On Wednesday 24 December 2008 00:31:50 Bill Hart wrote:
> On sage.math:
>
> cd tune
> make tune
>
> ./.libs/libspeed.a(gcd_bin.o): In function `__gmpn_ngcd_matrix_init':
> gcd_bin.c:(.text+0x0): multiple definition of `__gmpn_ngcd_matrix_init'
> gcd.o:gcd.c:(.text+0x0): first defined here
> ./.libs/libspeed.a(gcd_bin.o): In function `__gmpn_nhgcd_itch':
> gcd_bin.c:(.text+0x80): multiple definition of `__gmpn_nhgcd_itch'
> gcd.o:gcd.c:(.text+0x80): first defined here
> ./.libs/libspeed.a(gcd_bin.o): In function `__gmpn_nhgcd':
> gcd_bin.c:(.text+0xc4): multiple definition of `__gmpn_nhgcd'
> gcd.o:gcd.c:(.text+0xc4): first defined here
> ./.libs/libspeed.a(gcd_bin.o): In function `mpn_basic_gcd':
> gcd_bin.c:(.text+0x2ed): multiple definition of `mpn_basic_gcd'
> gcd.o:gcd.c:(.text+0x2ed): first defined here
On my K8-linux same problem with make speed and tune
./configure && make && make check passes OK , and I got bored running ./try
mpn_gcd :)
get this warning from the build though
ngcd.c: In function 'mpn_ngcd':
ngcd.c:75: warning: implicit declaration of function 'mpn_basic_gcd'
gcc -std=gnu99 -DHAVE_CONFIG_H -I. -I.. -D__GMP_WITHIN_GMP -I.. -O2 -m64
-march=k8 -mtune=k8 -c
gcd.c -fPIC -DPIC -o .libs/gcd.o
gcd.c: In function 'mpz_rgcd':
gcd.c:167: warning: implicit declaration of function 'mpn_rgcd'
gcd.c: In function 'mpz_bgcd':
gcd.c:171: warning: implicit declaration of function 'mpn_bgcd'
gcd.c: In function 'mpz_sgcd':
gcd.c:175: warning: implicit declaration of function 'mpn_sgcd'
gcd.c: In function 'mpz_ngcd':
gcd.c:179: warning: implicit declaration of function 'mpn_ngcd'
./configure --enable-alloca=debug --enable-assert && make && make check
passes OK
but
./configure --enable-alloca=debug --enable-assert --enable-nails=2 && make &&
make check
PASS: t-mul_i
PASS: t-tdiv
PASS: t-tdiv_ui
PASS: t-fdiv
PASS: t-fdiv_ui
PASS: t-cdiv_ui
nhgcd2.c:206: GNU MP assertion failed: h0 == h1
/bin/sh: line 4: 31599 Aborted ${dir}$tst
FAIL: t-gcd
PASS: t-gcd_ui
nhgcd2.c:206: GNU MP assertion failed: h0 == h1
/bin/sh: line 4: 31647 Aborted ${dir}$tst
FAIL: t-lcm
PASS: dive
PASS: dive_ui
PASS: t-sqrtrem
PASS: convert
PASS: io
>
> Bill.
>
> 2008/12/23 Jason Martin <jason.worth.mar...@gmail.com>:
> > Attached are some edited versions of
> >
> > mpn/generic/gcd.c
> >
> > and
> >
> > mpn/generic/ngcd.c
> >
> > Drop them in, test them for correctness and speed. Let me know what
> > breaks. When everyone is happy, I'll check them in to svn
> >
> > --jason
> >
> > Jason Worth Martin
> > Asst. Professor of Mathematics
> > http://www.math.jmu.edu/~martin
> >
> >
> >
> > /* Schönhage's 1987 algorithm, reorganized into hgcd form */
> >
> > #include <stdio.h> /* for NULL */
> >
> > #include "gmp.h"
> > #include "gmp-impl.h"
> > #include "longlong.h"
> >
> >
> > /* ******************************************************************
> > * Here we are including the original GMP version of mpn_gcd
> > * but we rename it as mpn_basic_gcd. It needs to be available
> > * for the ngcd algorithm to call in the base case.
> > *
> > * Preconditions [U = (up, usize) and V = (vp, vsize)]:
> > *
> > * 1. V is odd.
> > * 2. numbits(U) >= numbits(V).
> > *
> > * Both U and V are destroyed by the operation. The result is left at
> > vp, * and its size is returned.
> > *
> > * Ken Weber (kwe...@mat.ufrgs.br, kwe...@mcs.kent.edu)
> > *
> > * Funding for this work has been partially provided by Conselho
> > * Nacional de Desenvolvimento Cienti'fico e Tecnolo'gico (CNPq) do
> > * Brazil, Grant 301314194-2, and was done while I was a visiting
> > * reseacher in the Instituto de Matema'tica at Universidade Federal
> > * do Rio Grande do Sul (UFRGS).
> > *
> > * Refer to K. Weber, The accelerated integer GCD algorithm, ACM
> > * Transactions on Mathematical Software, v. 21 (March), 1995,
> > * pp. 111-122.
