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approximate dot product and matrix multiplication

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fredrik 2018-09-07 18:04:30 +02:00
parent fb9a514763
commit 7c98883478
24 changed files with 2276 additions and 299 deletions
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2
acb.h
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@ -651,6 +651,8 @@ void acb_dot_precise(acb_t res, const acb_t initial, int subtract,
void acb_dot(acb_t res, const acb_t initial, int subtract,
acb_srcptr x, slong xstep, acb_srcptr y, slong ystep, slong len, slong prec);
void acb_approx_dot(acb_t res, const acb_t initial, int subtract,
acb_srcptr x, slong xstep, acb_srcptr y, slong ystep, slong len, slong prec);
void acb_inv(acb_t z, const acb_t x, slong prec);

772
acb/approx_dot.c Normal file
View file

@ -0,0 +1,772 @@
/*
Copyright (C) 2018 Fredrik Johansson
This file is part of Arb.
Arb is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 2.1 of the License, or
(at your option) any later version. See <http://www.gnu.org/licenses/>.
*/
#include "acb.h"
/* We need uint64_t instead of mp_limb_t on 32-bit systems for
safe summation of 30-bit error bounds. */
#include <stdint.h>
/* The following macros are found in FLINT's longlong.h, but
the release version is out of date. */
/* x86 : 64 bit */
#if (GMP_LIMB_BITS == 64 && defined (__amd64__))
#define add_sssaaaaaa2(sh, sm, sl, ah, am, al, bh, bm, bl) \
__asm__ ("addq %8,%q2\n\tadcq %6,%q1\n\tadcq %4,%q0" \
: "=r" (sh), "=&r" (sm), "=&r" (sl) \
: "0" ((mp_limb_t)(ah)), "rme" ((mp_limb_t)(bh)), \
"1" ((mp_limb_t)(am)), "rme" ((mp_limb_t)(bm)), \
"2" ((mp_limb_t)(al)), "rme" ((mp_limb_t)(bl))) \
#define sub_dddmmmsss2(dh, dm, dl, mh, mm, ml, sh, sm, sl) \
__asm__ ("subq %8,%q2\n\tsbbq %6,%q1\n\tsbbq %4,%q0" \
: "=r" (dh), "=&r" (dm), "=&r" (dl) \
: "0" ((mp_limb_t)(mh)), "rme" ((mp_limb_t)(sh)), \
"1" ((mp_limb_t)(mm)), "rme" ((mp_limb_t)(sm)), \
"2" ((mp_limb_t)(ml)), "rme" ((mp_limb_t)(sl))) \
#endif /* x86_64 */
/* x86 : 32 bit */
#if (GMP_LIMB_BITS == 32 && (defined (__i386__) \
|| defined (__i486__) || defined(__amd64__)))
#define add_sssaaaaaa2(sh, sm, sl, ah, am, al, bh, bm, bl) \
__asm__ ("addl %8,%k2\n\tadcl %6,%k1\n\tadcl %4,%k0" \
: "=r" (sh), "=r" (sm), "=&r" (sl) \
: "0" ((mp_limb_t)(ah)), "g" ((mp_limb_t)(bh)), \
"1" ((mp_limb_t)(am)), "g" ((mp_limb_t)(bm)), \
"2" ((mp_limb_t)(al)), "g" ((mp_limb_t)(bl))) \
#define sub_dddmmmsss2(dh, dm, dl, mh, mm, ml, sh, sm, sl) \
__asm__ ("subl %8,%k2\n\tsbbl %6,%k1\n\tsbbl %4,%k0" \
: "=r" (dh), "=r" (dm), "=&r" (dl) \
: "0" ((mp_limb_t)(mh)), "g" ((mp_limb_t)(sh)), \
"1" ((mp_limb_t)(mm)), "g" ((mp_limb_t)(sm)), \
"2" ((mp_limb_t)(ml)), "g" ((mp_limb_t)(sl))) \
#endif /* x86 */
#if !defined(add_sssaaaaaa2)
#define add_sssaaaaaa2(sh, sm, sl, ah, am, al, bh, bm, bl) \
do { \
mp_limb_t __t, __u; \
add_ssaaaa(__t, sl, (mp_limb_t) 0, al, (mp_limb_t) 0, bl); \
add_ssaaaa(__u, sm, (mp_limb_t) 0, am, (mp_limb_t) 0, bm); \
add_ssaaaa(sh, sm, ah + bh, sm, __u, __t); \
} while (0)
#define sub_dddmmmsss2(dh, dm, dl, mh, mm, ml, sh, sm, sl) \
do { \
mp_limb_t __t, __u; \
sub_ddmmss(__t, dl, (mp_limb_t) 0, ml, (mp_limb_t) 0, sl); \
sub_ddmmss(__u, dm, (mp_limb_t) 0, mm, (mp_limb_t) 0, sm); \
sub_ddmmss(dh, dm, mh - sh, dm, __u, __t); \
} while (0)
#endif
void
_arb_dot_addmul_generic(mp_ptr sum, mp_ptr serr, mp_ptr tmp, mp_size_t sn,
mp_srcptr xptr, mp_size_t xn, mp_srcptr yptr, mp_size_t yn,
int negative, mp_bitcnt_t shift);
void
_arb_dot_add_generic(mp_ptr sum, mp_ptr serr, mp_ptr tmp, mp_size_t sn,
mp_srcptr xptr, mp_size_t xn,
int negative, mp_bitcnt_t shift);
static void
_arb_dot_output(arb_t res, mp_ptr sum, mp_size_t sn, int negative,
slong sum_exp, slong prec)
{
slong exp_fix;
if (sum[sn - 1] >= LIMB_TOP)
{
mpn_neg(sum, sum, sn);
negative ^= 1;
}
exp_fix = 0;
if (sum[sn - 1] == 0)
{
slong sum_exp2;
mp_size_t sn2;
sn2 = sn;
sum_exp2 = sum_exp;
while (sn2 > 0 && sum[sn2 - 1] == 0)
{
sum_exp2 -= FLINT_BITS;
sn2--;
}
if (sn2 == 0)
{
arf_zero(arb_midref(res));
}
else
{
_arf_set_round_mpn(arb_midref(res), &exp_fix, sum, sn2, negative, prec, ARF_RND_DOWN);
_fmpz_set_si_small(ARF_EXPREF(arb_midref(res)), exp_fix + sum_exp2);
}
}
else
{
if (sn == 2) /* unnecessary? */
_arf_set_round_uiui(arb_midref(res), &exp_fix, sum[1], sum[0], negative, prec, ARF_RND_DOWN);
else
_arf_set_round_mpn(arb_midref(res), &exp_fix, sum, sn, negative, prec, ARF_RND_DOWN);
_fmpz_set_si_small(ARF_EXPREF(arb_midref(res)), exp_fix + sum_exp);
}
}
/* xxx: don't use surrounding variables */
#define ARB_DOT_ADD(s_sum, s_serr, s_sn, s_sum_exp, s_subtract, xm) \
if (!arf_is_special(xm)) \
{ \
mp_srcptr xptr; \
xexp = ARF_EXP(xm); \
xn = ARF_SIZE(xm); \
xnegative = ARF_SGNBIT(xm); \
shift = s_sum_exp - xexp; \
if (shift >= s_sn * FLINT_BITS) \
{ \
} \
else \
{ \
xptr = (xn <= ARF_NOPTR_LIMBS) ? ARF_NOPTR_D(xm) : ARF_PTR_D(xm); \
_arb_dot_add_generic(s_sum, &s_serr, tmp, s_sn, xptr, xn, xnegative ^ s_subtract, shift); \
} \
} \
static void
_arf_complex_mul_gauss(arf_t e, arf_t f, const arf_t a, const arf_t b,
const arf_t c, const arf_t d)
{
mp_srcptr ap, bp, cp, dp;
int asgn, bsgn, csgn, dsgn;
mp_size_t an, bn, cn, dn;
slong aexp, bexp, cexp, dexp;
fmpz texp, uexp;
fmpz_t za, zb, zc, zd, t, u, v;
slong abot, bbot, cbot, dbot;
ARF_GET_MPN_READONLY(ap, an, a);
asgn = ARF_SGNBIT(a);
aexp = ARF_EXP(a);
ARF_GET_MPN_READONLY(bp, bn, b);
bsgn = ARF_SGNBIT(b);
bexp = ARF_EXP(b);
ARF_GET_MPN_READONLY(cp, cn, c);
csgn = ARF_SGNBIT(c);
cexp = ARF_EXP(c);
ARF_GET_MPN_READONLY(dp, dn, d);
dsgn = ARF_SGNBIT(d);
dexp = ARF_EXP(d);
/* Gauss multiplication
e = ac - bd
f = (a+b)(c+d) - ac - bd */
abot = aexp - an * FLINT_BITS;
bbot = bexp - bn * FLINT_BITS;
cbot = cexp - cn * FLINT_BITS;
dbot = dexp - dn * FLINT_BITS;
texp = FLINT_MIN(abot, bbot);
uexp = FLINT_MIN(cbot, dbot);
fmpz_init(za);
fmpz_init(zb);
fmpz_init(zc);
fmpz_init(zd);
fmpz_init(t);
fmpz_init(u);
fmpz_init(v);
fmpz_lshift_mpn(za, ap, an, asgn, abot - texp);
fmpz_lshift_mpn(zb, bp, bn, bsgn, bbot - texp);
fmpz_lshift_mpn(zc, cp, cn, csgn, cbot - uexp);
fmpz_lshift_mpn(zd, dp, dn, dsgn, dbot - uexp);
fmpz_add(t, za, zb);
fmpz_add(v, zc, zd);
fmpz_mul(u, t, v);
fmpz_mul(t, za, zc);
fmpz_mul(v, zb, zd);
fmpz_sub(u, u, t);
fmpz_sub(u, u, v);
fmpz_sub(t, t, v);
texp += uexp;
arf_set_fmpz_2exp(e, t, &texp);
arf_set_fmpz_2exp(f, u, &texp);
fmpz_clear(za);
fmpz_clear(zb);
fmpz_clear(zc);
fmpz_clear(zd);
fmpz_clear(t);
fmpz_clear(u);
fmpz_clear(v);
}
ARB_DLL extern slong acb_dot_gauss_dot_cutoff;
#define GAUSS_CUTOFF acb_dot_gauss_dot_cutoff
void
acb_approx_dot_simple(acb_t res, const acb_t initial, int subtract,
acb_srcptr x, slong xstep, acb_srcptr y, slong ystep, slong len, slong prec)
{
slong i;
if (len <= 0)
{
if (initial == NULL)
{
arf_zero(arb_midref(acb_realref(res)));
arf_zero(arb_midref(acb_imagref(res)));
}
else
{
arf_set_round(arb_midref(acb_realref(res)), arb_midref(acb_realref(initial)), prec, ARB_RND);
arf_set_round(arb_midref(acb_imagref(res)), arb_midref(acb_imagref(initial)), prec, ARB_RND);
}
return;
}
if (initial == NULL && len == 1)
{
arf_complex_mul(arb_midref(acb_realref(res)),
arb_midref(acb_imagref(res)),
arb_midref(acb_realref(x)),
arb_midref(acb_imagref(x)),
arb_midref(acb_realref(y)),
arb_midref(acb_imagref(y)), prec, ARB_RND);
}
else
{
arf_t e, f;
arf_init(e);
arf_init(f);
if (initial != NULL)
{
if (subtract)
{
arf_neg(arb_midref(acb_realref(res)), arb_midref(acb_realref(initial)));
arf_neg(arb_midref(acb_imagref(res)), arb_midref(acb_imagref(initial)));
}
else
{
arf_set(arb_midref(acb_realref(res)), arb_midref(acb_realref(initial)));
arf_set(arb_midref(acb_imagref(res)), arb_midref(acb_imagref(initial)));
}
}
for (i = 0; i < len; i++)
{
arf_complex_mul(e, f,
arb_midref(acb_realref(x + i * xstep)),
arb_midref(acb_imagref(x + i * xstep)),
arb_midref(acb_realref(y + i * ystep)),
arb_midref(acb_imagref(y + i * ystep)), prec, ARB_RND);
if (i == 0 && initial == NULL)
{
arf_set(arb_midref(acb_realref(res)), e);
arf_set(arb_midref(acb_imagref(res)), f);
}
else
{
arf_add(arb_midref(acb_realref(res)), arb_midref(acb_realref(res)), e, prec, ARB_RND);
arf_add(arb_midref(acb_imagref(res)), arb_midref(acb_imagref(res)), f, prec, ARB_RND);
}
}
arf_clear(e);
arf_clear(f);
}
if (subtract)
{
arf_neg(arb_midref(acb_realref(res)), arb_midref(acb_realref(res)));
arf_neg(arb_midref(acb_imagref(res)), arb_midref(acb_imagref(res)));
}
}
void
acb_approx_dot(acb_t res, const acb_t initial, int subtract, acb_srcptr x, slong xstep, acb_srcptr y, slong ystep, slong len, slong prec)
{
slong i, j, padding, extend;
slong xexp, yexp, exp;
slong re_nonzero, im_nonzero;
slong re_max_exp, re_min_exp, re_sum_exp;
slong im_max_exp, im_min_exp, im_sum_exp;
slong re_prec, im_prec;
int xnegative, ynegative;
mp_size_t xn, yn, re_sn, im_sn, alloc;
mp_bitcnt_t shift;
