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/* linalg/exponential.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002 Gerard Jungman, Brian Gough
*
* 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 Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/
/* Author: G. Jungman */
/* Calculate the matrix exponential, following
* Moler + Van Loan, SIAM Rev. 20, 801 (1978).
*/
#include <config.h>
#include <stdlib.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_mode.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_blas.h>
#include "gsl_linalg.h"
/* store one of the suggested choices for the
* Taylor series / square method from Moler + VanLoan
*/
struct moler_vanloan_optimal_suggestion
{
int k;
int j;
};
typedef struct moler_vanloan_optimal_suggestion mvl_suggestion_t;
/* table from Moler and Van Loan
* mvl_tab[gsl_mode_t][matrix_norm_group]
*/
static mvl_suggestion_t mvl_tab[3][6] =
{
/* double precision */
{
{ 5, 1 }, { 5, 4 }, { 7, 5 }, { 9, 7 }, { 10, 10 }, { 8, 14 }
},
/* single precision */
{
{ 2, 1 }, { 4, 0 }, { 7, 1 }, { 6, 5 }, { 5, 9 }, { 7, 11 }
},
/* approx precision */
{
{ 1, 0 }, { 3, 0 }, { 5, 1 }, { 4, 5 }, { 4, 8 }, { 2, 11 }
}
};
inline
static double
sup_norm(const gsl_matrix * A)
{
double min, max;
gsl_matrix_minmax(A, &min, &max);
return GSL_MAX_DBL(fabs(min), fabs(max));
}
static
mvl_suggestion_t
obtain_suggestion(const gsl_matrix * A, gsl_mode_t mode)
{
const unsigned int mode_prec = GSL_MODE_PREC(mode);
const double norm_A = sup_norm(A);
if(norm_A < 0.01) return mvl_tab[mode_prec][0];
else if(norm_A < 0.1) return mvl_tab[mode_prec][1];
else if(norm_A < 1.0) return mvl_tab[mode_prec][2];
else if(norm_A < 10.0) return mvl_tab[mode_prec][3];
else if(norm_A < 100.0) return mvl_tab[mode_prec][4];
else if(norm_A < 1000.0) return mvl_tab[mode_prec][5];
else
{
/* outside the table we simply increase the number
* of squarings, bringing the reduced matrix into
* the range of the table; this is obviously suboptimal,
* but that is the price paid for not having those extra
* table entries
*/
const double extra = log(1.01*norm_A/1000.0) / M_LN2;
const int extra_i = (unsigned int) ceil(extra);
mvl_suggestion_t s = mvl_tab[mode][5];
s.j += extra_i;
return s;
}
}
/* use series representation to calculate matrix exponential;
* this is used for small matrices; we use the sup_norm
* to measure the size of the terms in the expansion
*/
static void
matrix_exp_series(
const gsl_matrix * B,
gsl_matrix * eB,
int number_of_terms
)
{
int count;
gsl_matrix * temp = gsl_matrix_calloc(B->size1, B->size2);
/* init the Horner polynomial evaluation,
* eB = 1 + B/number_of_terms; we use
* eB to collect the partial results
*/
gsl_matrix_memcpy(eB, B);
gsl_matrix_scale(eB, 1.0/number_of_terms);
gsl_matrix_add_diagonal(eB, 1.0);
for(count = number_of_terms-1; count >= 1; --count)
{
/* mult_temp = 1 + B eB / count */
gsl_blas_dgemm(CblasNoTrans, CblasNoTrans, 1.0, B, eB, 0.0, temp);
gsl_matrix_scale(temp, 1.0/count);
gsl_matrix_add_diagonal(temp, 1.0);
/* transfer partial result out of temp */
gsl_matrix_memcpy(eB, temp);
}
/* now eB holds the full result; we're done */
gsl_matrix_free(temp);
}
int
gsl_linalg_exponential_ss(
const gsl_matrix * A,
gsl_matrix * eA,
gsl_mode_t mode
)
{
if(A->size1 != A->size2)
{
GSL_ERROR("cannot exponentiate a non-square matrix", GSL_ENOTSQR);
}
else if(A->size1 != eA->size1 || A->size2 != eA->size2)
{
GSL_ERROR("exponential of matrix must have same dimension as matrix", GSL_EBADLEN);
}
else
{
int i;
const mvl_suggestion_t sugg = obtain_suggestion(A, mode);
const double divisor = exp(M_LN2 * sugg.j);
gsl_matrix * reduced_A = gsl_matrix_alloc(A->size1, A->size2);
/* decrease A by the calculated divisor */
gsl_matrix_memcpy(reduced_A, A);
gsl_matrix_scale(reduced_A, 1.0/divisor);
/* calculate exp of reduced matrix; store in eA as temp */
matrix_exp_series(reduced_A, eA, sugg.k);
/* square repeatedly; use reduced_A for scratch */
for(i = 0; i < sugg.j; ++i)
{
gsl_blas_dgemm(CblasNoTrans, CblasNoTrans, 1.0, eA, eA, 0.0, reduced_A);
gsl_matrix_memcpy(eA, reduced_A);
}
gsl_matrix_free(reduced_A);
return GSL_SUCCESS;
}
}
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