> > *
> > * *****************************************************************/
> >
> >
> >
> > /* If MIN (usize, vsize) >= GCD_ACCEL_THRESHOLD, then the accelerated
> > algorithm is used, otherwise the binary algorithm is used. This may be
> > adjusted for different architectures. */
> > #ifndef GCD_ACCEL_THRESHOLD
> > #define GCD_ACCEL_THRESHOLD 5
> > #endif
> >
> > /* When U and V differ in size by more than BMOD_THRESHOLD, the
> > accelerated algorithm reduces using the bmod operation. Otherwise, the
> > k-ary reduction is used. 0 <= BMOD_THRESHOLD < GMP_NUMB_BITS. */
> > enum
> > {
> > BMOD_THRESHOLD = GMP_NUMB_BITS/2
> > };
> >
> >
> > /* Use binary algorithm to compute V <-- GCD (V, U) for usize, vsize ==
> > 2. Both U and V must be odd. */
> > static inline mp_size_t
> > gcd_2 (mp_ptr vp, mp_srcptr up)
> > {
> > mp_limb_t u0, u1, v0, v1;
> > mp_size_t vsize;
> >
> > u0 = up[0];
> > u1 = up[1];
> > v0 = vp[0];
> > v1 = vp[1];
> >
> > while (u1 != v1 && u0 != v0)
> > {
> > unsigned long int r;
> > if (u1 > v1)
> > {
> > u1 -= v1 + (u0 < v0);
> > u0 = (u0 - v0) & GMP_NUMB_MASK;
> > count_trailing_zeros (r, u0);
> > u0 = ((u1 << (GMP_NUMB_BITS - r)) & GMP_NUMB_MASK) | (u0 >> r);
> > u1 >>= r;
> > }
> > else /* u1 < v1. */
> > {
> > v1 -= u1 + (v0 < u0);
> > v0 = (v0 - u0) & GMP_NUMB_MASK;
> > count_trailing_zeros (r, v0);
> > v0 = ((v1 << (GMP_NUMB_BITS - r)) & GMP_NUMB_MASK) | (v0 >> r);
> > v1 >>= r;
> > }
> > }
> >
> > vp[0] = v0, vp[1] = v1, vsize = 1 + (v1 != 0);
> >
> > /* If U == V == GCD, done. Otherwise, compute GCD (V, |U - V|). */
> > if (u1 == v1 && u0 == v0)
> > return vsize;
> >
> > v0 = (u0 == v0) ? (u1 > v1) ? u1-v1 : v1-u1 : (u0 > v0) ? u0-v0 : v0-u0;
> > vp[0] = mpn_gcd_1 (vp, vsize, v0);
> >
> > return 1;
> > }
> >
> > /* The function find_a finds 0 < N < 2^GMP_NUMB_BITS such that there
> > exists 0 < |D| < 2^GMP_NUMB_BITS, and N == D * C mod 2^(2*GMP_NUMB_BITS).
> > In the reference article, D was computed along with N, but it is better
> > to compute D separately as D <-- N / C mod 2^(GMP_NUMB_BITS + 1),
> > treating the result as a twos' complement signed integer.
> >
> > Initialize N1 to C mod 2^(2*GMP_NUMB_BITS). According to the reference
> > article, N2 should be initialized to 2^(2*GMP_NUMB_BITS), but we use
> > 2^(2*GMP_NUMB_BITS) - N1 to start the calculations within double
> > precision. If N2 > N1 initially, the first iteration of the while loop
> > will swap them. In all other situations, N1 >= N2 is maintained. */
> >
> > #if HAVE_NATIVE_mpn_gcd_finda
> > #define find_a(cp) mpn_gcd_finda (cp)
> >
> > #else
> > static
> > #if ! defined (__i386__)
> > inline /* don't inline this for the x86 */
> > #endif
> > mp_limb_t
> > find_a (mp_srcptr cp)
> > {
> > unsigned long int leading_zero_bits = 0;
> >
> > mp_limb_t n1_l = cp[0]; /* N1 == n1_h * 2^GMP_NUMB_BITS + n1_l.
