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implement matrix exponentials

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This commit is contained in:
Fredrik Johansson 2013-11-11 18:50:53 +01:00
parent 7ce5df8690
commit b78e176b0d
9 changed files with 719 additions and 4 deletions
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20
doc/source/fmpcb_mat.rst
View file

@ -243,3 +243,23 @@ Gaussian elimination and solving
determinant of the remaining submatrix is bounded using
Hadamard's inequality.
Special functions
-------------------------------------------------------------------------------
.. function:: void fmpcb_mat_exp(fmpcb_mat_t B, const fmpcb_mat_t A, long prec)
Sets *B* to the exponential of the matrix *A*, defined by the Taylor series
.. math ::
\exp(A) = \sum_{k=0}^{\infty} \frac{A^k}{k!}.
The exponential function is evaluated using scaling followed by
rectangular splitting evaluation of the Taylor series.
Scaling amounts to picking a nonnegative integer *r* such that
the Taylor series converges quickly, and then evaluating
`\exp(A/2^r)^{2^r}`.
If `\|A/2^r\| \le c` and `N \ge 2c`, we bound the entrywise error
when truncating the Taylor series before term `N` by `2 c^N / N!`.

21
doc/source/fmprb_mat.rst
View file

@ -236,3 +236,24 @@ Gaussian elimination and solving
determinant of the remaining submatrix is bounded using
Hadamard's inequality.
Special functions
-------------------------------------------------------------------------------
.. function:: void fmprb_mat_exp(fmprb_mat_t B, const fmprb_mat_t A, long prec)
Sets *B* to the exponential of the matrix *A*, defined by the Taylor series
.. math ::
\exp(A) = \sum_{k=0}^{\infty} \frac{A^k}{k!}.
The exponential function is evaluated using scaling followed by
rectangular splitting evaluation of the Taylor series.
Scaling amounts to picking a nonnegative integer *r* such that
the Taylor series converges quickly, and then evaluating
`\exp(A/2^r)^{2^r}`.
If `\|A/2^r\| \le c` and `N \ge 2c`, we bound the entrywise error
when truncating the Taylor series before term `N` by `2 c^N / N!`.

12
fmpcb.h
View file

@ -249,6 +249,13 @@ fmpcb_trim(fmpcb_t z, const fmpcb_t x)
fmprb_trim(fmpcb_imagref(z), fmpcb_imagref(x));
}
static __inline__ void
fmpcb_add_error_fmpr(fmpcb_t x, const fmpr_t err)
{
fmprb_add_error_fmpr(fmpcb_realref(x), err);
fmprb_add_error_fmpr(fmpcb_imagref(x), err);
}
static __inline__ void
fmpcb_get_abs_ubound_fmpr(fmpr_t u, const fmpcb_t z, long prec)
{
@ -826,10 +833,7 @@ _fmpcb_vec_add_error_fmpr_vec(fmpcb_ptr res, fmpr_srcptr err, long len)
{
long i;
for (i = 0; i < len; i++)
{
fmprb_add_error_fmpr(fmpcb_realref(res + i), err + i);
fmprb_add_error_fmpr(fmpcb_imagref(res + i), err + i);
}
fmpcb_add_error_fmpr(res + i, err + i);
}
static __inline__ void

3
fmpcb_mat.h
View file

@ -282,6 +282,9 @@ int fmpcb_mat_inv(fmpcb_mat_t X, const fmpcb_mat_t A, long prec);
void fmpcb_mat_det(fmpcb_t det, const fmpcb_mat_t A, long prec);
/* Special functions */
void fmpcb_mat_exp(fmpcb_mat_t B, const fmpcb_mat_t A, long prec);
#ifdef __cplusplus
}

