2 #include <NTL/mat_ZZ.h>
3 #include <NTL/vec_ZZ.h>
4 #include <barvinok/barvinok.h>
5 #include <barvinok/evalue.h>
6 #include <barvinok/util.h>
8 #include "conversion.h"
9 #include "lattice_point.h"
10 #include "param_util.h"
15 #define ALLOC(type) (type*)malloc(sizeof(type))
17 /* returns an evalue that corresponds to
21 static evalue
*term(int param
, ZZ
& c
, Value
*den
= NULL
)
23 evalue
*EP
= new evalue();
25 value_set_si(EP
->d
,0);
26 EP
->x
.p
= new_enode(polynomial
, 2, param
+ 1);
27 evalue_set_si(&EP
->x
.p
->arr
[0], 0, 1);
28 value_init(EP
->x
.p
->arr
[1].x
.n
);
30 value_set_si(EP
->x
.p
->arr
[1].d
, 1);
32 value_assign(EP
->x
.p
->arr
[1].d
, *den
);
33 zz2value(c
, EP
->x
.p
->arr
[1].x
.n
);
37 /* returns an evalue that corresponds to
41 evalue
*multi_monom(vec_ZZ
& p
)
43 evalue
*X
= new evalue();
46 unsigned nparam
= p
.length()-1;
47 zz2value(p
[nparam
], X
->x
.n
);
48 value_set_si(X
->d
, 1);
49 for (int i
= 0; i
< nparam
; ++i
) {
52 evalue
*T
= term(i
, p
[i
]);
61 * Check whether mapping polyhedron P on the affine combination
62 * num yields a range that has a fixed quotient on integer
64 * If zero is true, then we are only interested in the quotient
65 * for the cases where the remainder is zero.
66 * Returns NULL if false and a newly allocated value if true.
68 static Value
*fixed_quotient(Polyhedron
*P
, vec_ZZ
& num
, Value d
, bool zero
)
71 int len
= num
.length();
72 Matrix
*T
= Matrix_Alloc(2, len
);
73 zz2values(num
, T
->p
[0]);
74 value_set_si(T
->p
[1][len
-1], 1);
75 Polyhedron
*I
= Polyhedron_Image(P
, T
, P
->NbConstraints
);
79 for (i
= 0; i
< I
->NbRays
; ++i
)
80 if (value_zero_p(I
->Ray
[i
][2])) {
88 int bounded
= line_minmax(I
, &min
, &max
);
92 mpz_cdiv_q(min
, min
, d
);
94 mpz_fdiv_q(min
, min
, d
);
95 mpz_fdiv_q(max
, max
, d
);
97 if (value_eq(min
, max
)) {
100 value_assign(*ret
, min
);
108 * Normalize linear expression coef modulo m
109 * Removes common factor and reduces coefficients
110 * Returns index of first non-zero coefficient or len
112 int normal_mod(Value
*coef
, int len
, Value
*m
)
117 Vector_Gcd(coef
, len
, &gcd
);
119 Vector_AntiScale(coef
, coef
, gcd
, len
);
121 value_division(*m
, *m
, gcd
);
128 for (j
= 0; j
< len
; ++j
)
129 mpz_fdiv_r(coef
[j
], coef
[j
], *m
);
130 for (j
= 0; j
< len
; ++j
)
131 if (value_notzero_p(coef
[j
]))
137 static bool mod_needed(Polyhedron
*PD
, vec_ZZ
& num
, Value d
, evalue
*E
)
139 Value
*q
= fixed_quotient(PD
, num
, d
, false);
144 value_oppose(*q
, *q
);
147 value_set_si(EV
.d
, 1);
149 value_multiply(EV
.x
.n
, *q
, d
);
151 free_evalue_refs(&EV
);
157 /* modifies coef argument ! */
158 static void fractional_part(Value
*coef
, int len
, Value d
, ZZ f
, evalue
*EP
,
159 Polyhedron
*PD
, bool up
)
165 /* f {{ x }} = f - f { -x } */
167 evalue_add_constant(EP
, m
);
168 Vector_Oppose(coef
, coef
, len
);
173 int j
= normal_mod(coef
, len
, &m
);
181 values2zz(coef
, num
, len
);
188 evalue_set_si(&tmp
, 0, 1);
192 while (j
< len
-1 && (num
[j
] == g
/2 || num
[j
] == 0))
194 if ((j
< len
-1 && num
[j
] > g
/2) || (j
== len
-1 && num
[j
] >= (g
+1)/2)) {
195 for (int k
= j
; k
< len
-1; ++k
)
198 num
[len
-1] = g
- 1 - num
[len
-1];
199 value_assign(tmp
.d
, m
);
201 zz2value(t
, tmp
.x
.n
);
207 ZZ t
= num
[len
-1] * f
;