> > */ mp_limb_t n1_h = cp[1];
> >
> > mp_limb_t n2_l = (-n1_l & GMP_NUMB_MASK); /* N2 == n2_h *
> > 2^GMP_NUMB_BITS + n2_l. */ mp_limb_t n2_h = (~n1_h & GMP_NUMB_MASK);
> >
> > /* Main loop. */
> > while (n2_h != 0) /* While N2 >= 2^GMP_NUMB_BITS. */
> > {
> > /* N1 <-- N1 % N2. */
> > if (((GMP_NUMB_HIGHBIT >> leading_zero_bits) & n2_h) == 0)
> > {
> > unsigned long int i;
> > count_leading_zeros (i, n2_h);
> > i -= GMP_NAIL_BITS;
> > i -= leading_zero_bits;
> > leading_zero_bits += i;
> > n2_h = ((n2_h << i) & GMP_NUMB_MASK) | (n2_l >> (GMP_NUMB_BITS -
> > i)); n2_l = (n2_l << i) & GMP_NUMB_MASK;
> > do
> > {
> > if (n1_h > n2_h || (n1_h == n2_h && n1_l >= n2_l))
> > {
> > n1_h -= n2_h + (n1_l < n2_l);
> > n1_l = (n1_l - n2_l) & GMP_NUMB_MASK;
> > }
> > n2_l = (n2_l >> 1) | ((n2_h << (GMP_NUMB_BITS - 1)) &
> > GMP_NUMB_MASK); n2_h >>= 1;
> > i -= 1;
> > }
> > while (i != 0);
> > }
> > if (n1_h > n2_h || (n1_h == n2_h && n1_l >= n2_l))
> > {
> > n1_h -= n2_h + (n1_l < n2_l);
> > n1_l = (n1_l - n2_l) & GMP_NUMB_MASK;
> > }
> >
> > MP_LIMB_T_SWAP (n1_h, n2_h);
> > MP_LIMB_T_SWAP (n1_l, n2_l);
> > }
> >
> > return n2_l;
> > }
> > #endif
> >
> >
> > mp_size_t
> > mpn_basic_gcd (mp_ptr gp, mp_ptr up, mp_size_t usize, mp_ptr vp,
> > mp_size_t vsize) {
> > mp_ptr orig_vp = vp;
> > mp_size_t orig_vsize = vsize;
> > int binary_gcd_ctr; /* Number of times binary gcd will
> > execute. */ mp_size_t scratch;
> > mp_ptr tp;
> > TMP_DECL;
> >
> > ASSERT (usize >= 1);
> > ASSERT (vsize >= 1);
> > ASSERT (usize >= vsize);
> > ASSERT (vp[0] & 1);
> > ASSERT (up[usize - 1] != 0);
> > ASSERT (vp[vsize - 1] != 0);
> > #if WANT_ASSERT
> > if (usize == vsize)
> > {
> > int uzeros, vzeros;
> > count_leading_zeros (uzeros, up[usize - 1]);
> > count_leading_zeros (vzeros, vp[vsize - 1]);
> > ASSERT (uzeros <= vzeros);
> > }
> > #endif
> > ASSERT (! MPN_OVERLAP_P (up, usize, vp, vsize));
> > ASSERT (MPN_SAME_OR_SEPARATE2_P (gp, vsize, up, usize));
> > ASSERT (MPN_SAME_OR_SEPARATE2_P (gp, vsize, vp, vsize));
> >
> > TMP_MARK;
> >
> > /* Use accelerated algorithm if vsize is over GCD_ACCEL_THRESHOLD.
> > Two EXTRA limbs for U and V are required for kary reduction. */
> > if (vsize >= GCD_ACCEL_THRESHOLD)
> > {
> > unsigned long int vbitsize, d;
> > mp_ptr orig_up = up;
> > mp_size_t orig_usize = usize;
> > mp_ptr anchor_up = (mp_ptr) TMP_ALLOC ((usize + 2) *
> > BYTES_PER_MP_LIMB);
> >
> > MPN_COPY (anchor_up, orig_up, usize);
> > up = anchor_up;
> >
> > count_leading_zeros (d, up[usize - 1]);
> > d -= GMP_NAIL_BITS;
> > d = usize * GMP_NUMB_BITS - d;
> > count_leading_zeros (vbitsize, vp[vsize - 1]);
> > vbitsize -= GMP_NAIL_BITS;
> > vbitsize = vsize * GMP_NUMB_BITS - vbitsize;
> > ASSERT (d >= vbitsize);
> > d = d - vbitsize + 1;
> >
> > /* Use bmod reduction to quickly discover whether V divides U. */
> > up[usize++] = 0; /* Insert leading zero.