199
fmpcb_mat/exp.c Normal file
View file

@ -0,0 +1,199 @@
/*=============================================================================
This file is part of ARB.
ARB 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.
ARB 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 ARB; if not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
=============================================================================*/
/******************************************************************************
Copyright (C) 2012 Fredrik Johansson
******************************************************************************/
#include "double_extras.h"
#include "fmpcb_mat.h"
long _fmprb_mat_exp_choose_N(const fmpr_t norm, long prec);
void _fmprb_mat_exp_bound(fmpr_t err, const fmpr_t norm, long N);
/* evaluates the truncated Taylor series (assumes no aliasing) */
void
_fmpcb_mat_exp_taylor(fmpcb_mat_t S, const fmpcb_mat_t A, long N, long prec)
{
if (N == 1)
{
fmpcb_mat_one(S);
}
else if (N == 2)
{
fmpcb_mat_one(S);
fmpcb_mat_add(S, S, A, prec);
}
else if (N == 3)
{
fmpcb_mat_t T;
fmpcb_mat_init(T, fmpcb_mat_nrows(A), fmpcb_mat_nrows(A));
fmpcb_mat_mul(T, A, A, prec);
fmpcb_mat_scalar_mul_2exp_si(T, T, -1);
fmpcb_mat_add(S, A, T, prec);
fmpcb_mat_one(T);
fmpcb_mat_add(S, S, T, prec);
fmpcb_mat_clear(T);
}
else
{
long i, lo, hi, m, w, dim;
fmpcb_mat_struct * pows;
fmpcb_mat_t T, U;
fmpz_t c, f;
dim = fmpcb_mat_nrows(A);
m = n_sqrt(N);
w = (N + m - 1) / m;
fmpz_init(c);
fmpz_init(f);
pows = flint_malloc(sizeof(fmpcb_mat_t) * (m + 1));
fmpcb_mat_init(T, dim, dim);
fmpcb_mat_init(U, dim, dim);
for (i = 0; i <= m; i++)
{
fmpcb_mat_init(pows + i, dim, dim);
if (i == 0)
fmpcb_mat_one(pows + i);
else if (i == 1)
fmpcb_mat_set(pows + i, A);
else
fmpcb_mat_mul(pows + i, pows + i - 1, A, prec);
}
fmpcb_mat_zero(S);
fmpz_one(f);
for (i = w - 1; i >= 0; i--)
{
lo = i * m;
hi = FLINT_MIN(N - 1, lo + m - 1);
fmpcb_mat_zero(T);
fmpz_one(c);
while (hi >= lo)
{
fmpcb_mat_scalar_addmul_fmpz(T, pows + hi - lo, c, prec);
if (hi != 0)
fmpz_mul_ui(c, c, hi);
hi--;
}
fmpcb_mat_mul(U, pows + m, S, prec);
fmpcb_mat_scalar_mul_fmpz(S, T, f, prec);
fmpcb_mat_add(S, S, U, prec);
fmpz_mul(f, f, c);
}
fmpcb_mat_scalar_div_fmpz(S, S, f, prec);
fmpz_clear(c);
fmpz_clear(f);
for (i = 0; i <= m; i++)
fmpcb_mat_clear(pows + i);
flint_free(pows);
fmpcb_mat_clear(T);
fmpcb_mat_clear(U);
}
}
void
fmpcb_mat_exp(fmpcb_mat_t B, const fmpcb_mat_t A, long prec)
{
long i, j, dim, wp, N, q, r;
fmpr_t norm, err;
fmpcb_mat_t T;
dim = fmpcb_mat_nrows(A);
if (dim != fmpcb_mat_ncols(A))
{
printf("fmpcb_mat_exp: a square matrix is required!\n");
abort();
}
if (dim == 0)
{
return;
}
else if (dim == 1)
{
fmpcb_exp(fmpcb_mat_entry(B, 0, 0), fmpcb_mat_entry(A, 0, 0), prec);
return;
}
wp = prec + 3 * FLINT_BIT_COUNT(prec);
fmpr_init(norm);
fmpr_init(err);
fmpcb_mat_init(T, dim, dim);
fmpcb_mat_bound_inf_norm(norm, A, FMPRB_RAD_PREC);
if (fmpr_is_zero(norm))
{
fmpcb_mat_one(B);
}
else
{
r = fmpr_abs_bound_lt_2exp_si(norm);
q = pow(wp, 0.25); /* wanted magnitude */
if (r > 2 * wp) /* too big */
r = 2 * wp;
else if (r < -q) /* tiny, no need to reduce */
r = 0;
else
r += q; /* reduce to magnitude 2^(-r) */
fmpcb_mat_scalar_mul_2exp_si(T, A, -r);
fmpr_mul_2exp_si(norm, norm, -r);
N = _fmprb_mat_exp_choose_N(norm, wp);
_fmprb_mat_exp_bound(err, norm, N);
_fmpcb_mat_exp_taylor(B, T, N, wp);
for (i = 0; i < dim; i++)
for (j = 0; j < dim; j++)
fmpcb_add_error_fmpr(fmpcb_mat_entry(B, i, j), err);
for (i = 0; i < r; i++)
{
fmpcb_mat_mul(T, B, B, wp);
fmpcb_mat_swap(T, B);
}
for (i = 0; i < dim; i++)
for (j = 0; j < dim; j++)
fmpcb_set_round(fmpcb_mat_entry(B, i, j),
fmpcb_mat_entry(B, i, j), prec);
}
fmpr_clear(norm);
fmpr_clear(err);
fmpcb_mat_clear(T);
}