arb_srcptr xi, yi;
arf_srcptr xm, ym;
mp_limb_t re_serr, im_serr; /* Sum over arithmetic errors */
mp_ptr tmp, re_sum, im_sum; /* Workspace */
slong xoff, yoff;
char * use_gauss;
ARF_ADD_TMP_DECL;
/* todo: fast fma and fmma (len=2) code */
if (len <= 1)
{
acb_approx_dot_simple(res, initial, subtract, x, xstep, y, ystep, len, prec);
return;
}
/* Number of nonzero midpoint terms in sum. */
re_nonzero = 0;
im_nonzero = 0;
/* Terms are bounded by 2^max_exp (with WORD_MIN = -infty) */
re_max_exp = WORD_MIN;
im_max_exp = WORD_MIN;
/* Used to reduce the precision. */
re_min_exp = WORD_MAX;
im_min_exp = WORD_MAX;
/* Account for the initial term. */
if (initial != NULL)
{
if (!ARF_IS_LAGOM(arb_midref(acb_realref(initial))) || !ARF_IS_LAGOM(arb_midref(acb_imagref(initial))))
{
acb_approx_dot_simple(res, initial, subtract, x, xstep, y, ystep, len, prec);
return;
}
xm = arb_midref(acb_realref(initial));
if (!arf_is_special(xm))
{
re_max_exp = ARF_EXP(xm);
re_nonzero++;
if (prec > 2 * FLINT_BITS)
re_min_exp = ARF_EXP(xm) - ARF_SIZE(xm) * FLINT_BITS;
}
xm = arb_midref(acb_imagref(initial));
if (!arf_is_special(xm))
{
im_max_exp = ARF_EXP(xm);
im_nonzero++;
if (prec > 2 * FLINT_BITS)
im_min_exp = ARF_EXP(xm) - ARF_SIZE(xm) * FLINT_BITS;
}
}
for (xoff = 0; xoff < 2; xoff++)
{
for (yoff = 0; yoff < 2; yoff++)
{
slong nonzero, max_exp, min_exp;
if (xoff == yoff)
{
nonzero = re_nonzero;
max_exp = re_max_exp;
min_exp = re_min_exp;
}
else
{
nonzero = im_nonzero;
max_exp = im_max_exp;
min_exp = im_min_exp;
}
/* Determine maximum exponents for the main sum and the radius sum. */
for (i = 0; i < len; i++)
{
xi = ((arb_srcptr) x) + 2 * i * xstep + xoff;
yi = ((arb_srcptr) y) + 2 * i * ystep + yoff;
/* Fallback for huge exponents or non-finite values. */
if (!ARF_IS_LAGOM(arb_midref(xi)) || !ARF_IS_LAGOM(arb_midref(yi)))
{
acb_approx_dot_simple(res, initial, subtract, x, xstep, y, ystep, len, prec);
return;
}
xm = arb_midref(xi);
ym = arb_midref(yi);
/* (xm+xr)(ym+yr) = xm ym + [xr ym + xm yr + xr yr] */
if (!arf_is_special(xm))
{
xexp = ARF_EXP(xm);
if (!arf_is_special(ym))
{
yexp = ARF_EXP(ym);
max_exp = FLINT_MAX(max_exp, xexp + yexp);
nonzero++;
if (prec > 2 * FLINT_BITS)
{
slong bot;
bot = (xexp + yexp) - (ARF_SIZE(xm) + ARF_SIZE(ym)) * FLINT_BITS;
min_exp = FLINT_MIN(min_exp, bot);
}
}
}
}
if (xoff == yoff)
{
re_nonzero = nonzero;
re_max_exp = max_exp;
re_min_exp = min_exp;
}
else
{
im_nonzero = nonzero;
im_max_exp = max_exp;
im_min_exp = min_exp;
}
}
}
re_prec = prec;
im_prec = prec;
if (re_max_exp == WORD_MIN && im_max_exp == WORD_MIN)
{
arf_zero(arb_midref(acb_realref(res)));
arf_zero(arb_midref(acb_imagref(res)));
return;
}
/* The midpoint sum is zero. */
if (re_max_exp == WORD_MIN)
{
re_prec = 2;
}
else
{
if (re_min_exp != WORD_MAX)
re_prec = FLINT_MIN(re_prec, re_max_exp - re_min_exp + MAG_BITS);
re_prec = FLINT_MAX(re_prec, 2);
}
if (im_max_exp == WORD_MIN)
{
im_prec = 2;
}
else
{
if (re_min_exp != WORD_MAX)
im_prec = FLINT_MIN(im_prec, im_max_exp - im_min_exp + MAG_BITS);
im_prec = FLINT_MAX(im_prec, 2);
}
extend = FLINT_BIT_COUNT(re_nonzero) + 1;
padding = 4 + FLINT_BIT_COUNT(len);
re_sn = (re_prec + extend + padding + FLINT_BITS - 1) / FLINT_BITS;
re_sn = FLINT_MAX(re_sn, 2);
re_sum_exp = re_max_exp + extend;
extend = FLINT_BIT_COUNT(im_nonzero) + 1;
padding = 4 + FLINT_BIT_COUNT(len);
im_sn = (im_prec + extend + padding + FLINT_BITS - 1) / FLINT_BITS;
im_sn = FLINT_MAX(im_sn, 2);
im_sum_exp = im_max_exp + extend;
/* We need sn + 1 limb for the sum (sn limbs + 1 dummy limb
for carry or borrow that avoids an extra branch). We need
2 * (sn + 2) limbs to store the product of two numbers
with up to (sn + 2) limbs, plus 1 extra limb for shifting
the product. */
alloc = (re_sn + 1) + (im_sn + 1) + 2 * (FLINT_MAX(re_sn, im_sn) + 2) + 1;
ARF_ADD_TMP_ALLOC(re_sum, alloc)
im_sum = re_sum + (re_sn + 1);
tmp = im_sum + (im_sn + 1);
/* Set sum to 0 */
re_serr = 0;
for (j = 0; j < re_sn + 1; j++)
re_sum[j] = 0;
im_serr = 0;
for (j = 0; j < im_sn + 1; j++)
im_sum[j] = 0;
if (initial != NULL)
{
xm = arb_midref(acb_realref(initial));
ARB_DOT_ADD(re_sum, re_serr, re_sn, re_sum_exp, subtract, xm);
xm = arb_midref(acb_imagref(initial));
ARB_DOT_ADD(im_sum, im_serr, im_sn, im_sum_exp, subtract, xm);
}
use_gauss = NULL;
if (re_prec >= GAUSS_CUTOFF * FLINT_BITS &&
im_prec >= GAUSS_CUTOFF * FLINT_BITS)
{
arf_t e, f;
for (i = 0; i < len; i++)
{
arb_srcptr ai, bi, ci, di;
mp_size_t an, bn, cn, dn;
slong aexp, bexp, cexp, dexp;
ai = ((arb_srcptr) x) + 2 * i * xstep;
bi = ((arb_srcptr) x) + 2 * i * xstep + 1;
ci = ((arb_srcptr) y) + 2 * i * ystep;
di = ((arb_srcptr) y) + 2 * i * ystep + 1;
an = ARF_SIZE(arb_midref(ai));
bn = ARF_SIZE(arb_midref(bi));
cn = ARF_SIZE(arb_midref(ci));
dn = ARF_SIZE(arb_midref(di));
aexp = ARF_EXP(arb_midref(ai));
bexp = ARF_EXP(arb_midref(bi));
cexp = ARF_EXP(arb_midref(ci));
dexp = ARF_EXP(arb_midref(di));
if (an >= GAUSS_CUTOFF && bn >= GAUSS_CUTOFF &&
bn >= GAUSS_CUTOFF && cn >= GAUSS_CUTOFF &&
FLINT_ABS(an - bn) <= 2 &&
FLINT_ABS(cn - dn) <= 2 &&
FLINT_ABS(aexp - bexp) <= 64 &&
FLINT_ABS(cexp - dexp) <= 64 &&
re_sum_exp - (aexp + cexp) < 0.1 * re_prec &&
im_sum_exp - (aexp + dexp) < 0.1 * im_prec &&
an + cn < 2.2 * re_sn && an + dn < 2.2 * im_sn)
{
if (use_gauss == NULL)
{
use_gauss = flint_calloc(len, sizeof(char));
arf_init(e);
arf_init(f);
}
use_gauss[i] = 1;
_arf_complex_mul_gauss(e, f, arb_midref(ai), arb_midref(bi), arb_midref(ci), arb_midref(di));
ARB_DOT_ADD(re_sum, re_serr, re_sn, re_sum_exp, 0, e);
ARB_DOT_ADD(im_sum, im_serr, im_sn, im_sum_exp, 0, f);
}
}
if (use_gauss != NULL)
{
arf_clear(e);
arf_clear(f);
}
}
for (xoff = 0; xoff < 2; xoff++)
{
for (yoff = 0; yoff < 2; yoff++)
{
slong sum_exp;
mp_ptr sum;
mp_size_t sn;
mp_limb_t serr;
int flipsign;
if (xoff == yoff)
{
sum_exp = re_sum_exp;
sum = re_sum;
sn = re_sn;
if (re_max_exp == WORD_MIN)
continue;
}
else
{
sum_exp = im_sum_exp;
sum = im_sum;
sn = im_sn;
if (im_max_exp == WORD_MIN)
continue;
}
serr = 0;
flipsign = (xoff + yoff == 2);
for (i = 0; i < len; i++)
{
xi = ((arb_srcptr) x) + 2 * i * xstep + xoff;
yi = ((arb_srcptr) y) + 2 * i * ystep + yoff;
xm = arb_midref(xi);
ym = arb_midref(yi);
/* The midpoints of x[i] and y[i] are both nonzero. */
if (!arf_is_special(xm) && !arf_is_special(ym))
{
xexp = ARF_EXP(xm);
xn = ARF_SIZE(xm);
xnegative = ARF_SGNBIT(xm);
yexp = ARF_EXP(ym);
yn = ARF_SIZE(ym);
ynegative = ARF_SGNBIT(ym);
exp = xexp + yexp;
shift = sum_exp - exp;
if (shift >= sn * FLINT_BITS)
{
}
else if (xn <= 2 && yn <= 2 && sn <= 3)
{
mp_limb_t x1, x0, y1, y0;
mp_limb_t u3, u2, u1, u0;
if (xn == 1 && yn == 1)
{
x0 = ARF_NOPTR_D(xm)[0];
y0 = ARF_NOPTR_D(ym)[0];
umul_ppmm(u3, u2, x0, y0);
u1 = u0 = 0;
}
else if (xn == 2 && yn == 2)
{
x0 = ARF_NOPTR_D(xm)[0];
x1 = ARF_NOPTR_D(xm)[1];
y0 = ARF_NOPTR_D(ym)[0];
y1 = ARF_NOPTR_D(ym)[1];
nn_mul_2x2(u3, u2, u1, u0, x1, x0, y1, y0);
}
else if (xn == 1)
{
x0 = ARF_NOPTR_D(xm)[0];
y0 = ARF_NOPTR_D(ym)[0];
y1 = ARF_NOPTR_D(ym)[1];
nn_mul_2x1(u3, u2, u1, y1, y0, x0);
u0 = 0;
}
else
{
x0 = ARF_NOPTR_D(xm)[0];
x1 = ARF_NOPTR_D(xm)[1];
y0 = ARF_NOPTR_D(ym)[0];
nn_mul_2x1(u3, u2, u1, x1, x0, y0);
u0 = 0;
}
if (sn == 2)
{
if (shift < FLINT_BITS)
{
u2 = (u2 >> shift) | (u3 << (FLINT_BITS - shift));
u3 = (u3 >> shift);
}
else if (shift == FLINT_BITS)
{
u2 = u3;
u3 = 0;
}
else /* FLINT_BITS < shift < 2 * FLINT_BITS */
{
u2 = (u3 >> (shift - FLINT_BITS));
u3 = 0;
}
if (xnegative ^ ynegative ^ flipsign)
sub_ddmmss(sum[1], sum[0], sum[1], sum[0], u3, u2);
else
add_ssaaaa(sum[1], sum[0], sum[1], sum[0], u3, u2);
}
else if (sn == 3)
{
if (shift < FLINT_BITS)
{
u1 = (u1 >> shift) | (u2 << (FLINT_BITS - shift));
u2 = (u2 >> shift) | (u3 << (FLINT_BITS - shift));
u3 = (u3 >> shift);
}
else if (shift == FLINT_BITS)
{
u1 = u2;
u2 = u3;
u3 = 0;
}
else if (shift < 2 * FLINT_BITS)
{
u1 = (u3 << (2 * FLINT_BITS - shift)) | (u2 >> (shift - FLINT_BITS));
u2 = (u3 >> (shift - FLINT_BITS));
u3 = 0;
}
else if (shift == 2 * FLINT_BITS)
{
u1 = u3;
u2 = 0;
u3 = 0;
}
else /* 2 * FLINT_BITS < shift < 3 * FLINT_BITS */
{
u1 = (u3 >> (shift - 2 * FLINT_BITS));
u2 = 0;
u3 = 0;
}
if (xnegative ^ ynegative ^ flipsign)
sub_dddmmmsss2(sum[2], sum[1], sum[0], sum[2], sum[1], sum[0], u3, u2, u1);
else
add_sssaaaaaa2(sum[2], sum[1], sum[0], sum[2], sum[1], sum[0], u3, u2, u1);
}
}
else
{
mp_srcptr xptr, yptr;
xptr = (xn <= ARF_NOPTR_LIMBS) ? ARF_NOPTR_D(xm) : ARF_PTR_D(xm);
yptr = (yn <= ARF_NOPTR_LIMBS) ? ARF_NOPTR_D(ym) : ARF_PTR_D(ym);
if (use_gauss == NULL || use_gauss[i] == 0)
_arb_dot_addmul_generic(sum, &serr, tmp, sn, xptr, xn, yptr, yn, xnegative ^ ynegative ^ flipsign, shift);
}
}
}
}
}
_arb_dot_output(acb_realref(res), re_sum, re_sn, subtract, re_sum_exp, re_prec);
_arb_dot_output(acb_imagref(res), im_sum, im_sn, subtract, im_sum_exp, im_prec);
ARF_ADD_TMP_FREE(re_sum, alloc);
if (use_gauss != NULL)
flint_free(use_gauss);
}