> > */ mpn_bdivmod (up, up, usize, vp, vsize, d);
> >
> > /* Now skip U/V mod 2^d and any low zero limbs. */
> > d /= GMP_NUMB_BITS, up += d, usize -= d;
> > while (usize != 0 && up[0] == 0)
> > up++, usize--;
> >
> > if (usize == 0) /* GCD == ORIG_V. */
> > goto done;
> >
> > vp = (mp_ptr) TMP_ALLOC ((vsize + 2) * BYTES_PER_MP_LIMB);
> > MPN_COPY (vp, orig_vp, vsize);
> >
> > do /* Main loop. */
> > {
> > /* mpn_com_n can't be used here because anchor_up and up may
> > partially overlap */
> > if ((up[usize - 1] & GMP_NUMB_HIGHBIT) != 0) /* U < 0; take
> > twos' compl. */ {
> > mp_size_t i;
> > anchor_up[0] = -up[0] & GMP_NUMB_MASK;
> > for (i = 1; i < usize; i++)
> > anchor_up[i] = (~up[i] & GMP_NUMB_MASK);
> > up = anchor_up;
> > }
> >
> > MPN_NORMALIZE_NOT_ZERO (up, usize);
> >
> > if ((up[0] & 1) == 0) /* Result even; remove
> > twos. */ {
> > unsigned int r;
> > count_trailing_zeros (r, up[0]);
> > mpn_rshift (anchor_up, up, usize, r);
> > usize -= (anchor_up[usize - 1] == 0);
> > }
> > else if (anchor_up != up)
> > MPN_COPY_INCR (anchor_up, up, usize);
> >
> > MPN_PTR_SWAP (anchor_up,usize, vp,vsize);
> > up = anchor_up;
> >
> > if (vsize <= 2) /* Kary can't handle < 2 limbs and
> > */ break; /* isn't efficient for == 2 limbs. */
> >
> > d = vbitsize;
> > count_leading_zeros (vbitsize, vp[vsize - 1]);
> > vbitsize -= GMP_NAIL_BITS;
> > vbitsize = vsize * GMP_NUMB_BITS - vbitsize;
> > d = d - vbitsize + 1;
> >
> > if (d > BMOD_THRESHOLD) /* Bmod reduction. */
> > {
> > up[usize++] = 0;
> > mpn_bdivmod (up, up, usize, vp, vsize, d);
> > d /= GMP_NUMB_BITS, up += d, usize -= d;
> > }
> > else /* Kary reduction. */
> > {
> > mp_limb_t bp[2], cp[2];
> >
> > /* C <-- V/U mod 2^(2*GMP_NUMB_BITS). */
> > {
> > mp_limb_t u_inv, hi, lo;
> > modlimb_invert (u_inv, up[0]);
> > cp[0] = (vp[0] * u_inv) & GMP_NUMB_MASK;
> > umul_ppmm (hi, lo, cp[0], up[0] << GMP_NAIL_BITS);
> > lo >>= GMP_NAIL_BITS;
> > cp[1] = (vp[1] - hi - cp[0] * up[1]) * u_inv &
> > GMP_NUMB_MASK; }
> >
> > /* U <-- find_a (C) * U. */
> > up[usize] = mpn_mul_1 (up, up, usize, find_a (cp));
> > usize++;
> >
> > /* B <-- A/C == U/V mod 2^(GMP_NUMB_BITS + 1).
> > bp[0] <-- U/V mod 2^GMP_NUMB_BITS and
> > bp[1] <-- ( (U - bp[0] * V)/2^GMP_NUMB_BITS ) / V mod 2
> >
> > Like V/U above, but simplified because only the low bit
> > of bp[1] is wanted. */
> > {
> > mp_limb_t v_inv, hi, lo;
> > modlimb_invert (v_inv, vp[0]);
> > bp[0] = (up[0] * v_inv) & GMP_NUMB_MASK;
> > umul_ppmm (hi, lo, bp[0], vp[0] << GMP_NAIL_BITS);
> > lo >>= GMP_NAIL_BITS;
> > bp[1] = (up[1] + hi + (bp[0] & vp[1])) & 1;
> > }
> >
> > up[usize++] = 0;
> > if (bp[1] != 0) /* B < 0: U <-- U + (-B) * V. */
> > {
> > mp_limb_t c = mpn_addmul_1 (up, vp, vsize, -bp[0] &
> > GMP_NUMB_MASK); mpn_add_1 (up + vsize, up + vsize, usize - vsize, c); }
> > else /* B >= 0: U <-- U - B * V. */
> > {
> > mp_limb_t b = mpn_submul_1 (up, vp, vsize, bp[0]);
> > mpn_sub_1 (up + vsize, up + vsize, usize - vsize, b);
> > }
> >
> > up += 2, usize -= 2; /* At least two low limbs are zero.
> > */ }
> >
> > /* Must remove low zero limbs before complementing. */
> > while (usize != 0 && up[0] == 0)
> > up++, usize--;
> > }
> > while (usize != 0);
> >
> > /* Compute GCD (ORIG_V, GCD (ORIG_U, V)). Binary will execute
> > twice. */ up = orig_up, usize = orig_usize;
> > binary_gcd_ctr = 2;
> > }
> > else
> > binary_gcd_ctr = 1;
> >
> > scratch = MPN_NGCD_LEHMER_ITCH (vsize);
> > if (usize + 1 > scratch)
> > scratch = usize + 1;
> >
> > tp = TMP_ALLOC_LIMBS (scratch);
> >
> > /* Finish up with the binary algorithm. Executes once or twice. */
> > for ( ; binary_gcd_ctr--; up = orig_vp, usize = orig_vsize)
> > {
> > #if 1
> > if (usize > vsize)
> > {
> > /* FIXME: Could use mpn_bdivmod. */
> > mp_size_t rsize;
> >
> > mpn_tdiv_qr (tp + vsize, tp, 0, up, usize, vp, vsize);
> > rsize = vsize;
> > MPN_NORMALIZE (tp, rsize);
> > if (rsize == 0)
> > continue;
> >
> > MPN_COPY (up, tp, vsize);
> > }
> > vsize = mpn_ngcd_lehmer (vp, up, vp, vsize, tp);
> > #else
> > if (usize > 2) /* First make U close to V in size. */
> > {
> > unsigned long int vbitsize, d;
> > count_leading_zeros (d, up[usize - 1]);
> > d -= GMP_NAIL_BITS;
> > d = usize * GMP_NUMB_BITS - d;
> > count_leading_zeros (vbitsize, vp[vsize - 1]);
> > vbitsize -= GMP_NAIL_BITS;
> > vbitsize = vsize * GMP_NUMB_BITS - vbitsize;
> > d = d - vbitsize - 1;
> > if (d != -(unsigned long int)1 && d > 2)
> > {
> > mpn_bdivmod (up, up, usize, vp, vsize, d); /* Result > 0.
