *   L  := lower left triangle matrix
 *   L' := the transposed form of L.
 *
 */

#include <config.h>

#include <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_linalg.h>

static inline 
double
quiet_sqrt (double x)  
     /* avoids runtime error, for checking matrix for positive definiteness */
{
  return (x >= 0) ? sqrt(x) : GSL_NAN;
}

int
gsl_linalg_cholesky_decomp (gsl_matrix * A)
{
  const size_t M = A->size1;
  const size_t N = A->size2;

  if (M != N)
    {
      GSL_ERROR("cholesky decomposition requires square matrix", GSL_ENOTSQR);
    }
  else
    {
      size_t i,j,k;
      int status = 0;

      /* Do the first 2 rows explicitly.  It is simple, and faster.  And
       * one can return if the matrix has only 1 or 2 rows.  
       */

      double A_00 = gsl_matrix_get (A, 0, 0);
      
      double L_00 = quiet_sqrt(A_00);
      
      if (A_00 <= 0)
        {
          status = GSL_EDOM ;
        }

      gsl_matrix_set (A, 0, 0, L_00);
  
      if (M > 1)
        {
          double A_10 = gsl_matrix_get (A, 1, 0);
          double A_11 = gsl_matrix_get (A, 1, 1);
          
          double L_10 = A_10 / L_00;
          double diag = A_11 - L_10 * L_10;
          double L_11 = quiet_sqrt(diag);
          
          if (diag <= 0)
            {
              status = GSL_EDOM;
            }

          gsl_matrix_set (A, 1, 0, L_10);        
          gsl_matrix_set (A, 1, 1, L_11);
        }
      
      for (k = 2; k < M; k++)
        {
          double A_kk = gsl_matrix_get (A, k, k);
          
          for (i = 0; i < k; i++)
            {
              double sum = 0;

              double A_ki = gsl_matrix_get (A, k, i);
              double A_ii = gsl_matrix_get (A, i, i);

              gsl_vector_view ci = gsl_matrix_row (A, i);
              gsl_vector_view ck = gsl_matrix_row (A, k);

              if (i > 0) {
                gsl_vector_view di = gsl_vector_subvector(&ci.vector, 0, i);
                gsl_vector_view dk = gsl_vector_subvector(&ck.vector, 0, i);
                
                gsl_blas_ddot (&di.vector, &dk.vector, &sum);
              }

              A_ki = (A_ki - sum) / A_ii;
              gsl_matrix_set (A, k, i, A_ki);
            } 

          {
            gsl_vector_view ck = gsl_matrix_row (A, k);
            gsl_vector_view dk = gsl_vector_subvector (&ck.vector, 0, k);
            
            double sum = gsl_blas_dnrm2 (&dk.vector);
            double diag = A_kk - sum * sum;

            double L_kk = quiet_sqrt(diag);
            
            if (diag <= 0)
              {
                status = GSL_EDOM;
              }
            
            gsl_matrix_set (A, k, k, L_kk);
          }
        }

      /* Now copy the transposed lower triangle to the upper triangle,
       * the diagonal is common.  
       */
      
      for (i = 1; i < M; i++)
        {
          for (j = 0; j < i; j++)
            {
              double A_ij = gsl_matrix_get (A, i, j);
              gsl_matrix_set (A, j, i, A_ij);
            }
        } 
      
      if (status == GSL_EDOM)
        {
          GSL_ERROR ("matrix must be positive definite", GSL_EDOM);
        }
      
      return GSL_SUCCESS;
    }
}


int
gsl_linalg_cholesky_solve (const gsl_matrix * LLT,
                           const gsl_vector * b,
                           gsl_vector * x)
{
  if (LLT->size1 != LLT->size2)
    {
      GSL_ERROR ("cholesky matrix must be square", GSL_ENOTSQR);
    }
  else if (LLT->size1 != b->size)
    {
      GSL_ERROR ("matrix size must match b size", GSL_EBADLEN);
    }
  else if (LLT->size2 != x->size)
    {
      GSL_ERROR ("matrix size must match solution size", GSL_EBADLEN);
    }
  else
    {
      /* Copy x <- b */

      gsl_vector_memcpy (x, b);

      /* Solve for c using forward-substitution, L c = b */

      gsl_blas_dtrsv (CblasLower, CblasNoTrans, CblasNonUnit, LLT, x);

      /* Perform back-substitution, U x = c */

      gsl_blas_dtrsv (CblasUpper, CblasNoTrans, CblasNonUnit, LLT, x);


      return GSL_SUCCESS;
    }
}

int
gsl_linalg_cholesky_svx (const gsl_matrix * LLT,
                         gsl_vector * x)
{
  if (LLT->size1 != LLT->size2)
    {
      GSL_ERROR ("cholesky matrix must be square", GSL_ENOTSQR);
    }
  else if (LLT->size2 != x->size)
    {
      GSL_ERROR ("matrix size must match solution size", GSL_EBADLEN);
    }
  else
    {
      /* Solve for c using forward-substitution, L c = b */

      gsl_blas_dtrsv (CblasLower, CblasNoTrans, CblasNonUnit, LLT, x);

      /* Perform back-substitution, U x = c */

      gsl_blas_dtrsv (CblasUpper, CblasNoTrans, CblasNonUnit, LLT, x);

      return GSL_SUCCESS;
    }
}

/*
gsl_linalg_cholesky_invert()
  Compute the inverse of a symmetric positive definite matrix in
Cholesky form.