105
fmpcb_mat/test/t-exp.c Normal file
View file

@ -0,0 +1,105 @@
/*=============================================================================
This file is part of ARB.
ARB 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.
ARB 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 ARB; if not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
=============================================================================*/
/******************************************************************************
Copyright (C) 2013 Fredrik Johansson
******************************************************************************/
#include "fmpcb_mat.h"
int main()
{
long iter;
flint_rand_t state;
printf("exp....");
fflush(stdout);
flint_randinit(state);
/* check exp(A)*exp(c*A) = exp((1+c)*A) */
for (iter = 0; iter < 500; iter++)
{
fmpcb_mat_t A, E, F, EF, G;
fmpq_mat_t Q;
fmpcb_t c, d;
long n, qbits, prec;
n = n_randint(state, 5);
qbits = 2 + n_randint(state, 300);
prec = 2 + n_randint(state, 300);
fmpcb_init(c);
fmpcb_init(d);
fmpq_mat_init(Q, n, n);
fmpcb_mat_init(A, n, n);
fmpcb_mat_init(E, n, n);
fmpcb_mat_init(F, n, n);
fmpcb_mat_init(EF, n, n);
fmpcb_mat_init(G, n, n);
fmpq_mat_randtest(Q, state, qbits);
fmpcb_mat_set_fmpq_mat(A, Q, prec);
fmpcb_mat_exp(E, A, prec);
fmpcb_randtest(c, state, prec, 10);
fmpcb_mat_scalar_mul_fmpcb(F, A, c, prec);
fmpcb_mat_exp(F, F, prec);
fmpcb_add_ui(d, c, 1, prec);
fmpcb_mat_scalar_mul_fmpcb(G, A, d, prec);
fmpcb_mat_exp(G, G, prec);
fmpcb_mat_mul(EF, E, F, prec);
if (!fmpcb_mat_overlaps(EF, G))
{
printf("FAIL\n\n");
printf("n = %ld, prec = %ld\n", n, prec);
printf("c = \n"); fmpcb_printd(c, 15); printf("\n\n");
printf("A = \n"); fmpcb_mat_printd(A, 15); printf("\n\n");
printf("E = \n"); fmpcb_mat_printd(E, 15); printf("\n\n");
printf("F = \n"); fmpcb_mat_printd(F, 15); printf("\n\n");
printf("E*F = \n"); fmpcb_mat_printd(EF, 15); printf("\n\n");
printf("G = \n"); fmpcb_mat_printd(G, 15); printf("\n\n");
abort();
}
fmpcb_clear(c);
fmpcb_clear(d);
fmpq_mat_clear(Q);
fmpcb_mat_clear(A);
fmpcb_mat_clear(E);
fmpcb_mat_clear(F);
fmpcb_mat_clear(EF);
fmpcb_mat_clear(G);
}
flint_randclear(state);
flint_cleanup();
printf("PASS\n");
return EXIT_SUCCESS;
}

4
fmprb_mat.h
View file

@ -254,6 +254,10 @@ int fmprb_mat_inv(fmprb_mat_t X, const fmprb_mat_t A, long prec);
void fmprb_mat_det(fmprb_t det, const fmprb_mat_t A, long prec);
/* Special functions */
void fmprb_mat_exp(fmprb_mat_t B, const fmprb_mat_t A, long prec);
#ifdef __cplusplus
}
#endif