> > */ d /= (unsigned long int)GMP_NUMB_BITS, up += d, usize -= d; }
> > }
> >
> > /* Start binary GCD. */
> > do
> > {
> > mp_size_t zeros;
> >
> > /* Make sure U is odd. */
> > MPN_NORMALIZE (up, usize);
> > while (up[0] == 0)
> > up += 1, usize -= 1;
> > if ((up[0] & 1) == 0)
> > {
> > unsigned int r;
> > count_trailing_zeros (r, up[0]);
> > mpn_rshift (up, up, usize, r);
> > usize -= (up[usize - 1] == 0);
> > }
> >
> > /* Keep usize >= vsize. */
> > if (usize < vsize)
> > MPN_PTR_SWAP (up, usize, vp, vsize);
> >
> > if (usize <= 2) /* Double
> > precision. */ {
> > if (vsize == 1)
> > vp[0] = mpn_gcd_1 (up, usize, vp[0]);
> > else
> > vsize = gcd_2 (vp, up);
> > break; /* Binary GCD
> > done. */ }
> >
> > /* Count number of low zero limbs of U - V. */
> > for (zeros = 0; up[zeros] == vp[zeros] && ++zeros != vsize; )
> > continue;
> >
> > /* If U < V, swap U and V; in any case, subtract V from U. */
> > if (zeros == vsize) /* Subtract done.
> > */ up += zeros, usize -= zeros;
> > else if (usize == vsize)
> > {
> > mp_size_t size = vsize;
> > do
> > size--;
> > while (up[size] == vp[size]);
> > if (up[size] < vp[size]) /* usize == vsize.
> > */ MP_PTR_SWAP (up, vp);
> > up += zeros, usize = size + 1 - zeros;
> > mpn_sub_n (up, up, vp + zeros, usize);
> > }
> > else
> > {
> > mp_size_t size = vsize - zeros;
> > up += zeros, usize -= zeros;
> > if (mpn_sub_n (up, up, vp + zeros, size))
> > {
> > while (up[size] == 0) /* Propagate
> > borrow. */ up[size++] = -(mp_limb_t)1;
> > up[size] -= 1;
> > }
> > }
> > }
> > while (usize); /* End binary GCD.
> > */ #endif
> > }
> >
> > done:
> > if (vp != gp)
> > MPN_COPY_INCR (gp, vp, vsize);
> > TMP_FREE;
> > return vsize;
> > }
> >
> >
> >
> > /* ******************************************************************
> > * END of original GMP mpn_gcd
> > * *****************************************************************/
> >
> >
> >
> >
> >
> > /* For input of size n, matrix elements are of size at most ceil(n/2)
> > - 1, but we need one limb extra. */
> >
> > void
> > mpn_ngcd_matrix_init (struct ngcd_matrix *M, mp_size_t n, mp_ptr p)
> > {
> > mp_size_t s = (n+1)/2;
> > M->alloc = s;
> > M->n = 1;
> > MPN_ZERO (p, 4 * s);
> > M->p[0][0] = p;
> > M->p[0][1] = p + s;
> > M->p[1][0] = p + 2 * s;
> > M->p[1][1] = p + 3 * s;
> > M->tp = p + 4*s;
> >
> > M->p[0][0][0] = M->p[1][1][0] = 1;
> > }
> >
> > #define NHGCD_BASE_ITCH MPN_NGCD_STEP_ITCH
> >
> > /* Reduces a,b until |a-b| fits in n/2 + 1 limbs. Constructs matrix M
> > with elements of size at most (n+1)/2 - 1. Returns new size of a,
> > b, or zero if no reduction is possible. */
> > static mp_size_t
> > nhgcd_base (mp_ptr ap, mp_ptr bp, mp_size_t n,
> > struct ngcd_matrix *M, mp_ptr tp)
> > {
> > mp_size_t s = n/2 + 1;
> > mp_size_t nn;
> >
> > ASSERT (n > s);
> > ASSERT (ap[n-1] > 0 || bp[n-1] > 0);
> >
> > nn = mpn_ngcd_step (n, ap, bp, s, M, tp);
> > if (!nn)
> > return 0;
> >
> > for (;;)
> > {
> > n = nn;
> > ASSERT (n > s);
> > nn = mpn_ngcd_step (n, ap, bp, s, M, tp);
> > if (!nn )
> > return n;
> > }
> > }
> >
> > /* Size analysis for nhgcd:
> >
> > For the recursive calls, we have n1 <= ceil(n / 2). Then the
> > storage need is determined by the storage for the recursive call
> > computing M1, and ngcd_matrix_adjust and ngcd_matrix_mul calls that use
> > M1 (after this, the storage needed for M1 can be recycled).