261
acb/test/t-approx_dot.c Normal file
View file

@ -0,0 +1,261 @@
/*
Copyright (C) 2018 Fredrik Johansson
This file is part of Arb.
Arb is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 2.1 of the License, or
(at your option) any later version. See <http://www.gnu.org/licenses/>.
*/
#include "acb.h"
ARB_DLL extern slong acb_dot_gauss_dot_cutoff;
int main()
{
slong iter;
flint_rand_t state;
flint_printf("approx_dot....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 100000 * arb_test_multiplier(); iter++)
{
acb_ptr x, y;
acb_t s1, s2, z;
slong i, len, prec, xbits, ybits, ebits;
int initial, subtract, revx, revy;
if (n_randint(state, 100) == 0)
len = n_randint(state, 100);
else if (n_randint(state, 10) == 0)
len = n_randint(state, 10);
else
len = n_randint(state, 3);
acb_dot_gauss_dot_cutoff = 3 + n_randint(state, 30);
if (n_randint(state, 10) != 0 || len > 10)
{
prec = 2 + n_randint(state, 500);
xbits = 2 + n_randint(state, 500);
ybits = 2 + n_randint(state, 500);
}
else
{
prec = 2 + n_randint(state, 4000);
xbits = 2 + n_randint(state, 4000);
ybits = 2 + n_randint(state, 4000);
}
if (n_randint(state, 100) == 0)
ebits = 2 + n_randint(state, 100);
else
ebits = 2 + n_randint(state, 10);
initial = n_randint(state, 2);
subtract = n_randint(state, 2);
revx = n_randint(state, 2);
revy = n_randint(state, 2);
x = _acb_vec_init(len);
y = _acb_vec_init(len);
acb_init(s1);
acb_init(s2);
acb_init(z);
switch (n_randint(state, 3))
{
case 0:
for (i = 0; i < len; i++)
{
acb_randtest(x + i, state, xbits, ebits);
acb_randtest(y + i, state, ybits, ebits);
}
break;
/* Test with cancellation */
case 1:
for (i = 0; i < len; i++)
{
if (i <= len / 2)
{
acb_randtest(x + i, state, xbits, ebits);
acb_randtest(y + i, state, ybits, ebits);
}
else
{
acb_neg(x + i, x + len - i - 1);
acb_set(y + i, y + len - i - 1);
}
}
break;
default:
for (i = 0; i < len; i++)
{
if (i <= len / 2)
{
acb_randtest(x + i, state, xbits, ebits);
acb_randtest(y + i, state, ybits, ebits);
}
else
{
acb_neg_round(x + i, x + len - i - 1, 2 + n_randint(state, 500));
acb_set_round(y + i, y + len - i - 1, 2 + n_randint(state, 500));
}
}
break;
}
acb_randtest(s1, state, 200, 100);
acb_randtest(s2, state, 200, 100);
acb_randtest(z, state, xbits, ebits);
acb_approx_dot(s1, initial ? z : NULL, subtract,
revx ? (x + len - 1) : x, revx ? -1 : 1,
revy ? (y + len - 1) : y, revy ? -1 : 1,
len, prec);
mag_zero(arb_radref(acb_realref(s1)));
mag_zero(arb_radref(acb_imagref(s1)));
/* With the fast algorithm, we expect identical results when
reversing the vectors. */
if (ebits <= 12)
{
acb_approx_dot(s2, initial ? z : NULL, subtract,
!revx ? (x + len - 1) : x, !revx ? -1 : 1,
!revy ? (y + len - 1) : y, !revy ? -1 : 1,
len, prec);
mag_zero(arb_radref(acb_realref(s2)));
mag_zero(arb_radref(acb_imagref(s2)));
if (!acb_equal(s1, s2))
{
flint_printf("FAIL (reversal)\n\n");
flint_printf("iter = %wd, len = %wd, prec = %wd, ebits = %wd\n\n", iter, len, prec, ebits);
if (initial)
{
flint_printf("z = ", i); acb_printn(z, 100, ARB_STR_MORE); flint_printf(" (%wd)\n\n", acb_bits(z));
}
for (i = 0; i < len; i++)
{
flint_printf("x[%wd] = ", i); acb_printn(x + i, 100, ARB_STR_MORE); flint_printf(" (%wd)\n", acb_bits(x + i));
flint_printf("y[%wd] = ", i); acb_printn(y + i, 100, ARB_STR_MORE); flint_printf(" (%wd)\n", acb_bits(y + i));
}
flint_printf("\n\n");
flint_printf("s1 = "); acb_printn(s1, 100, ARB_STR_MORE); flint_printf("\n\n");
flint_printf("s2 = "); acb_printn(s2, 100, ARB_STR_MORE); flint_printf("\n\n");
flint_abort();
}
}
/* Verify that radii are ignored */
for (i = 0; i < len; i++)
{
arb_get_mid_arb(acb_realref(x + i), acb_realref(x + i));
arb_get_mid_arb(acb_imagref(x + i), acb_imagref(x + i));
arb_get_mid_arb(acb_realref(y + i), acb_realref(y + i));
arb_get_mid_arb(acb_imagref(y + i), acb_imagref(y + i));
}
arb_get_mid_arb(acb_realref(z), acb_realref(z));
arb_get_mid_arb(acb_imagref(z), acb_imagref(z));
acb_approx_dot(s2, initial ? z : NULL, subtract,
revx ? (x + len - 1) : x, revx ? -1 : 1,
revy ? (y + len - 1) : y, revy ? -1 : 1,
len, prec);
mag_zero(arb_radref(acb_realref(s2)));
mag_zero(arb_radref(acb_imagref(s2)));
if (!acb_equal(s1, s2))
{
flint_printf("FAIL (radii)\n\n");
flint_printf("iter = %wd, len = %wd, prec = %wd, ebits = %wd\n\n", iter, len, prec, ebits);
if (initial)
{
flint_printf("z = ", i); acb_printn(z, 100, ARB_STR_MORE); flint_printf(" (%wd)\n\n", acb_bits(z));
}
for (i = 0; i < len; i++)
{
flint_printf("x[%wd] = ", i); acb_printn(x + i, 100, ARB_STR_MORE); flint_printf(" (%wd)\n", acb_bits(x + i));
flint_printf("y[%wd] = ", i); acb_printn(y + i, 100, ARB_STR_MORE); flint_printf(" (%wd)\n", acb_bits(y + i));
}
flint_printf("\n\n");
flint_printf("s1 = "); acb_printn(s1, 100, ARB_STR_MORE); flint_printf("\n\n");
flint_printf("s2 = "); acb_printn(s2, 100, ARB_STR_MORE); flint_printf("\n\n");
flint_abort();
}
/* Compare with acb_dot */
acb_approx_dot(s2, initial ? z : NULL, subtract,
revx ? (x + len - 1) : x, revx ? -1 : 1,
revy ? (y + len - 1) : y, revy ? -1 : 1,
len, prec);
{
mag_t err, xx, yy;
mag_init(err);
mag_init(xx);
mag_init(yy);
if (initial)
acb_get_mag(err, z);
for (i = 0; i < len; i++)
{
acb_get_mag(xx, revx ? x + len - 1 - i : x + i);
acb_get_mag(yy, revx ? y + len - 1 - i : y + i);
mag_addmul(err, xx, yy);
}
mag_mul_2exp_si(err, err, -prec + 2);
acb_add_error_mag(s2, err);
if (!acb_contains(s2, s1))
{
flint_printf("FAIL (inclusion)\n\n");
flint_printf("iter = %wd, len = %wd, prec = %wd, ebits = %wd\n\n", iter, len, prec, ebits);
if (initial)
{
flint_printf("z = ", i); acb_printn(z, 100, ARB_STR_MORE); flint_printf(" (%wd)\n\n", acb_bits(z));
}
for (i = 0; i < len; i++)
{
flint_printf("x[%wd] = ", i); acb_printn(x + i, 100, ARB_STR_MORE); flint_printf(" (%wd)\n", acb_bits(x + i));
flint_printf("y[%wd] = ", i); acb_printn(y + i, 100, ARB_STR_MORE); flint_printf(" (%wd)\n", acb_bits(y + i));
}
flint_printf("\n\n");
flint_printf("s1 = "); acb_printn(s1, 100, ARB_STR_MORE); flint_printf("\n\n");
flint_printf("s2 = "); acb_printn(s2, 100, ARB_STR_MORE); flint_printf("\n\n");
flint_abort();
}
mag_clear(err);
mag_clear(xx);
mag_clear(yy);
}
acb_clear(s1);
acb_clear(s2);
acb_clear(z);
_acb_vec_clear(x, len);
_acb_vec_clear(y, len);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}

1
acb_mat.h
View file

@ -389,6 +389,7 @@ int acb_mat_solve(acb_mat_t X, const acb_mat_t A, const acb_mat_t B, slong prec)
int acb_mat_solve_precond(acb_mat_t X, const acb_mat_t A, const acb_mat_t B, slong prec);
void acb_mat_approx_mul(acb_mat_t C, const acb_mat_t A, const acb_mat_t B, slong prec);
void acb_mat_approx_solve_triu(acb_mat_t X, const acb_mat_t U, const acb_mat_t B, int unit, slong prec);
void acb_mat_approx_solve_tril(acb_mat_t X, const acb_mat_t L, const acb_mat_t B, int unit, slong prec);
int acb_mat_approx_lu(slong * P, acb_mat_t LU, const acb_mat_t A, slong prec);

4
acb_mat/approx_lu.c
View file

@ -194,8 +194,7 @@ acb_mat_approx_lu_recursive(slong * P, acb_mat_t LU, const acb_mat_t A, slong pr
/* acb_mat_submul(A11, A11, A10, A01, prec); */
acb_mat_t T;
acb_mat_init(T, A10->r, A01->c);
acb_mat_mul(T, A10, A01, prec);
acb_mat_get_mid(T, T);
acb_mat_approx_mul(T, A10, A01, prec);
acb_mat_sub(A11, A11, T, prec);
acb_mat_get_mid(A11, A11);
acb_mat_clear(T);
@ -227,4 +226,3 @@ acb_mat_approx_lu(slong * P, acb_mat_t LU, const acb_mat_t A, slong prec)
else
return acb_mat_approx_lu_recursive(P, LU, A, prec);
}