254
fmprb_mat/exp.c Normal file
View file

@ -0,0 +1,254 @@
/*=============================================================================
This file is part of ARB.
ARB 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.
ARB 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 ARB; if not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
=============================================================================*/
/******************************************************************************
Copyright (C) 2013 Fredrik Johansson
******************************************************************************/
#include "double_extras.h"
#include "fmprb_mat.h"
#define LOG2_OVER_E 0.25499459743395350926
void fmpr_gamma_ui_lbound(fmpr_t x, ulong n, long prec);
long
_fmprb_mat_exp_choose_N(const fmpr_t norm, long prec)
{
if (fmpr_is_special(norm) || fmpr_cmp_2exp_si(norm, 30) > 0)
{
return 1;
}
else if (fmpr_cmp_2exp_si(norm, -300) < 0)
{
long N = -fmpr_abs_bound_lt_2exp_si(norm);
return (prec + N - 1) / N;
}
else
{
double c, t;
c = fmpr_get_d(norm, FMPR_RND_UP);
t = d_lambertw(prec * LOG2_OVER_E / c);
t = c * exp(t + 1.0);
return FLINT_MIN((long) (t + 1.0), 2 * prec);
}
}
void
_fmprb_mat_exp_bound(fmpr_t err, const fmpr_t norm, long N)
{
fmpr_t t, u;
fmpr_init(t);
fmpr_init(u);
fmpr_set_si_2exp_si(t, N, -1);
/* bound by geometric series when N >= 2*c <=> N/2 >= c */
if (N > 0 && fmpr_cmp(t, norm) >= 0)
{
/* 2 c^N / N! */
fmpr_pow_sloppy_ui(t, norm, N, FMPRB_RAD_PREC, FMPR_RND_UP);
fmpr_gamma_ui_lbound(u, N + 1, FMPRB_RAD_PREC);
fmpr_div(err, t, u, FMPRB_RAD_PREC, FMPR_RND_UP);
fmpr_mul_2exp_si(err, err, 1);
}
else
{
fmpr_pos_inf(err);
}
fmpr_clear(t);
fmpr_clear(u);
}
/* evaluates the truncated Taylor series (assumes no aliasing) */
void
_fmprb_mat_exp_taylor(fmprb_mat_t S, const fmprb_mat_t A, long N, long prec)
{
if (N == 1)
{
fmprb_mat_one(S);
}
else if (N == 2)
{
fmprb_mat_one(S);
fmprb_mat_add(S, S, A, prec);
}
else if (N == 3)
{
fmprb_mat_t T;
fmprb_mat_init(T, fmprb_mat_nrows(A), fmprb_mat_nrows(A));
fmprb_mat_mul(T, A, A, prec);
fmprb_mat_scalar_mul_2exp_si(T, T, -1);
fmprb_mat_add(S, A, T, prec);
fmprb_mat_one(T);
fmprb_mat_add(S, S, T, prec);
fmprb_mat_clear(T);
}
else
{
long i, lo, hi, m, w, dim;
fmprb_mat_struct * pows;
fmprb_mat_t T, U;
fmpz_t c, f;
dim = fmprb_mat_nrows(A);
m = n_sqrt(N);
w = (N + m - 1) / m;
fmpz_init(c);
fmpz_init(f);
pows = flint_malloc(sizeof(fmprb_mat_t) * (m + 1));
fmprb_mat_init(T, dim, dim);
fmprb_mat_init(U, dim, dim);
for (i = 0; i <= m; i++)
{
fmprb_mat_init(pows + i, dim, dim);
if (i == 0)
fmprb_mat_one(pows + i);
else if (i == 1)
fmprb_mat_set(pows + i, A);
else
fmprb_mat_mul(pows + i, pows + i - 1, A, prec);
}
fmprb_mat_zero(S);
fmpz_one(f);
for (i = w - 1; i >= 0; i--)
{
lo = i * m;
hi = FLINT_MIN(N - 1, lo + m - 1);
fmprb_mat_zero(T);
fmpz_one(c);
while (hi >= lo)
{
fmprb_mat_scalar_addmul_fmpz(T, pows + hi - lo, c, prec);
if (hi != 0)
fmpz_mul_ui(c, c, hi);
hi--;
}
fmprb_mat_mul(U, pows + m, S, prec);
fmprb_mat_scalar_mul_fmpz(S, T, f, prec);
fmprb_mat_add(S, S, U, prec);
fmpz_mul(f, f, c);
}
fmprb_mat_scalar_div_fmpz(S, S, f, prec);
fmpz_clear(c);
fmpz_clear(f);
for (i = 0; i <= m; i++)
fmprb_mat_clear(pows + i);
flint_free(pows);
fmprb_mat_clear(T);
fmprb_mat_clear(U);
}
}
void
fmprb_mat_exp(fmprb_mat_t B, const fmprb_mat_t A, long prec)
{
long i, j, dim, wp, N, q, r;
fmpr_t norm, err;
fmprb_mat_t T;
dim = fmprb_mat_nrows(A);
if (dim != fmprb_mat_ncols(A))
{
printf("fmprb_mat_exp: a square matrix is required!\n");
abort();
}
if (dim == 0)
{
return;
}
else if (dim == 1)
{
fmprb_exp(fmprb_mat_entry(B, 0, 0), fmprb_mat_entry(A, 0, 0), prec);
return;
}
wp = prec + 3 * FLINT_BIT_COUNT(prec);
fmpr_init(norm);
fmpr_init(err);
fmprb_mat_init(T, dim, dim);
fmprb_mat_bound_inf_norm(norm, A, FMPRB_RAD_PREC);
if (fmpr_is_zero(norm))
{
fmprb_mat_one(B);
}
else
{
r = fmpr_abs_bound_lt_2exp_si(norm);
q = pow(wp, 0.25); /* wanted magnitude */
if (r > 2 * wp) /* too big */
r = 2 * wp;
else if (r < -q) /* tiny, no need to reduce */
r = 0;
else
r += q; /* reduce to magnitude 2^(-r) */
fmprb_mat_scalar_mul_2exp_si(T, A, -r);
fmpr_mul_2exp_si(norm, norm, -r);
N = _fmprb_mat_exp_choose_N(norm, wp);
_fmprb_mat_exp_bound(err, norm, N);
_fmprb_mat_exp_taylor(B, T, N, wp);
for (i = 0; i < dim; i++)
for (j = 0; j < dim; j++)
fmprb_add_error_fmpr(fmprb_mat_entry(B, i, j), err);
for (i = 0; i < r; i++)
{
fmprb_mat_mul(T, B, B, wp);
fmprb_mat_swap(T, B);
}
for (i = 0; i < dim; i++)
for (j = 0; j < dim; j++)
fmprb_set_round(fmprb_mat_entry(B, i, j),
fmprb_mat_entry(B, i, j), prec);
}
fmpr_clear(norm);
fmpr_clear(err);
fmprb_mat_clear(T);
}