> >
> > Let S(r) denote the required storage. For M1 we need 5 * ceil(n1/2)
> > = 5 * ceil(n/4), and for the ngcd_matrix_adjust call, we need n + 2. In
> > total, 5 * ceil(n/4) + n + 2 <= 9 ceil(n/4) + 2.
> >
> > For the recursive call, we need S(n1) = S(ceil(n/2)).
> >
> > S(n) <= 9*ceil(n/4) + 2 + S(ceil(n/2))
> > <= 9*(ceil(n/4) + ... + ceil(n/2^(1+k))) + 2k + S(ceil(n/2^k))
> > <= 9*(2 ceil(n/4) + k) + 2k + S(n/2^k)
> > <= 18 ceil(n/4) + 11k + S(n/2^k)
> >
> > */
> >
> > mp_size_t
> > mpn_nhgcd_itch (mp_size_t n)
> > {
> > unsigned k;
> > mp_size_t nn;
> >
> > /* Inefficient way to almost compute
> > log_2(n/NHGCD_BASE_THRESHOLD) */
> > for (k = 0, nn = n;
> > ABOVE_THRESHOLD (nn, NHGCD_THRESHOLD);
> > nn = (nn + 1) / 2)
> > k++;
> >
> > if (k == 0)
> > return NHGCD_BASE_ITCH (n);
> >
> > return 18 * ((n+3) / 4) + 11 * k
> > + NHGCD_BASE_ITCH (NHGCD_THRESHOLD);
> > }
> >
> > /* Reduces a,b until |a-b| fits in n/2 + 1 limbs. Constructs matrix M
> > with elements of size at most (n+1)/2 - 1. Returns new size of a,
> > b, or zero if no reduction is possible. */
> >
> > mp_size_t
> > mpn_nhgcd (mp_ptr ap, mp_ptr bp, mp_size_t n,
> > struct ngcd_matrix *M, mp_ptr tp)
> > {
> > mp_size_t s = n/2 + 1;
> > mp_size_t n2 = (3*n)/4 + 1;
> >
> > mp_size_t p, nn;
> > unsigned count;
> > int success = 0;
> >
> > ASSERT (n > s);
> > ASSERT ((ap[n-1] | bp[n-1]) > 0);
> >
> > ASSERT ((n+1)/2 - 1 < M->alloc);
> >
> > if (BELOW_THRESHOLD (n, NHGCD_THRESHOLD))
> > return nhgcd_base (ap, bp, n, M, tp);
> >
> > p = n/2;
> > nn = mpn_nhgcd (ap + p, bp + p, n - p, M, tp);
> > if (nn > 0)
> > {
> > /* Needs 2*(p + M->n) <= 2*(floor(n/2) + ceil(n/2) - 1)
> > = 2 (n - 1) */
> > n = mpn_ngcd_matrix_adjust (M, p + nn, ap, bp, p, tp);
> > success = 1;
> > }
> > count = 0;
> > while (n > n2)
> > {
> > count++;
> > /* Needs n + 1 storage */
> > nn = mpn_ngcd_step (n, ap, bp, s, M, tp);
> > if (!nn)
> > return success ? n : 0;
> > n = nn;
> > success = 1;
> > }
> >
> > if (n > s + 2)
> > {
> > struct ngcd_matrix M1;
> > mp_size_t scratch;
> >
> > p = 2*s - n + 1;
> > scratch = MPN_NGCD_MATRIX_INIT_ITCH (n-p);
> >
> > mpn_ngcd_matrix_init(&M1, n - p, tp);
> > nn = mpn_nhgcd (ap + p, bp + p, n - p, &M1, tp + scratch);
> > if (nn > 0)
> > {
> > /* Needs 2 (p + M->n) <= 2 (2*s - n2 + 1 + n2 - s - 1)
> > = 2*s <= 2*(floor(n/2) + 1) <= n + 2. */
> > n = mpn_ngcd_matrix_adjust (&M1, p + nn, ap, bp, p, tp +
> > scratch); /* Needs M->n <= n2 - s - 1 < n/4 */
> > mpn_ngcd_matrix_mul (M, &M1, tp + scratch);
> > success = 1;
> > }
> > }
> >
> > /* FIXME: This really is the base case */
> > for (count = 0;; count++)
> > {
> > /* Needs s+3 < n */
> > nn = mpn_ngcd_step (n, ap, bp, s, M, tp);
> > if (!nn)
> > return success ? n : 0;
> >
> > n = nn;
> > success = 1;
> > }
> > }
> >
> > #define EVEN_P(x) (((x) & 1) == 0)
> >
> > mp_size_t
> > mpn_gcd (mp_ptr gp, mp_ptr ap, mp_size_t an, mp_ptr bp, mp_size_t n)
> > {
> > mp_size_t init_scratch;
> > mp_size_t scratch;
> > mp_ptr tp;
> > TMP_DECL;
> >
> > ASSERT (an >= n);
> >
> > if (BELOW_THRESHOLD (n, NGCD_THRESHOLD))
> > return mpn_basic_gcd (gp, ap, an, bp, n);
> >
> > init_scratch = MPN_NGCD_MATRIX_INIT_ITCH ((n+1)/2);
> > scratch = mpn_nhgcd_itch ((n+1)/2);
> >