157
acb_mat/approx_mul.c Normal file
View file

@ -0,0 +1,157 @@
/*
Copyright (C) 2018 Fredrik Johansson
This file is part of Arb.
Arb is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 2.1 of the License, or
(at your option) any later version. See <http://www.gnu.org/licenses/>.
*/
#include "acb_mat.h"
void
acb_mat_approx_mul_classical(acb_mat_t C, const acb_mat_t A, const acb_mat_t B, slong prec)
{
slong ar, br, bc, i, j;
ar = acb_mat_nrows(A);
br = acb_mat_nrows(B);
bc = acb_mat_ncols(B);
if (br == 0)
{
for (i = 0; i < ar; i++)
{
for (j = 0; j < bc; j++)
{
arf_zero(arb_midref(acb_realref(acb_mat_entry(C, i, j))));
arf_zero(arb_midref(acb_imagref(acb_mat_entry(C, i, j))));
}
}
return;
}
if (A == C || B == C)
{
acb_mat_t T;
acb_mat_init(T, ar, bc);
acb_mat_approx_mul_classical(T, A, B, prec);
acb_mat_swap(T, C);
acb_mat_clear(T);
return;
}
if (br == 1)
{
for (i = 0; i < ar; i++)
{
for (j = 0; j < bc; j++)
{
arf_complex_mul(
arb_midref(acb_realref(acb_mat_entry(C, i, j))),
arb_midref(acb_imagref(acb_mat_entry(C, i, j))),
arb_midref(acb_realref(acb_mat_entry(A, i, 0))),
arb_midref(acb_imagref(acb_mat_entry(A, i, 0))),
arb_midref(acb_realref(acb_mat_entry(B, 0, j))),
arb_midref(acb_imagref(acb_mat_entry(B, 0, j))), prec, ARB_RND);
}
}
}
else
{
acb_ptr tmp;
TMP_INIT;
TMP_START;
tmp = TMP_ALLOC(sizeof(acb_struct) * br * bc);
for (i = 0; i < br; i++)
for (j = 0; j < bc; j++)
tmp[j * br + i] = *acb_mat_entry(B, i, j);
for (i = 0; i < ar; i++)
{
for (j = 0; j < bc; j++)
{
acb_approx_dot(acb_mat_entry(C, i, j), NULL, 0,
A->rows[i], 1, tmp + j * br, 1, br, prec);
}
}
TMP_END;
}
}
int
acb_mat_is_exact(const acb_mat_t A)
{
slong i, j;
for (i = 0; i < acb_mat_nrows(A); i++)
for (j = 0; j < acb_mat_ncols(A); j++)
if (!mag_is_zero(arb_radref(acb_realref(acb_mat_entry(A, i, j)))) ||
!mag_is_zero(arb_radref(acb_imagref(acb_mat_entry(A, i, j)))))
return 0;
return 1;
}
void
acb_mat_approx_mul(acb_mat_t C, const acb_mat_t A, const acb_mat_t B, slong prec)
{
slong cutoff;
/* todo: detect small-integer matrices */
if (prec <= 2 * FLINT_BITS)
cutoff = 120;
else if (prec <= 16 * FLINT_BITS)
cutoff = 60;
else
cutoff = 40;
if (acb_mat_nrows(A) <= cutoff || acb_mat_ncols(A) <= cutoff ||
acb_mat_ncols(B) <= cutoff)
{
acb_mat_approx_mul_classical(C, A, B, prec);
}
else
{
if (acb_mat_is_exact(A) && acb_mat_is_exact(B))
{
acb_mat_mul(C, A, B, prec);
}
else
{
acb_mat_t AM, BM;
if (acb_mat_is_exact(A))
{
acb_mat_init(BM, acb_mat_nrows(B), acb_mat_ncols(B));
acb_mat_get_mid(BM, B);
acb_mat_mul(C, A, BM, prec);
acb_mat_clear(BM);
}
else if (acb_mat_is_exact(A))
{
acb_mat_init(AM, acb_mat_nrows(A), acb_mat_ncols(A));
acb_mat_get_mid(AM, A);
acb_mat_mul(C, AM, B, prec);
acb_mat_clear(AM);
}
else
{
acb_mat_init(BM, acb_mat_nrows(B), acb_mat_ncols(B));
acb_mat_get_mid(BM, B);
acb_mat_init(AM, acb_mat_nrows(A), acb_mat_ncols(A));
acb_mat_get_mid(AM, A);
acb_mat_mul(C, AM, BM, prec);
acb_mat_clear(AM);
acb_mat_clear(BM);
}
}
acb_mat_get_mid(C, C);
}
}

77
acb_mat/approx_solve_lu_precomp.c
View file

@ -11,48 +11,11 @@
#include "acb_mat.h"
static void
_acb_approx_mul(acb_t res, const acb_t x, const acb_t y, slong prec)
{
arf_complex_mul(arb_midref(acb_realref(res)), arb_midref(acb_imagref(res)),
arb_midref(acb_realref(x)), arb_midref(acb_imagref(x)),
arb_midref(acb_realref(y)), arb_midref(acb_imagref(y)), prec, ARB_RND);
}
static void
_acb_approx_submul(acb_t z, const acb_t x, const acb_t y, acb_t t, slong prec)
{
_acb_approx_mul(t, x, y, prec);
arf_sub(arb_midref(acb_realref(z)),
arb_midref(acb_realref(z)),
arb_midref(acb_realref(t)), prec, ARB_RND);
arf_sub(arb_midref(acb_imagref(z)),
arb_midref(acb_imagref(z)),
arb_midref(acb_imagref(t)), prec, ARB_RND);
}
/* note: the tmp variable t should have zero radius */
static void
_acb_approx_div(acb_t z, const acb_t x, const acb_t y, acb_t t, slong prec)
{
arf_set(arb_midref(acb_realref(t)), arb_midref(acb_realref(y)));
arf_set(arb_midref(acb_imagref(t)), arb_midref(acb_imagref(y)));
acb_inv(t, t, prec);
mag_zero(arb_radref(acb_realref(t)));
mag_zero(arb_radref(acb_imagref(t)));
_acb_approx_mul(z, x, t, prec);
}
void
acb_mat_approx_solve_lu_precomp(acb_mat_t X, const slong * perm,
const acb_mat_t A, const acb_mat_t B, slong prec)
{
acb_t t;
slong i, j, c, n, m;
slong i, c, n, m;
n = acb_mat_nrows(X);
m = acb_mat_ncols(X);
@ -84,44 +47,6 @@ acb_mat_approx_solve_lu_precomp(acb_mat_t X, const slong * perm,
}
acb_mat_get_mid(X, X);
/* todo: solve_tril and solve_triu have some overhead; should be
able to eliminate the basecase code below */
if (n >= 8 && m >= 8)
{
acb_mat_approx_solve_tril(X, A, X, 1, prec);
acb_mat_approx_solve_triu(X, A, X, 0, prec);
return;
}
acb_init(t);
for (c = 0; c < m; c++)
{
/* solve Ly = b */
for (i = 1; i < n; i++)
{
for (j = 0; j < i; j++)
{
_acb_approx_submul(acb_mat_entry(X, i, c),
acb_mat_entry(A, i, j), acb_mat_entry(X, j, c), t, prec);
}
}
/* solve Ux = y */
for (i = n - 1; i >= 0; i--)
{
for (j = i + 1; j < n; j++)
{
_acb_approx_submul(acb_mat_entry(X, i, c),
acb_mat_entry(A, i, j), acb_mat_entry(X, j, c), t, prec);
}
_acb_approx_div(acb_mat_entry(X, i, c), acb_mat_entry(X, i, c),
acb_mat_entry(A, i, i), t, prec);
}
}
acb_clear(t);
}

54
acb_mat/approx_solve_tril.c
View file

@ -11,13 +11,6 @@
#include "acb_mat.h"
static void
acb_approx_set(acb_t z, const acb_t x)
{
arf_set(arb_midref(acb_realref(z)), arb_midref(acb_realref(x)));
arf_set(arb_midref(acb_imagref(z)), arb_midref(acb_imagref(x)));
}
static void
acb_approx_mul(acb_t res, const acb_t x, const acb_t y, slong prec)
{
@ -26,26 +19,6 @@ acb_approx_mul(acb_t res, const acb_t x, const acb_t y, slong prec)
arb_midref(acb_realref(y)), arb_midref(acb_imagref(y)), prec, ARB_RND);
}
static void
acb_approx_sub(acb_t z, const acb_t x, const acb_t y, slong prec)
{
arf_sub(arb_midref(acb_realref(z)), arb_midref(acb_realref(x)), arb_midref(acb_realref(y)), prec, ARF_RND_DOWN);
arf_sub(arb_midref(acb_imagref(z)), arb_midref(acb_imagref(x)), arb_midref(acb_imagref(y)), prec, ARF_RND_DOWN);
}
static void
acb_approx_addmul(acb_t z, const acb_t x, const acb_t y, acb_t t, slong prec)
{
acb_approx_mul(t, x, y, prec);
arf_add(arb_midref(acb_realref(z)),
arb_midref(acb_realref(z)),
arb_midref(acb_realref(t)), prec, ARB_RND);
arf_add(arb_midref(acb_imagref(z)),
arb_midref(acb_imagref(z)),
arb_midref(acb_imagref(t)), prec, ARB_RND);
}
/* note: the tmp variable t should have zero radius */
static void
acb_approx_div(acb_t z, const acb_t x, const acb_t y, acb_t t, slong prec)
@ -65,7 +38,7 @@ void
acb_mat_approx_solve_tril_classical(acb_mat_t X,
const acb_mat_t L, const acb_mat_t B, int unit, slong prec)
{
slong i, j, k, n, m;
slong i, j, n, m;
acb_ptr tmp;
acb_t s, t;
@ -74,29 +47,28 @@ acb_mat_approx_solve_tril_classical(acb_mat_t X,
acb_init(s);
acb_init(t);
tmp = _acb_vec_init(n);
tmp = flint_malloc(sizeof(acb_struct) * n);
for (i = 0; i < m; i++)
{
for (j = 0; j < n; j++)
acb_approx_set(tmp + j, acb_mat_entry(X, j, i));
tmp[j] = *acb_mat_entry(X, j, i);
for (j = 0; j < n; j++)
{
acb_zero(s);
for (k = 0; k < j; k++)
acb_approx_addmul(s, L->rows[j] + k, tmp + k, t, prec);
acb_approx_sub(s, acb_mat_entry(B, j, i), s, prec);
acb_approx_dot(s, acb_mat_entry(B, j, i), 1, L->rows[j], 1, tmp, 1, j, prec);
if (!unit)
acb_approx_div(s, s, acb_mat_entry(L, j, j), t, prec);
acb_approx_set(tmp + j, s);
acb_approx_div(tmp + j, s, acb_mat_entry(L, j, j), t, prec);
else
acb_swap(tmp + j, s);
}
for (j = 0; j < n; j++)
acb_approx_set(acb_mat_entry(X, j, i), tmp + j);
*acb_mat_entry(X, j, i) = tmp[j];
}
_acb_vec_clear(tmp, n);
flint_free(tmp);
acb_clear(s);
acb_clear(t);
}
@ -133,8 +105,7 @@ acb_mat_approx_solve_tril_recursive(acb_mat_t X,
/* acb_mat_submul(XY, BY, LC, XX); */
acb_mat_init(T, LC->r, BX->c);
acb_mat_mul(T, LC, XX, prec);
acb_mat_get_mid(T, T);
acb_mat_approx_mul(T, LC, XX, prec);
acb_mat_sub(XY, BY, T, prec);
acb_mat_get_mid(XY, XY);
acb_mat_clear(T);
@ -154,9 +125,8 @@ void
acb_mat_approx_solve_tril(acb_mat_t X, const acb_mat_t L,
const acb_mat_t B, int unit, slong prec)
{
if (B->r < 8 || B->c < 8)
if (B->r < 40 || B->c < 40)
acb_mat_approx_solve_tril_classical(X, L, B, unit, prec);
else
acb_mat_approx_solve_tril_recursive(X, L, B, unit, prec);
}