105
fmprb_mat/test/t-exp.c Normal file
View file

@ -0,0 +1,105 @@
/*=============================================================================
This file is part of ARB.
ARB 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.
ARB 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 ARB; if not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
=============================================================================*/
/******************************************************************************
Copyright (C) 2013 Fredrik Johansson
******************************************************************************/
#include "fmprb_mat.h"
int main()
{
long iter;
flint_rand_t state;
printf("exp....");
fflush(stdout);
flint_randinit(state);
/* check exp(A)*exp(c*A) = exp((1+c)*A) */
for (iter = 0; iter < 1000; iter++)
{
fmprb_mat_t A, E, F, EF, G;
fmpq_mat_t Q;
fmprb_t c, d;
long n, qbits, prec;
n = n_randint(state, 5);
qbits = 2 + n_randint(state, 300);
prec = 2 + n_randint(state, 300);
fmprb_init(c);
fmprb_init(d);
fmpq_mat_init(Q, n, n);
fmprb_mat_init(A, n, n);
fmprb_mat_init(E, n, n);
fmprb_mat_init(F, n, n);
fmprb_mat_init(EF, n, n);
fmprb_mat_init(G, n, n);
fmpq_mat_randtest(Q, state, qbits);
fmprb_mat_set_fmpq_mat(A, Q, prec);
fmprb_mat_exp(E, A, prec);
fmprb_randtest(c, state, prec, 10);
fmprb_mat_scalar_mul_fmprb(F, A, c, prec);
fmprb_mat_exp(F, F, prec);
fmprb_add_ui(d, c, 1, prec);
fmprb_mat_scalar_mul_fmprb(G, A, d, prec);
fmprb_mat_exp(G, G, prec);
fmprb_mat_mul(EF, E, F, prec);
if (!fmprb_mat_overlaps(EF, G))
{
printf("FAIL\n\n");
printf("n = %ld, prec = %ld\n", n, prec);
printf("c = \n"); fmprb_printd(c, 15); printf("\n\n");
printf("A = \n"); fmprb_mat_printd(A, 15); printf("\n\n");
printf("E = \n"); fmprb_mat_printd(E, 15); printf("\n\n");
printf("F = \n"); fmprb_mat_printd(F, 15); printf("\n\n");
printf("E*F = \n"); fmprb_mat_printd(EF, 15); printf("\n\n");
printf("G = \n"); fmprb_mat_printd(G, 15); printf("\n\n");
abort();
}
fmprb_clear(c);
fmprb_clear(d);
fmpq_mat_clear(Q);
fmprb_mat_clear(A);
fmprb_mat_clear(E);
fmprb_mat_clear(F);
fmprb_mat_clear(EF);
fmprb_mat_clear(G);
}
flint_randclear(state);
flint_cleanup();
printf("PASS\n");
return EXIT_SUCCESS;
}