> > /* Space needed for mpn_ngcd_matrix_adjust */
> > if (scratch < 2*n)
> > scratch = 2*n;
> >
> > TMP_MARK;
> >
> > if (an + 1 > init_scratch + scratch)
> > tp = TMP_ALLOC_LIMBS (an + 1);
> > else
> > tp = TMP_ALLOC_LIMBS (init_scratch + scratch);
> >
> > if (an > n)
> > {
> > mp_ptr rp = tp;
> > mp_ptr qp = rp + n;
> >
> > mpn_tdiv_qr (qp, rp, 0, ap, an, bp, n);
> > MPN_COPY (ap, rp, n);
> > an = n;
> > MPN_NORMALIZE (ap, an);
> > if (an == 0)
> > {
> > MPN_COPY (gp, bp, n);
> > TMP_FREE;
> > return n;
> > }
> > }
> >
> > while (ABOVE_THRESHOLD (n, NGCD_THRESHOLD))
> > {
> > struct ngcd_matrix M;
> > mp_size_t p = n/2;
> > mp_size_t nn;
> >
> > mpn_ngcd_matrix_init (&M, n - p, tp);
> > nn = mpn_nhgcd (ap + p, bp + p, n - p, &M, tp + init_scratch);
> > if (nn > 0)
> > /* Needs 2*(p + M->n) <= 2*(floor(n/2) + ceil(n/2) - 1)
> > = 2(n-1) */
> > n = mpn_ngcd_matrix_adjust (&M, p + nn, ap, bp, p, tp +
> > init_scratch);
> >
> > else
> > {
> > mp_size_t gn;
> > n = mpn_ngcd_subdiv_step (gp, &gn, ap, bp, n, tp);
> > if (n == 0)
> > {
> > TMP_FREE;
> > return gn;
> > }
> > }
> > }
> >
> > ASSERT (ap[n-1] > 0 || bp[n-1] > 0);
> > #if 0
> > /* FIXME: We may want to use lehmer on some systems. */
> > n = mpn_ngcd_lehmer (gp, ap, bp, n, tp);
> >
> > TMP_FREE;
> > return n;
> > #endif
> >
> > if (ap[n-1] < bp[n-1])
> > MP_PTR_SWAP (ap, bp);
> >
> > an = n;
> > MPN_NORMALIZE (bp, n);
> >
> > if (n == 0)
> > {
> > MPN_COPY (gp, ap, an);
> > TMP_FREE;
> > return an;
> > }
> >
> > if (EVEN_P (bp[0]))
> > {
> > /* Then a must be odd (since the calling convention implies that
> > there's no common factor of 2) */
> > ASSERT (!EVEN_P (ap[0]));
> >
> > while (bp[0] == 0)
> > {
> > bp++;
> > n--;
> > }
> >
> > if (EVEN_P(bp[0]))
> > {
> > int count;
> > count_trailing_zeros (count, bp[0]);
> > ASSERT_NOCARRY (mpn_rshift (bp, bp, n, count));
> > n -= (bp[n-1] == 0);
> > }
> > }
> >
> > TMP_FREE;
> > return mpn_basic_gcd (gp, ap, an, bp, n);
> > }
> >
> > /* Schönhage's 1987 algorithm, reorganized into hgcd form */
> >
> > #include <stdio.h> /* for NULL */
> >
> > #include "gmp.h"
> > #include "gmp-impl.h"
> > #include "longlong.h"
> >
> >
> >
> >
> >
> >
> > /* For input of size n, matrix elements are of size at most ceil(n/2)
> > - 1, but we need one limb extra. */
> >
> > void
> > mpn_ngcd_matrix_init (struct ngcd_matrix *M, mp_size_t n, mp_ptr p);
> >
> > #define NHGCD_BASE_ITCH MPN_NGCD_STEP_ITCH
> >
> > /* Reduces a,b until |a-b| fits in n/2 + 1 limbs. Constructs matrix M
> > with elements of size at most (n+1)/2 - 1. Returns new size of a,
> > b, or zero if no reduction is possible. */
> > static mp_size_t
> > nhgcd_base (mp_ptr ap, mp_ptr bp, mp_size_t n,
> > struct ngcd_matrix *M, mp_ptr tp);
> >
> > /* Size analysis for nhgcd:
> >
> > For the recursive calls, we have n1 <= ceil(n / 2). Then the
> > storage need is determined by the storage for the recursive call
> > computing M1, and ngcd_matrix_adjust and ngcd_matrix_mul calls that use
> > M1 (after this, the storage needed for M1 can be recycled).
> >
> > Let S(r) denote the required storage. For M1 we need 5 * ceil(n1/2)
> > = 5 * ceil(n/4), and for the ngcd_matrix_adjust call, we need n + 2. In
> > total, 5 * ceil(n/4) + n + 2 <= 9 ceil(n/4) + 2.