Inputs: LLT - matrix in cholesky form on input
              A^{-1} = L^{-t} L^{-1} on output

Return: success or error
*/

int
gsl_linalg_cholesky_invert(gsl_matrix * LLT)
{
  if (LLT->size1 != LLT->size2)
    {
      GSL_ERROR ("cholesky matrix must be square", GSL_ENOTSQR);
    }
  else
    {
      size_t N = LLT->size1;
      size_t i, j;
      double sum;
      gsl_vector_view v1, v2;

      /* invert the lower triangle of LLT */
      for (i = 0; i < N; ++i)
        {
          double ajj;

          j = N - i - 1;

          gsl_matrix_set(LLT, j, j, 1.0 / gsl_matrix_get(LLT, j, j));
          ajj = -gsl_matrix_get(LLT, j, j);

          if (j < N - 1)
            {
              gsl_matrix_view m;
              
              m = gsl_matrix_submatrix(LLT, j + 1, j + 1,
                                       N - j - 1, N - j - 1);
              v1 = gsl_matrix_subcolumn(LLT, j, j + 1, N - j - 1);

              gsl_blas_dtrmv(CblasLower, CblasNoTrans, CblasNonUnit,
                             &m.matrix, &v1.vector);

              gsl_blas_dscal(ajj, &v1.vector);
            }
        } /* for (i = 0; i < N; ++i) */

      /*
       * The lower triangle of LLT now contains L^{-1}. Now compute
       * A^{-1} = L^{-t} L^{-1}
       *
       * The (ij) element of A^{-1} is column i of L^{-1} dotted into
       * column j of L^{-1}
       */

      for (i = 0; i < N; ++i)
        {
          for (j = i + 1; j < N; ++j)
            {
              v1 = gsl_matrix_subcolumn(LLT, i, j, N - j);
              v2 = gsl_matrix_subcolumn(LLT, j, j, N - j);

              /* compute Ainv_{ij} = sum_k Linv_{ki} Linv_{kj} */
              gsl_blas_ddot(&v1.vector, &v2.vector, &sum);

              /* store in upper triangle */
              gsl_matrix_set(LLT, i, j, sum);
            }

          /* now compute the diagonal element */
          v1 = gsl_matrix_subcolumn(LLT, i, i, N - i);
          gsl_blas_ddot(&v1.vector, &v1.vector, &sum);
          gsl_matrix_set(LLT, i, i, sum);
        }

      /* copy the transposed upper triangle to the lower triangle */

      for (j = 1; j < N; j++)
        {
          for (i = 0; i < j; i++)
            {
              double A_ij = gsl_matrix_get (LLT, i, j);
              gsl_matrix_set (LLT, j, i, A_ij);
            }
        } 

      return GSL_SUCCESS;
    }
} /* gsl_linalg_cholesky_invert() */

int
gsl_linalg_cholesky_decomp_unit(gsl_matrix * A, gsl_vector * D)
{
  const size_t N = A->size1;
  size_t i, j;

  /* initial Cholesky */
  int stat_chol = gsl_linalg_cholesky_decomp(A);

  if(stat_chol == GSL_SUCCESS)
  {
    /* calculate D from diagonal part of initial Cholesky */
    for(i = 0; i < N; ++i)
    {
      const double C_ii = gsl_matrix_get(A, i, i);
      gsl_vector_set(D, i, C_ii*C_ii);
    }

    /* multiply initial Cholesky by 1/sqrt(D) on the right */
    for(i = 0; i < N; ++i)
    {
      for(j = 0; j < N; ++j)
      {
        gsl_matrix_set(A, i, j, gsl_matrix_get(A, i, j) / sqrt(gsl_vector_get(D, j)));
      }
    }

    /* Because the initial Cholesky contained both L and transpose(L),
       the result of the multiplication is not symmetric anymore;
       but the lower triangle _is_ correct. Therefore we reflect
       it to the upper triangle and declare victory.
       */
    for(i = 0; i < N; ++i)
      for(j = i + 1; j < N; ++j)
        gsl_matrix_set(A, i, j, gsl_matrix_get(A, j, i));
  }

  return stat_chol;
}

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