54
acb_mat/approx_solve_triu.c
View file

@ -11,13 +11,6 @@
#include "acb_mat.h"
static void
acb_approx_set(acb_t z, const acb_t x)
{
arf_set(arb_midref(acb_realref(z)), arb_midref(acb_realref(x)));
arf_set(arb_midref(acb_imagref(z)), arb_midref(acb_imagref(x)));
}
static void
acb_approx_mul(acb_t res, const acb_t x, const acb_t y, slong prec)
{
@ -26,26 +19,6 @@ acb_approx_mul(acb_t res, const acb_t x, const acb_t y, slong prec)
arb_midref(acb_realref(y)), arb_midref(acb_imagref(y)), prec, ARB_RND);
}
static void
acb_approx_sub(acb_t z, const acb_t x, const acb_t y, slong prec)
{
arf_sub(arb_midref(acb_realref(z)), arb_midref(acb_realref(x)), arb_midref(acb_realref(y)), prec, ARF_RND_DOWN);
arf_sub(arb_midref(acb_imagref(z)), arb_midref(acb_imagref(x)), arb_midref(acb_imagref(y)), prec, ARF_RND_DOWN);
}
static void
acb_approx_addmul(acb_t z, const acb_t x, const acb_t y, acb_t t, slong prec)
{
acb_approx_mul(t, x, y, prec);
arf_add(arb_midref(acb_realref(z)),
arb_midref(acb_realref(z)),
arb_midref(acb_realref(t)), prec, ARB_RND);
arf_add(arb_midref(acb_imagref(z)),
arb_midref(acb_imagref(z)),
arb_midref(acb_imagref(t)), prec, ARB_RND);
}
/* note: the tmp variable t should have zero radius */
static void
acb_approx_div(acb_t z, const acb_t x, const acb_t y, acb_t t, slong prec)
@ -65,40 +38,37 @@ void
acb_mat_approx_solve_triu_classical(acb_mat_t X, const acb_mat_t U,
const acb_mat_t B, int unit, slong prec)
{
slong i, j, k, n, m;
slong i, j, n, m;
acb_ptr tmp;
acb_t s, t;
n = U->r;
m = B->c;
tmp = _acb_vec_init(n);
acb_init(s);
acb_init(t);
tmp = flint_malloc(sizeof(acb_struct) * n);
for (i = 0; i < m; i++)
{
for (j = 0; j < n; j++)
acb_approx_set(tmp + j, acb_mat_entry(X, j, i));
tmp[j] = *acb_mat_entry(X, j, i);
for (j = n - 1; j >= 0; j--)
{
acb_zero(s);
for (k = 0; k < n - j - 1; k++)
acb_approx_addmul(s, U->rows[j] + j + 1 + k, tmp + j + 1 + k, t, prec);
acb_approx_dot(s, acb_mat_entry(B, j, i), 1, U->rows[j] + j + 1, 1, tmp + j + 1, 1, n - j - 1, prec);
acb_approx_sub(s, acb_mat_entry(B, j, i), s, prec);
if (!unit)
acb_approx_div(s, s, acb_mat_entry(U, j, j), t, prec);
acb_approx_set(tmp + j, s);
acb_approx_div(tmp + j, s, arb_mat_entry(U, j, j), t, prec);
else
acb_swap(tmp + j, s);
}
for (j = 0; j < n; j++)
acb_approx_set(acb_mat_entry(X, j, i), tmp + j);
*acb_mat_entry(X, j, i) = tmp[j];
}
_acb_vec_clear(tmp, n);
flint_free(tmp);
acb_clear(s);
acb_clear(t);
}
@ -134,8 +104,7 @@ acb_mat_approx_solve_triu_recursive(acb_mat_t X,
acb_mat_approx_solve_triu(XY, UD, BY, unit, prec);
acb_mat_init(T, UB->r, XY->c);
acb_mat_mul(T, UB, XY, prec);
acb_mat_get_mid(T, T);
acb_mat_approx_mul(T, UB, XY, prec);
acb_mat_sub(XX, BX, T, prec);
acb_mat_get_mid(XX, XX);
acb_mat_clear(T);
@ -155,9 +124,8 @@ void
acb_mat_approx_solve_triu(acb_mat_t X, const acb_mat_t U,
const acb_mat_t B, int unit, slong prec)
{
if (B->r < 8 || B->c < 8)
if (B->r < 40 || B->c < 40)
acb_mat_approx_solve_triu_classical(X, U, B, unit, prec);
else
acb_mat_approx_solve_triu_recursive(X, U, B, unit, prec);
}

33
acb_mat/solve_lu_precomp.c
View file

@ -15,7 +15,7 @@ void
acb_mat_solve_lu_precomp(acb_mat_t X, const slong * perm,
const acb_mat_t A, const acb_mat_t B, slong prec)
{
slong i, c, n, m;
slong i, j, c, n, m;
n = acb_mat_nrows(X);
m = acb_mat_ncols(X);
@ -46,6 +46,37 @@ acb_mat_solve_lu_precomp(acb_mat_t X, const slong * perm,
}
}
/* solve_tril and solve_triu have some overhead */
if (n >= 4)
{
acb_mat_solve_tril(X, A, X, 1, prec);
acb_mat_solve_triu(X, A, X, 0, prec);
return;
}
for (c = 0; c < m; c++)
{
/* solve Ly = b */
for (i = 1; i < n; i++)
{
for (j = 0; j < i; j++)
{
acb_submul(acb_mat_entry(X, i, c),
acb_mat_entry(A, i, j), acb_mat_entry(X, j, c), prec);
}
}
/* solve Ux = y */
for (i = n - 1; i >= 0; i--)
{
for (j = i + 1; j < n; j++)
{
acb_submul(acb_mat_entry(X, i, c),
acb_mat_entry(A, i, j), acb_mat_entry(X, j, c), prec);
}
acb_div(acb_mat_entry(X, i, c), acb_mat_entry(X, i, c),
acb_mat_entry(A, i, i), prec);
}
}
}

3
arb.h
View file

@ -464,6 +464,9 @@ void arb_dot_precise(arb_t res, const arb_t initial, int subtract,
void arb_dot(arb_t res, const arb_t initial, int subtract,
arb_srcptr x, slong xstep, arb_srcptr y, slong ystep, slong len, slong prec);
void arb_approx_dot(arb_t res, const arb_t initial, int subtract,
arb_srcptr x, slong xstep, arb_srcptr y, slong ystep, slong len, slong prec);
void arb_div(arb_t z, const arb_t x, const arb_t y, slong prec);
void arb_div_arf(arb_t z, const arb_t x, const arf_t y, slong prec);
void arb_div_si(arb_t z, const arb_t x, slong y, slong prec);