> >
> > For the recursive call, we need S(n1) = S(ceil(n/2)).
> >
> > S(n) <= 9*ceil(n/4) + 2 + S(ceil(n/2))
> > <= 9*(ceil(n/4) + ... + ceil(n/2^(1+k))) + 2k + S(ceil(n/2^k))
> > <= 9*(2 ceil(n/4) + k) + 2k + S(n/2^k)
> > <= 18 ceil(n/4) + 11k + S(n/2^k)
> >
> > */
> >
> > mp_size_t
> > mpn_nhgcd_itch (mp_size_t n);
> >
> >
> > /* Reduces a,b until |a-b| fits in n/2 + 1 limbs. Constructs matrix M
> > with elements of size at most (n+1)/2 - 1. Returns new size of a,
> > b, or zero if no reduction is possible. */
> >
> > mp_size_t
> > mpn_nhgcd (mp_ptr ap, mp_ptr bp, mp_size_t n,
> > struct ngcd_matrix *M, mp_ptr tp);
> >
> >
> > #define EVEN_P(x) (((x) & 1) == 0)
> >
> > mp_size_t
> > mpn_ngcd (mp_ptr gp, mp_ptr ap, mp_size_t an, mp_ptr bp, mp_size_t n)
> > {
> > mp_size_t init_scratch;
> > mp_size_t scratch;
> > mp_ptr tp;
> > TMP_DECL;
> >
> > ASSERT (an >= n);
> >
> > if (BELOW_THRESHOLD (n, NGCD_THRESHOLD))
> > return mpn_basic_gcd (gp, ap, an, bp, n);
> >
> > init_scratch = MPN_NGCD_MATRIX_INIT_ITCH ((n+1)/2);
> > scratch = mpn_nhgcd_itch ((n+1)/2);
> >
> > /* Space needed for mpn_ngcd_matrix_adjust */
> > if (scratch < 2*n)
> > scratch = 2*n;
> >
> > TMP_MARK;
> >
> > if (an + 1 > init_scratch + scratch)
> > tp = TMP_ALLOC_LIMBS (an + 1);
> > else
> > tp = TMP_ALLOC_LIMBS (init_scratch + scratch);
> >
> > if (an > n)
> > {
> > mp_ptr rp = tp;
> > mp_ptr qp = rp + n;
> >
> > mpn_tdiv_qr (qp, rp, 0, ap, an, bp, n);
> > MPN_COPY (ap, rp, n);
> > an = n;
> > MPN_NORMALIZE (ap, an);
> > if (an == 0)
> > {
> > MPN_COPY (gp, bp, n);
> > TMP_FREE;
> > return n;
> > }
> > }
> >
> > while (ABOVE_THRESHOLD (n, NGCD_THRESHOLD))
> > {
> > struct ngcd_matrix M;
> > mp_size_t p = n/2;
> > mp_size_t nn;
> >
> > mpn_ngcd_matrix_init (&M, n - p, tp);
> > nn = mpn_nhgcd (ap + p, bp + p, n - p, &M, tp + init_scratch);
> > if (nn > 0)
> > /* Needs 2*(p + M->n) <= 2*(floor(n/2) + ceil(n/2) - 1)
> > = 2(n-1) */
> > n = mpn_ngcd_matrix_adjust (&M, p + nn, ap, bp, p, tp +
> > init_scratch);
> >
> > else
> > {
> > mp_size_t gn;
> > n = mpn_ngcd_subdiv_step (gp, &gn, ap, bp, n, tp);
> > if (n == 0)
> > {
> > TMP_FREE;
> > return gn;
> > }
> > }
> > }
> >
> > ASSERT (ap[n-1] > 0 || bp[n-1] > 0);
> > #if 0
> > /* FIXME: We may want to use lehmer on some systems. */
> > n = mpn_ngcd_lehmer (gp, ap, bp, n, tp);
> >
> > TMP_FREE;
> > return n;
> > #endif
> >
> > if (ap[n-1] < bp[n-1])
> > MP_PTR_SWAP (ap, bp);
> >
> > an = n;
> > MPN_NORMALIZE (bp, n);
> >
> > if (n == 0)
> > {
> > MPN_COPY (gp, ap, an);
> > TMP_FREE;
> > return an;
> > }
> >
> > if (EVEN_P (bp[0]))
> > {
> > /* Then a must be odd (since the calling convention implies that
> > there's no common factor of 2) */
> > ASSERT (!EVEN_P (ap[0]));
> >
> > while (bp[0] == 0)
> > {
> > bp++;
> > n--;
> > }
> >
> > if (EVEN_P(bp[0]))
> > {
> > int count;
> > count_trailing_zeros (count, bp[0]);
> > ASSERT_NOCARRY (mpn_rshift (bp, bp, n, count));
> > n -= (bp[n-1] == 0);
> > }
> > }
> >
> > TMP_FREE;
> > return mpn_basic_gcd (gp, ap, an, bp, n);
> > }
>
>
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