527
arb/approx_dot.c Normal file
View file

@ -0,0 +1,527 @@
/*
Copyright (C) 2018 Fredrik Johansson
This file is part of Arb.
Arb is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 2.1 of the License, or
(at your option) any later version. See <http://www.gnu.org/licenses/>.
*/
#include "arb.h"
/* We need uint64_t instead of mp_limb_t on 32-bit systems for
safe summation of 30-bit error bounds. */
#include <stdint.h>
/* The following macros are found in FLINT's longlong.h, but
the release version is out of date. */
/* x86 : 64 bit */
#if (GMP_LIMB_BITS == 64 && defined (__amd64__))
#define add_sssaaaaaa2(sh, sm, sl, ah, am, al, bh, bm, bl) \
__asm__ ("addq %8,%q2\n\tadcq %6,%q1\n\tadcq %4,%q0" \
: "=r" (sh), "=&r" (sm), "=&r" (sl) \
: "0" ((mp_limb_t)(ah)), "rme" ((mp_limb_t)(bh)), \
"1" ((mp_limb_t)(am)), "rme" ((mp_limb_t)(bm)), \
"2" ((mp_limb_t)(al)), "rme" ((mp_limb_t)(bl))) \
#define sub_dddmmmsss2(dh, dm, dl, mh, mm, ml, sh, sm, sl) \
__asm__ ("subq %8,%q2\n\tsbbq %6,%q1\n\tsbbq %4,%q0" \
: "=r" (dh), "=&r" (dm), "=&r" (dl) \
: "0" ((mp_limb_t)(mh)), "rme" ((mp_limb_t)(sh)), \
"1" ((mp_limb_t)(mm)), "rme" ((mp_limb_t)(sm)), \
"2" ((mp_limb_t)(ml)), "rme" ((mp_limb_t)(sl))) \
#endif /* x86_64 */
/* x86 : 32 bit */
#if (GMP_LIMB_BITS == 32 && (defined (__i386__) \
|| defined (__i486__) || defined(__amd64__)))
#define add_sssaaaaaa2(sh, sm, sl, ah, am, al, bh, bm, bl) \
__asm__ ("addl %8,%k2\n\tadcl %6,%k1\n\tadcl %4,%k0" \
: "=r" (sh), "=r" (sm), "=&r" (sl) \
: "0" ((mp_limb_t)(ah)), "g" ((mp_limb_t)(bh)), \
"1" ((mp_limb_t)(am)), "g" ((mp_limb_t)(bm)), \
"2" ((mp_limb_t)(al)), "g" ((mp_limb_t)(bl))) \
#define sub_dddmmmsss2(dh, dm, dl, mh, mm, ml, sh, sm, sl) \
__asm__ ("subl %8,%k2\n\tsbbl %6,%k1\n\tsbbl %4,%k0" \
: "=r" (dh), "=r" (dm), "=&r" (dl) \
: "0" ((mp_limb_t)(mh)), "g" ((mp_limb_t)(sh)), \
"1" ((mp_limb_t)(mm)), "g" ((mp_limb_t)(sm)), \
"2" ((mp_limb_t)(ml)), "g" ((mp_limb_t)(sl))) \
#endif /* x86 */
#if !defined(add_sssaaaaaa2)
#define add_sssaaaaaa2(sh, sm, sl, ah, am, al, bh, bm, bl) \
do { \
mp_limb_t __t, __u; \
add_ssaaaa(__t, sl, (mp_limb_t) 0, al, (mp_limb_t) 0, bl); \
add_ssaaaa(__u, sm, (mp_limb_t) 0, am, (mp_limb_t) 0, bm); \
add_ssaaaa(sh, sm, ah + bh, sm, __u, __t); \
} while (0)
#define sub_dddmmmsss2(dh, dm, dl, mh, mm, ml, sh, sm, sl) \
do { \
mp_limb_t __t, __u; \
sub_ddmmss(__t, dl, (mp_limb_t) 0, ml, (mp_limb_t) 0, sl); \
sub_ddmmss(__u, dm, (mp_limb_t) 0, mm, (mp_limb_t) 0, sm); \
sub_ddmmss(dh, dm, mh - sh, dm, __u, __t); \
} while (0)
#endif
void
_arb_dot_addmul_generic(mp_ptr sum, mp_ptr serr, mp_ptr tmp, mp_size_t sn,
mp_srcptr xptr, mp_size_t xn, mp_srcptr yptr, mp_size_t yn,
int negative, mp_bitcnt_t shift);
void
_arb_dot_add_generic(mp_ptr sum, mp_ptr serr, mp_ptr tmp, mp_size_t sn,
mp_srcptr xptr, mp_size_t xn,
int negative, mp_bitcnt_t shift);
void
arb_approx_dot_simple(arb_t res, const arb_t initial, int subtract,
arb_srcptr x, slong xstep, arb_srcptr y, slong ystep, slong len, slong prec)
{
slong i;
if (len <= 0)
{
if (initial == NULL)
arf_zero(arb_midref(res));
else
arf_set_round(arb_midref(res), arb_midref(initial), prec, ARB_RND);
return;
}
if (initial == NULL)
{
arf_mul(arb_midref(res), arb_midref(x), arb_midref(y), prec, ARB_RND);
}
else
{
if (subtract)
arf_neg(arb_midref(res), arb_midref(initial));
else
arf_set(arb_midref(res), arb_midref(initial));
arf_addmul(arb_midref(res), arb_midref(x), arb_midref(y), prec, ARB_RND);
}
for (i = 1; i < len; i++)
arf_addmul(arb_midref(res), arb_midref(x + i * xstep), arb_midref(y + i * ystep), prec, ARB_RND);
if (subtract)
arf_neg(arb_midref(res), arb_midref(res));
}
void
arb_approx_dot(arb_t res, const arb_t initial, int subtract, arb_srcptr x, slong xstep, arb_srcptr y, slong ystep, slong len, slong prec)
{
slong i, j, nonzero, padding, extend;
slong xexp, yexp, exp, max_exp, min_exp, sum_exp;
int xnegative, ynegative;
mp_size_t xn, yn, sn, alloc;
mp_bitcnt_t shift;
arb_srcptr xi, yi;
arf_srcptr xm, ym;
mp_limb_t serr; /* Sum over arithmetic errors - not used, but need dummy for calls */
mp_ptr tmp, sum; /* Workspace */
ARF_ADD_TMP_DECL;
/* todo: fast fma and fmma (len=2) code */
if (len <= 1)
{
if (initial == NULL)
{
if (len <= 0)
arf_zero(arb_midref(res));
else
{
if (subtract)
arf_neg_mul(arb_midref(res), arb_midref(x), arb_midref(y), prec, ARB_RND);
else
arf_mul(arb_midref(res), arb_midref(x), arb_midref(y), prec, ARB_RND);
}
return;
}
else if (len <= 0)
{
arf_set_round(arb_midref(res), arb_midref(initial), prec, ARB_RND);
return;
}
}
/* Number of nonzero midpoint terms in sum. */
nonzero = 0;
/* Terms are bounded by 2^max_exp (with WORD_MIN = -infty) */
max_exp = WORD_MIN;
/* Used to reduce the precision. */
min_exp = WORD_MAX;
/* Account for the initial term. */
if (initial != NULL)
{
if (!ARF_IS_LAGOM(arb_midref(initial)))
{
arb_approx_dot_simple(res, initial, subtract, x, xstep, y, ystep, len, prec);
return;
}
xm = arb_midref(initial);
if (!arf_is_special(xm))
{
max_exp = ARF_EXP(xm);
nonzero++;
if (prec > 2 * FLINT_BITS)
min_exp = ARF_EXP(xm) - ARF_SIZE(xm) * FLINT_BITS;
}
}
/* Determine maximum exponents for the main sum and the radius sum. */
for (i = 0; i < len; i++)
{
xi = x + i * xstep;
yi = y + i * ystep;
/* Fallback for huge exponents or non-finite values. */
if (!ARF_IS_LAGOM(arb_midref(xi)) || !ARF_IS_LAGOM(arb_midref(yi)))
{
arb_approx_dot_simple(res, initial, subtract, x, xstep, y, ystep, len, prec);
return;
}
xm = arb_midref(xi);
ym = arb_midref(yi);
if (!arf_is_special(xm))
{
xexp = ARF_EXP(xm);
if (!arf_is_special(ym))
{
yexp = ARF_EXP(ym);
max_exp = FLINT_MAX(max_exp, xexp + yexp);
nonzero++;
if (prec > 2 * FLINT_BITS)
{
slong bot;
bot = (xexp + yexp) - (ARF_SIZE(xm) + ARF_SIZE(ym)) * FLINT_BITS;
min_exp = FLINT_MIN(min_exp, bot);
}
}
}
}
/* The midpoint sum is zero. */
if (max_exp == WORD_MIN)
{
arf_zero(arb_midref(res));
return;
}
else
{
/* Reduce precision based on actual sizes. */
if (min_exp != WORD_MAX)
prec = FLINT_MIN(prec, max_exp - min_exp + MAG_BITS);
prec = FLINT_MAX(prec, 2);
}
/* Extend sum so that we can use two's complement addition. */
extend = FLINT_BIT_COUNT(nonzero) + 1;
/* Extra bits to improve accuracy (optional). */
padding = 4 + FLINT_BIT_COUNT(len);
/* Number of limbs. */
sn = (prec + extend + padding + FLINT_BITS - 1) / FLINT_BITS;
/* Avoid having to make a special case for sn = 1. */
sn = FLINT_MAX(sn, 2);
/* Exponent for the main sum. */
sum_exp = max_exp + extend;
/* We need sn + 1 limb for the sum (sn limbs + 1 dummy limb
for carry or borrow that avoids an extra branch). We need
2 * (sn + 2) limbs to store the product of two numbers
with up to (sn + 2) limbs, plus 1 extra limb for shifting
the product. */
alloc = (sn + 1) + 2 * (sn + 2) + 1;
ARF_ADD_TMP_ALLOC(sum, alloc)
tmp = sum + (sn + 1);
/* Set sum to 0 */
serr = 0;
for (j = 0; j < sn + 1; j++)
sum[j] = 0;
if (initial != NULL)
{
xm = arb_midref(initial);
if (!arf_is_special(xm))
{
mp_srcptr xptr;
xexp = ARF_EXP(xm);
xn = ARF_SIZE(xm);
xnegative = ARF_SGNBIT(xm);
shift = sum_exp - xexp;
if (shift < sn * FLINT_BITS)
{
xptr = (xn <= ARF_NOPTR_LIMBS) ? ARF_NOPTR_D(xm) : ARF_PTR_D(xm);
_arb_dot_add_generic(sum, &serr, tmp, sn, xptr, xn, xnegative ^ subtract, shift);
}
}
}
for (i = 0; i < len; i++)
{
xi = x + i * xstep;
yi = y + i * ystep;
xm = arb_midref(xi);
ym = arb_midref(yi);
/* The midpoints of x[i] and y[i] are both nonzero. */
if (!arf_is_special(xm) && !arf_is_special(ym))
{
xexp = ARF_EXP(xm);
xn = ARF_SIZE(xm);
xnegative = ARF_SGNBIT(xm);
yexp = ARF_EXP(ym);
yn = ARF_SIZE(ym);
ynegative = ARF_SGNBIT(ym);
exp = xexp + yexp;
shift = sum_exp - exp;
if (shift >= sn * FLINT_BITS)
{
/* do nothing */
}
#if 0
else if (xn == 1 && yn == 1 && sn == 2 && shift < FLINT_BITS) /* Fastest path. */
{
mp_limb_t hi, lo, x0, y0;
x0 = ARF_NOPTR_D(xm)[0];
y0 = ARF_NOPTR_D(ym)[0];
umul_ppmm(hi, lo, x0, y0);
lo = (lo >> shift) | (hi << (FLINT_BITS - shift));
hi = (hi >> shift);
if (xnegative ^ ynegative)
sub_ddmmss(sum[1], sum[0], sum[1], sum[0], hi, lo);
else
add_ssaaaa(sum[1], sum[0], sum[1], sum[0], hi, lo);
}
else if (xn == 2 && yn == 2 && shift < FLINT_BITS && sn <= 3)
{
mp_limb_t x1, x0, y1, y0;
mp_limb_t u3, u2, u1, u0;
x0 = ARF_NOPTR_D(xm)[0];
x1 = ARF_NOPTR_D(xm)[1];
y0 = ARF_NOPTR_D(ym)[0];
y1 = ARF_NOPTR_D(ym)[1];
nn_mul_2x2(u3, u2, u1, u0, x1, x0, y1, y0);
u1 = (u1 >> shift) | (u2 << (FLINT_BITS - shift));
u2 = (u2 >> shift) | (u3 << (FLINT_BITS - shift));
u3 = (u3 >> shift);
if (sn == 2)
{
if (xnegative ^ ynegative)
sub_ddmmss(sum[1], sum[0], sum[1], sum[0], u3, u2);
else
add_ssaaaa(sum[1], sum[0], sum[1], sum[0], u3, u2);
}
else
{
if (xnegative ^ ynegative)
sub_dddmmmsss2(sum[2], sum[1], sum[0], sum[2], sum[1], sum[0], u3, u2, u1);
else
add_sssaaaaaa2(sum[2], sum[1], sum[0], sum[2], sum[1], sum[0], u3, u2, u1);
}
}
#endif
else if (xn <= 2 && yn <= 2 && sn <= 3)
{
mp_limb_t x1, x0, y1, y0;
mp_limb_t u3, u2, u1, u0;
if (xn == 1 && yn == 1)
{
x0 = ARF_NOPTR_D(xm)[0];
y0 = ARF_NOPTR_D(ym)[0];
umul_ppmm(u3, u2, x0, y0);
u1 = u0 = 0;
}
else if (xn == 2 && yn == 2)
{
x0 = ARF_NOPTR_D(xm)[0];
x1 = ARF_NOPTR_D(xm)[1];
y0 = ARF_NOPTR_D(ym)[0];
y1 = ARF_NOPTR_D(ym)[1];
nn_mul_2x2(u3, u2, u1, u0, x1, x0, y1, y0);
}
else if (xn == 1)
{
x0 = ARF_NOPTR_D(xm)[0];
y0 = ARF_NOPTR_D(ym)[0];
y1 = ARF_NOPTR_D(ym)[1];
nn_mul_2x1(u3, u2, u1, y1, y0, x0);
u0 = 0;
}
else
{
x0 = ARF_NOPTR_D(xm)[0];
x1 = ARF_NOPTR_D(xm)[1];
y0 = ARF_NOPTR_D(ym)[0];
nn_mul_2x1(u3, u2, u1, x1, x0, y0);
u0 = 0;
}
if (sn == 2)
{
if (shift < FLINT_BITS)
{
u2 = (u2 >> shift) | (u3 << (FLINT_BITS - shift));
u3 = (u3 >> shift);
}
else if (shift == FLINT_BITS)
{
u2 = u3;
u3 = 0;
}
else /* FLINT_BITS < shift < 2 * FLINT_BITS */
{
u2 = (u3 >> (shift - FLINT_BITS));
u3 = 0;
}
if (xnegative ^ ynegative)
sub_ddmmss(sum[1], sum[0], sum[1], sum[0], u3, u2);
else
add_ssaaaa(sum[1], sum[0], sum[1], sum[0], u3, u2);
}
else if (sn == 3)
{
if (shift < FLINT_BITS)
{
u1 = (u1 >> shift) | (u2 << (FLINT_BITS - shift));
u2 = (u2 >> shift) | (u3 << (FLINT_BITS - shift));
u3 = (u3 >> shift);
}
else if (shift == FLINT_BITS)
{
u1 = u2;
u2 = u3;
u3 = 0;
}
else if (shift < 2 * FLINT_BITS)
{
u1 = (u3 << (2 * FLINT_BITS - shift)) | (u2 >> (shift - FLINT_BITS));
u2 = (u3 >> (shift - FLINT_BITS));
u3 = 0;
}
else if (shift == 2 * FLINT_BITS)
{
u1 = u3;
u2 = 0;
u3 = 0;
}
else /* 2 * FLINT_BITS < shift < 3 * FLINT_BITS */
{
u1 = (u3 >> (shift - 2 * FLINT_BITS));
u2 = 0;
u3 = 0;
}
if (xnegative ^ ynegative)
sub_dddmmmsss2(sum[2], sum[1], sum[0], sum[2], sum[1], sum[0], u3, u2, u1);
else
add_sssaaaaaa2(sum[2], sum[1], sum[0], sum[2], sum[1], sum[0], u3, u2, u1);
}
}
else
{
mp_srcptr xptr, yptr;
xptr = (xn <= ARF_NOPTR_LIMBS) ? ARF_NOPTR_D(xm) : ARF_PTR_D(xm);
yptr = (yn <= ARF_NOPTR_LIMBS) ? ARF_NOPTR_D(ym) : ARF_PTR_D(ym);
_arb_dot_addmul_generic(sum, &serr, tmp, sn, xptr, xn, yptr, yn, xnegative ^ ynegative, shift);
}
}
}
xnegative = 0;
if (sum[sn - 1] >= LIMB_TOP)
{
mpn_neg(sum, sum, sn);
xnegative = 1;
}
if (sum[sn - 1] == 0)
{
slong sum_exp2;
mp_size_t sn2;
sn2 = sn;
sum_exp2 = sum_exp;
while (sn2 > 0 && sum[sn2 - 1] == 0)
{
sum_exp2 -= FLINT_BITS;
sn2--;
}
if (sn2 == 0)
{
arf_zero(arb_midref(res));
}
else
{
_arf_set_round_mpn(arb_midref(res), &exp, sum, sn2, xnegative ^ subtract, prec, ARF_RND_DOWN);
_fmpz_set_si_small(ARF_EXPREF(arb_midref(res)), exp + sum_exp2);
}
}
else
{
if (sn == 2)
_arf_set_round_uiui(arb_midref(res), &exp, sum[1], sum[0], xnegative ^ subtract, prec, ARF_RND_DOWN);
else
_arf_set_round_mpn(arb_midref(res), &exp, sum, sn, xnegative ^ subtract, prec, ARF_RND_DOWN);
_fmpz_set_si_small(ARF_EXPREF(arb_midref(res)), exp + sum_exp);
}
ARF_ADD_TMP_FREE(sum, alloc);
}

251
arb/test/t-approx_dot.c Normal file
View file

@ -0,0 +1,251 @@
/*
Copyright (C) 2018 Fredrik Johansson
This file is part of Arb.
Arb is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 2.1 of the License, or
(at your option) any later version. See <http://www.gnu.org/licenses/>.
*/
#include "arb.h"
int main()
{
slong iter;
flint_rand_t state;
flint_printf("approx_dot....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 100000 * arb_test_multiplier(); iter++)
{
arb_ptr x, y;
arb_t s1, s2, z;
slong i, len, prec, xbits, ybits, ebits;
int initial, subtract, revx, revy;
if (n_randint(state, 100) == 0)
len = n_randint(state, 100);
else if (n_randint(state, 10) == 0)
len = n_randint(state, 10);
else
len = n_randint(state, 3);
if (n_randint(state, 10) != 0 || len > 10)
{
prec = 2 + n_randint(state, 500);
xbits = 2 + n_randint(state, 500);
ybits = 2 + n_randint(state, 500);
}
else
{
prec = 2 + n_randint(state, 4000);
xbits = 2 + n_randint(state, 4000);
ybits = 2 + n_randint(state, 4000);
}
if (n_randint(state, 100) == 0)
ebits = 2 + n_randint(state, 100);
else
ebits = 2 + n_randint(state, 10);
initial = n_randint(state, 2);
subtract = n_randint(state, 2);
revx = n_randint(state, 2);
revy = n_randint(state, 2);
x = _arb_vec_init(len);
y = _arb_vec_init(len);
arb_init(s1);
arb_init(s2);
arb_init(z);
switch (n_randint(state, 3))
{
case 0:
for (i = 0; i < len; i++)
{
arb_randtest(x + i, state, xbits, ebits);
arb_randtest(y + i, state, ybits, ebits);
}
break;
/* Test with cancellation */
case 1:
for (i = 0; i < len; i++)
{
if (i <= len / 2)
{
arb_randtest(x + i, state, xbits, ebits);
arb_randtest(y + i, state, ybits, ebits);
}
else
{
arb_neg(x + i, x + len - i - 1);
arb_set(y + i, y + len - i - 1);
}
}
break;
default:
for (i = 0; i < len; i++)
{
if (i <= len / 2)
{
arb_randtest(x + i, state, xbits, ebits);
arb_randtest(y + i, state, ybits, ebits);
}
else
{
arb_neg_round(x + i, x + len - i - 1, 2 + n_randint(state, 500));
arb_set_round(y + i, y + len - i - 1, 2 + n_randint(state, 500));
}
}
break;
}
arb_randtest(s1, state, 200, 100);
arb_randtest(s2, state, 200, 100);
arb_randtest(z, state, xbits, ebits);
arb_approx_dot(s1, initial ? z : NULL, subtract,
revx ? (x + len - 1) : x, revx ? -1 : 1,
revy ? (y + len - 1) : y, revy ? -1 : 1,
len, prec);
mag_zero(arb_radref(s1));
/* With the fast algorithm, we expect identical results when
reversing the vectors. */
if (ebits <= 12)
{
arb_approx_dot(s2, initial ? z : NULL, subtract,
!revx ? (x + len - 1) : x, !revx ? -1 : 1,
!revy ? (y + len - 1) : y, !revy ? -1 : 1,
len, prec);
mag_zero(arb_radref(s2));
if (!arb_equal(s1, s2))
{
flint_printf("FAIL (reversal)\n\n");
flint_printf("iter = %wd, len = %wd, prec = %wd, ebits = %wd\n\n", iter, len, prec, ebits);
if (initial)
{
flint_printf("z = ", i); arb_printn(z, 100, ARB_STR_MORE); flint_printf(" (%wd)\n\n", arb_bits(z));
}
for (i = 0; i < len; i++)
{
flint_printf("x[%wd] = ", i); arb_printn(x + i, 100, ARB_STR_MORE); flint_printf(" (%wd)\n", arb_bits(x + i));
flint_printf("y[%wd] = ", i); arb_printn(y + i, 100, ARB_STR_MORE); flint_printf(" (%wd)\n", arb_bits(y + i));
}
flint_printf("\n\n");
flint_printf("s1 = "); arb_printn(s1, 100, ARB_STR_MORE); flint_printf("\n\n");
flint_printf("s2 = "); arb_printn(s2, 100, ARB_STR_MORE); flint_printf("\n\n");
flint_abort();
}
}
/* Verify that radii are ignored */
for (i = 0; i < len; i++)
{
arb_get_mid_arb(x + i, x + i);
arb_get_mid_arb(y + i, y + i);
}
arb_get_mid_arb(z, z);
arb_approx_dot(s2, initial ? z : NULL, subtract,
revx ? (x + len - 1) : x, revx ? -1 : 1,
revy ? (y + len - 1) : y, revy ? -1 : 1,
len, prec);
mag_zero(arb_radref(s2));
if (!arb_equal(s1, s2))
{
flint_printf("FAIL (radii)\n\n");
flint_printf("iter = %wd, len = %wd, prec = %wd, ebits = %wd\n\n", iter, len, prec, ebits);
if (initial)
{
flint_printf("z = ", i); arb_printn(z, 100, ARB_STR_MORE); flint_printf(" (%wd)\n\n", arb_bits(z));
}
for (i = 0; i < len; i++)
{
flint_printf("x[%wd] = ", i); arb_printn(x + i, 100, ARB_STR_MORE); flint_printf(" (%wd)\n", arb_bits(x + i));
flint_printf("y[%wd] = ", i); arb_printn(y + i, 100, ARB_STR_MORE); flint_printf(" (%wd)\n", arb_bits(y + i));
}
flint_printf("\n\n");
flint_printf("s1 = "); arb_printn(s1, 100, ARB_STR_MORE); flint_printf("\n\n");
flint_printf("s2 = "); arb_printn(s2, 100, ARB_STR_MORE); flint_printf("\n\n");
flint_abort();
}
/* Compare with arb_dot */
arb_approx_dot(s2, initial ? z : NULL, subtract,
revx ? (x + len - 1) : x, revx ? -1 : 1,
revy ? (y + len - 1) : y, revy ? -1 : 1,
len, prec);
{
mag_t err, xx, yy;
mag_init(err);
mag_init(xx);
mag_init(yy);
if (initial)
arb_get_mag(err, z);
for (i = 0; i < len; i++)
{
arb_get_mag(xx, revx ? x + len - 1 - i : x + i);
arb_get_mag(yy, revx ? y + len - 1 - i : y + i);
mag_addmul(err, xx, yy);
}
mag_mul_2exp_si(err, err, -prec + 2);
arb_add_error_mag(s2, err);
if (!arb_contains(s2, s1))
{
flint_printf("FAIL (inclusion)\n\n");
flint_printf("iter = %wd, len = %wd, prec = %wd, ebits = %wd\n\n", iter, len, prec, ebits);
if (initial)
{
flint_printf("z = ", i); arb_printn(z, 100, ARB_STR_MORE); flint_printf(" (%wd)\n\n", arb_bits(z));
}
for (i = 0; i < len; i++)
{
flint_printf("x[%wd] = ", i); arb_printn(x + i, 100, ARB_STR_MORE); flint_printf(" (%wd)\n", arb_bits(x + i));
flint_printf("y[%wd] = ", i); arb_printn(y + i, 100, ARB_STR_MORE); flint_printf(" (%wd)\n", arb_bits(y + i));
}
flint_printf("\n\n");
flint_printf("s1 = "); arb_printn(s1, 100, ARB_STR_MORE); flint_printf("\n\n");
flint_printf("s2 = "); arb_printn(s2, 100, ARB_STR_MORE); flint_printf("\n\n");
flint_abort();
}
mag_clear(err);
mag_clear(xx);
mag_clear(yy);
}
arb_clear(s1);
arb_clear(s2);
arb_clear(z);
_arb_vec_clear(x, len);
_arb_vec_clear(y, len);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}

1
arb_mat.h
View file

@ -351,6 +351,7 @@ int arb_mat_solve_precond(arb_mat_t X, const arb_mat_t A, const arb_mat_t B, slo
int arb_mat_solve_preapprox(arb_mat_t X, const arb_mat_t A,
const arb_mat_t B, const arb_mat_t R, const arb_mat_t T, slong prec);
void arb_mat_approx_mul(arb_mat_t C, const arb_mat_t A, const arb_mat_t B, slong prec);
void arb_mat_approx_solve_triu(arb_mat_t X, const arb_mat_t U, const arb_mat_t B, int unit, slong prec);
void arb_mat_approx_solve_tril(arb_mat_t X, const arb_mat_t L, const arb_mat_t B, int unit, slong prec);
int arb_mat_approx_lu(slong * P, arb_mat_t LU, const arb_mat_t A, slong prec);

10
arb_mat/approx_lu.c
View file

@ -85,11 +85,11 @@ arb_mat_approx_lu_classical(slong * P, arb_mat_t LU, const arb_mat_t A, slong pr
else if (r != row)
arb_mat_swap_rows(LU, P, row, r);
arf_set(d, arb_midref(a[row] + col));
arf_ui_div(d, 1, arb_midref(a[row] + col), prec, ARB_RND);
for (j = row + 1; j < m; j++)
{
arf_div(arb_midref(e), arb_midref(a[j] + col), d, prec, ARB_RND);
arf_mul(arb_midref(e), arb_midref(a[j] + col), d, prec, ARB_RND);
arb_neg(e, e);
_arb_vec_approx_scalar_addmul(a[j] + col,
a[row] + col, n - col, e, prec);
@ -156,11 +156,10 @@ arb_mat_approx_lu_recursive(slong * P, arb_mat_t LU, const arb_mat_t A, slong pr
arb_mat_approx_solve_tril(A01, A00, A01, 1, prec);
{
/* arb_mat_submul(A11, A11, A10, A01, prec); */
/* arb_mat_approx_submul(A11, A11, A10, A01, prec); */
arb_mat_t T;
arb_mat_init(T, A10->r, A01->c);
arb_mat_mul(T, A10, A01, prec);
arb_mat_get_mid(T, T);
arb_mat_approx_mul(T, A10, A01, prec);
arb_mat_sub(A11, A11, T, prec);
arb_mat_get_mid(A11, A11);
arb_mat_clear(T);
@ -192,4 +191,3 @@ arb_mat_approx_lu(slong * P, arb_mat_t LU, const arb_mat_t A, slong prec)
else
return arb_mat_approx_lu_recursive(P, LU, A, prec);
}

152
arb_mat/approx_mul.c Normal file
View file

@ -0,0 +1,152 @@
/*
Copyright (C) 2018 Fredrik Johansson
This file is part of Arb.
Arb is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 2.1 of the License, or
(at your option) any later version. See <http://www.gnu.org/licenses/>.
*/
#include "arb_mat.h"
void
arb_mat_approx_mul_classical(arb_mat_t C, const arb_mat_t A, const arb_mat_t B, slong prec)
{
slong ar, br, bc, i, j, k;
ar = arb_mat_nrows(A);
br = arb_mat_nrows(B);
bc = arb_mat_ncols(B);
if (br == 0)
{
arb_mat_zero(C);
return;
}
if (A == C || B == C)
{
arb_mat_t T;
arb_mat_init(T, ar, bc);
arb_mat_approx_mul_classical(T, A, B, prec);
arb_mat_swap(T, C);
arb_mat_clear(T);
return;
}
if (br <= 2)
{
for (i = 0; i < ar; i++)
{
for (j = 0; j < bc; j++)
{
arf_mul(arb_midref(arb_mat_entry(C, i, j)),
arb_midref(arb_mat_entry(A, i, 0)),
arb_midref(arb_mat_entry(B, 0, j)), prec, ARB_RND);
for (k = 1; k < br; k++)
{
arf_addmul(arb_midref(arb_mat_entry(C, i, j)),
arb_midref(arb_mat_entry(A, i, k)),
arb_midref(arb_mat_entry(B, k, j)), prec, ARB_RND);
}
}
}
}
else
{
arb_ptr tmp;
TMP_INIT;
TMP_START;
tmp = TMP_ALLOC(sizeof(arb_struct) * br * bc);
for (i = 0; i < br; i++)
for (j = 0; j < bc; j++)
tmp[j * br + i] = *arb_mat_entry(B, i, j);
for (i = 0; i < ar; i++)
{
for (j = 0; j < bc; j++)
{
arb_approx_dot(arb_mat_entry(C, i, j), NULL, 0,
A->rows[i], 1, tmp + j * br, 1, br, prec);
}
}
TMP_END;
}
}
int
arb_mat_is_exact(const arb_mat_t A)
{
slong i, j;
for (i = 0; i < arb_mat_nrows(A); i++)
for (j = 0; j < arb_mat_ncols(A); j++)
if (!mag_is_zero(arb_radref(arb_mat_entry(A, i, j))))
return 0;
return 1;
}
void
arb_mat_approx_mul(arb_mat_t C, const arb_mat_t A, const arb_mat_t B, slong prec)
{
slong cutoff;
/* todo: detect small-integer matrices */
if (prec <= 2 * FLINT_BITS)
cutoff = 120;
else if (prec <= 16 * FLINT_BITS)
cutoff = 60;
else
cutoff = 40;
if (arb_mat_nrows(A) <= cutoff || arb_mat_ncols(A) <= cutoff ||
arb_mat_ncols(B) <= cutoff)
{
arb_mat_approx_mul_classical(C, A, B, prec);
}
else
{
if (arb_mat_is_exact(A) && arb_mat_is_exact(B))
{
arb_mat_mul(C, A, B, prec);
}
else
{
arb_mat_t AM, BM;
if (arb_mat_is_exact(A))
{
arb_mat_init(BM, arb_mat_nrows(B), arb_mat_ncols(B));
arb_mat_get_mid(BM, B);
arb_mat_mul(C, A, BM, prec);
arb_mat_clear(BM);
}
else if (arb_mat_is_exact(A))
{
arb_mat_init(AM, arb_mat_nrows(A), arb_mat_ncols(A));
arb_mat_get_mid(AM, A);
arb_mat_mul(C, AM, B, prec);
arb_mat_clear(AM);
}
else
{
arb_mat_init(BM, arb_mat_nrows(B), arb_mat_ncols(B));
arb_mat_get_mid(BM, B);
arb_mat_init(AM, arb_mat_nrows(A), arb_mat_ncols(A));
arb_mat_get_mid(AM, A);
arb_mat_mul(C, AM, BM, prec);
arb_mat_clear(AM);
arb_mat_clear(BM);
}
}
arb_mat_get_mid(C, C);
}
}

49
arb_mat/approx_solve_lu_precomp.c
View file

@ -12,24 +12,11 @@
#include "arb_mat.h"
static void
arb_approx_submul(arb_t z, const arb_t x, const arb_t y, slong prec)
{
arf_submul(arb_midref(z),
arb_midref(x), arb_midref(y), prec, ARF_RND_DOWN);
}
static void
arb_approx_div(arb_t z, const arb_t x, const arb_t y, slong prec)
{
arf_div(arb_midref(z), arb_midref(x), arb_midref(y), prec, ARB_RND);
}
void
arb_mat_approx_solve_lu_precomp(arb_mat_t X, const slong * perm,
const arb_mat_t A, const arb_mat_t B, slong prec)
{
slong i, j, c, n, m;
slong i, c, n, m;
n = arb_mat_nrows(X);
m = arb_mat_ncols(X);
@ -61,40 +48,6 @@ arb_mat_approx_solve_lu_precomp(arb_mat_t X, const slong * perm,
}
arb_mat_get_mid(X, X);
/* todo: solve_tril and solve_triu have some overhead; should be
able to eliminate the basecase code below */
if (n >= 8 && m >= 8)
{
arb_mat_approx_solve_tril(X, A, X, 1, prec);
arb_mat_approx_solve_triu(X, A, X, 0, prec);
return;
}
for (c = 0; c < m; c++)
{
/* solve Ly = b */
for (i = 1; i < n; i++)
{
for (j = 0; j < i; j++)
{
arb_approx_submul(arb_mat_entry(X, i, c),
arb_mat_entry(A, i, j), arb_mat_entry(X, j, c), prec);
}
}
/* solve Ux = y */
for (i = n - 1; i >= 0; i--)
{
for (j = i + 1; j < n; j++)
{
arb_approx_submul(arb_mat_entry(X, i, c),
arb_mat_entry(A, i, j), arb_mat_entry(X, j, c), prec);
}
arb_approx_div(arb_mat_entry(X, i, c), arb_mat_entry(X, i, c),
arb_mat_entry(A, i, i), prec);
}
}
}

47
arb_mat/approx_solve_tril.c
View file

@ -11,26 +11,6 @@
#include "arb_mat.h"
static void
arb_approx_set(arb_t z, const arb_t x)
{
arf_set(arb_midref(z), arb_midref(x));
}
static void
arb_approx_sub(arb_t z, const arb_t x, const arb_t y, slong prec)
{
arf_sub(arb_midref(z),
arb_midref(x), arb_midref(y), prec, ARF_RND_DOWN);
}
static void
arb_approx_addmul(arb_t z, const arb_t x, const arb_t y, slong prec)
{
arf_addmul(arb_midref(z),
arb_midref(x), arb_midref(y), prec, ARF_RND_DOWN);
}
static void
arb_approx_div(arb_t z, const arb_t x, const arb_t y, slong prec)
{
@ -41,7 +21,7 @@ void
arb_mat_approx_solve_tril_classical(arb_mat_t X,
const arb_mat_t L, const arb_mat_t B, int unit, slong prec)
{
slong i, j, k, n, m;
slong i, j, n, m;
arb_ptr tmp;
arb_t s;
@ -49,29 +29,28 @@ arb_mat_approx_solve_tril_classical(arb_mat_t X,
m = B->c;
arb_init(s);
tmp = _arb_vec_init(n);
tmp = flint_malloc(sizeof(arb_struct) * n);
for (i = 0; i < m; i++)
{
for (j = 0; j < n; j++)
arb_approx_set(tmp + j, arb_mat_entry(X, j, i));
tmp[j] = *arb_mat_entry(X, j, i);
for (j = 0; j < n; j++)
{
arb_zero(s);
for (k = 0; k < j; k++)
arb_approx_addmul(s, L->rows[j] + k, tmp + k, prec);
arb_approx_sub(s, arb_mat_entry(B, j, i), s, prec);
arb_approx_dot(s, arb_mat_entry(B, j, i), 1, L->rows[j], 1, tmp, 1, j, prec);
if (!unit)
arb_approx_div(s, s, arb_mat_entry(L, j, j), prec);
arb_approx_set(tmp + j, s);
arb_approx_div(tmp + j, s, arb_mat_entry(L, j, j), prec);
else
arb_swap(tmp + j, s);
}
for (j = 0; j < n; j++)
arb_approx_set(arb_mat_entry(X, j, i), tmp + j);
*arb_mat_entry(X, j, i) = tmp[j];
}
_arb_vec_clear(tmp, n);
flint_free(tmp);
arb_clear(s);
}
@ -107,8 +86,7 @@ arb_mat_approx_solve_tril_recursive(arb_mat_t X,
/* arb_mat_submul(XY, BY, LC, XX); */
arb_mat_init(T, LC->r, BX->c);
arb_mat_mul(T, LC, XX, prec);
arb_mat_get_mid(T, T);
arb_mat_approx_mul(T, LC, XX, prec);
arb_mat_sub(XY, BY, T, prec);
arb_mat_get_mid(XY, XY);
arb_mat_clear(T);
@ -128,9 +106,8 @@ void
arb_mat_approx_solve_tril(arb_mat_t X, const arb_mat_t L,
const arb_mat_t B, int unit, slong prec)
{
if (B->r < 8 || B->c < 8)
if (B->r < 40 || B->c < 40)
arb_mat_approx_solve_tril_classical(X, L, B, unit, prec);
else
arb_mat_approx_solve_tril_recursive(X, L, B, unit, prec);
}

47
arb_mat/approx_solve_triu.c
View file

@ -11,26 +11,6 @@
#include "arb_mat.h"
static void
arb_approx_set(arb_t z, const arb_t x)
{
arf_set(arb_midref(z), arb_midref(x));
}
static void
arb_approx_sub(arb_t z, const arb_t x, const arb_t y, slong prec)
{
arf_sub(arb_midref(z),
arb_midref(x), arb_midref(y), prec, ARF_RND_DOWN);
}
static void
arb_approx_addmul(arb_t z, const arb_t x, const arb_t y, slong prec)
{
arf_addmul(arb_midref(z),
arb_midref(x), arb_midref(y), prec, ARF_RND_DOWN);
}
static void
arb_approx_div(arb_t z, const arb_t x, const arb_t y, slong prec)
{
@ -41,39 +21,36 @@ void
arb_mat_approx_solve_triu_classical(arb_mat_t X, const arb_mat_t U,
const arb_mat_t B, int unit, slong prec)
{
slong i, j, k, n, m;
slong i, j, n, m;
arb_ptr tmp;
arb_t s;
n = U->r;
m = B->c;
tmp = _arb_vec_init(n);
arb_init(s);
tmp = flint_malloc(sizeof(arb_struct) * n);
for (i = 0; i < m; i++)
{
for (j = 0; j < n; j++)
arb_approx_set(tmp + j, arb_mat_entry(X, j, i));
tmp[j] = *arb_mat_entry(X, j, i);
for (j = n - 1; j >= 0; j--)
{
arb_zero(s);
for (k = 0; k < n - j - 1; k++)
arb_approx_addmul(s, U->rows[j] + j + 1 + k, tmp + j + 1 + k, prec);
arb_approx_dot(s, arb_mat_entry(B, j, i), 1, U->rows[j] + j + 1, 1, tmp + j + 1, 1, n - j - 1, prec);
arb_approx_sub(s, arb_mat_entry(B, j, i), s, prec);
if (!unit)
arb_approx_div(s, s, arb_mat_entry(U, j, j), prec);
arb_approx_set(tmp + j, s);
arb_approx_div(tmp + j, s, arb_mat_entry(U, j, j), prec);
else
arb_swap(tmp + j, s);
}
for (j = 0; j < n; j++)
arb_approx_set(arb_mat_entry(X, j, i), tmp + j);
*arb_mat_entry(X, j, i) = tmp[j];
}
_arb_vec_clear(tmp, n);
flint_free(tmp);
arb_clear(s);
}
@ -108,8 +85,7 @@ arb_mat_approx_solve_triu_recursive(arb_mat_t X,
arb_mat_approx_solve_triu(XY, UD, BY, unit, prec);
arb_mat_init(T, UB->r, XY->c);
arb_mat_mul(T, UB, XY, prec);
arb_mat_get_mid(T, T);
arb_mat_approx_mul(T, UB, XY, prec);
arb_mat_sub(XX, BX, T, prec);
arb_mat_get_mid(XX, XX);
arb_mat_clear(T);
@ -129,9 +105,8 @@ void
arb_mat_approx_solve_triu(arb_mat_t X, const arb_mat_t U,
const arb_mat_t B, int unit, slong prec)
{
if (B->r < 8 || B->c < 8)
if (B->r < 40 || B->c < 40)
arb_mat_approx_solve_triu_classical(X, U, B, unit, prec);
else
arb_mat_approx_solve_triu_recursive(X, U, B, unit, prec);
}

33
arb_mat/solve_lu_precomp.c
View file

@ -15,7 +15,7 @@ void
arb_mat_solve_lu_precomp(arb_mat_t X, const slong * perm,
const arb_mat_t A, const arb_mat_t B, slong prec)
{
slong i, c, n, m;
slong i, j, c, n, m;
n = arb_mat_nrows(X);
m = arb_mat_ncols(X);
@ -46,6 +46,37 @@ arb_mat_solve_lu_precomp(arb_mat_t X, const slong * perm,
}
}
/* solve_tril and solve_triu have some overhead */
if (n >= 4)
{
arb_mat_solve_tril(X, A, X, 1, prec);
arb_mat_solve_triu(X, A, X, 0, prec);
return;
}
for (c = 0; c < m; c++)
{
/* solve Ly = b */
for (i = 1; i < n; i++)
{
for (j = 0; j < i; j++)
{
arb_submul(arb_mat_entry(X, i, c),
arb_mat_entry(A, i, j), arb_mat_entry(X, j, c), prec);
}
}
/* solve Ux = y */
for (i = n - 1; i >= 0; i--)
{
for (j = i + 1; j < n; j++)
{
arb_submul(arb_mat_entry(X, i, c),
arb_mat_entry(A, i, j), arb_mat_entry(X, j, c), prec);
}
arb_div(arb_mat_entry(X, i, c), arb_mat_entry(X, i, c),
arb_mat_entry(A, i, i), prec);
}
}
}

6
doc/source/acb.rst
View file

@ -539,6 +539,12 @@ Dot product
final rounding. This can be extremely slow and is only intended
for testing.
.. function:: void acb_approx_dot(acb_t res, const acb_t s, int subtract, acb_srcptr x, slong xstep, acb_srcptr y, slong ystep, slong len, slong prec)
Computes an approximate dot product *without error bounds*.
The radii of the inputs are ignored (only the midpoints are read)
and only the midpoint of the output is written.
Mathematical constants
-------------------------------------------------------------------------------

6
doc/source/acb_mat.rst
View file

@ -487,6 +487,12 @@ Component and error operations
Approximate solving
-------------------------------------------------------------------------------
.. function:: void acb_mat_approx_mul(acb_mat_t res, const acb_mat_t mat1, const acb_mat_t mat2, slong prec)
Approximate matrix multiplication. The input radii are ignored and
the output matrix is set to an approximate floating-point result.
The radii in the output matrix will *not* necessarily be zeroed.
.. function:: void acb_mat_approx_solve_triu(acb_mat_t X, const acb_mat_t U, const acb_mat_t B, int unit, slong prec)
.. function:: void acb_mat_approx_solve_tril(acb_mat_t X, const acb_mat_t L, const acb_mat_t B, int unit, slong prec)

6
doc/source/arb.rst
View file

@ -822,6 +822,12 @@ Dot product
final rounding. This can be extremely slow and is only intended
for testing.
.. function:: void arb_approx_dot(arb_t res, const arb_t s, int subtract, arb_srcptr x, slong xstep, arb_srcptr y, slong ystep, slong len, slong prec)
Computes an approximate dot product *without error bounds*.
The radii of the inputs are ignored (only the midpoints are read)
and only the midpoint of the output is written.
Powers and roots
-------------------------------------------------------------------------------

6
doc/source/arb_mat.rst
View file

@ -663,6 +663,12 @@ Component and error operations
Approximate solving
-------------------------------------------------------------------------------
.. function:: void arb_mat_approx_mul(arb_mat_t res, const arb_mat_t mat1, const arb_mat_t mat2, slong prec)
Approximate matrix multiplication. The input radii are ignored and
the output matrix is set to an approximate floating-point result.
The radii in the output matrix will *not* necessarily be zeroed.
.. function:: void arb_mat_approx_solve_triu(arb_mat_t X, const arb_mat_t U, const arb_mat_t B, int unit, slong prec)
.. function:: void arb_mat_approx_solve_tril(arb_mat_t X, const arb_mat_t L, const arb_mat_t B, int unit, slong prec)