208 zz2value(t
, tmp
.x
.n
);
209 value_assign(tmp
.d
, m
);
212 evalue
*E
= multi_monom(num
);
216 if (PD
&& !mod_needed(PD
, num
, m
, E
)) {
219 value_assign(EV
.d
, m
);
224 value_set_si(EV
.x
.n
, 1);
225 value_assign(EV
.d
, m
);
228 value_set_si(EV
.d
, 0);
229 EV
.x
.p
= new_enode(fractional
, 3, -1);
230 evalue_copy(&EV
.x
.p
->arr
[0], E
);
231 evalue_set_si(&EV
.x
.p
->arr
[1], 0, 1);
232 value_init(EV
.x
.p
->arr
[2].x
.n
);
233 zz2value(f
, EV
.x
.p
->arr
[2].x
.n
);
234 value_set_si(EV
.x
.p
->arr
[2].d
, 1);
239 free_evalue_refs(&EV
);
244 free_evalue_refs(&tmp
);
250 static void ceil(Value
*coef
, int len
, Value d
, ZZ
& f
,
251 evalue
*EP
, barvinok_options
*options
)
253 Vector_Oppose(coef
, coef
, len
);
254 fractional_part(coef
, len
, d
, f
, EP
, NULL
, false);
255 if (options
->lookup_table
)
256 evalue_mod2table(EP
, len
-1);
259 evalue
* bv_ceil3(Value
*coef
, int len
, Value d
, Polyhedron
*P
)
261 Vector
*val
= Vector_Alloc(len
);
266 Vector_Scale(coef
, val
->p
, t
, len
);
267 value_absolute(t
, d
);
270 values2zz(val
->p
, num
, len
);
271 evalue
*EP
= multi_monom(num
);
276 value_set_si(tmp
.x
.n
, 1);
277 value_assign(tmp
.d
, t
);
283 Vector_Oppose(val
->p
, val
->p
, len
);
284 fractional_part(val
->p
, len
, t
, one
, EP
, P
, false);
287 /* copy EP to malloc'ed evalue */
288 evalue
*E
= ALLOC(evalue
);
292 free_evalue_refs(&tmp
);
298 void lattice_point(Value
* values
, const mat_ZZ
& rays
, vec_ZZ
& vertex
, int *closed
)
300 unsigned dim
= rays
.NumRows();
301 if (value_one_p(values
[dim
]) && !closed
)
302 values2zz(values
, vertex
, dim
);
304 Matrix
* Rays
= rays2matrix(rays
);
305 Matrix
*inv
= Matrix_Alloc(Rays
->NbRows
, Rays
->NbColumns
);
306 int ok
= Matrix_Inverse(Rays
, inv
);
309 Rays
= rays2matrix(rays
);
310 Vector
*lambda
= Vector_Alloc(dim
+1);
311 Vector_Matrix_Product(values
, inv
, lambda
->p
);
313 for (int j
= 0; j
< dim
; ++j
)
314 if (!closed
|| closed
[j
])
315 mpz_cdiv_q(lambda
->p
[j
], lambda
->p
[j
], lambda
->p
[dim
]);
317 value_addto(lambda
->p
[j
], lambda
->p
[j
], lambda
->p
[dim
]);
318 mpz_fdiv_q(lambda
->p
[j
], lambda
->p
[j
], lambda
->p
[dim
]);
320 value_set_si(lambda
->p
[dim
], 1);
321 Vector
*A
= Vector_Alloc(dim
+1);
322 Vector_Matrix_Product(lambda
->p
, Rays
, A
->p
);
325 values2zz(A
->p
, vertex
, dim
);
330 #define FORALL_COSETS(det,D,i,k) \
332 Vector *k = Vector_Alloc(D->NbRows+1); \
333 value_set_si(k->p[D->NbRows], 1); \
334 for (unsigned long i = 0; i < det; ++i) { \
335 unsigned long _fc_val = i; \
336 for (int j = 0; j < D->NbRows; ++j) { \
337 value_set_si(k->p[j], _fc_val % mpz_get_ui(D->p[j][j]));\
338 _fc_val /= mpz_get_ui(D->p[j][j]); \
340 #define END_FORALL_COSETS \
345 /* Compute the lattice points in the vertex cone at "values" with rays "rays".
346 * The lattice points are returned in "vertex".
348 * Rays has the generators as rows and so does W.
349 * We first compute { m-v, u_i^* } with m = k W, where k runs through
352 * [k 1] [ d1*W 0 ] [ U' 0 ] = [k 1] T2
354 * where d1 and d2 are the denominators of v and U^{-1}=U'/d2.
355 * Then lambda = { k } (componentwise)
356 * We compute x - floor(x) = {x} = { a/b } as fdiv_r(a,b)/b
357 * For open rays/facets, we need values in (0,1] rather than [0,1),
358 * so we compute {{x}} = x - ceil(x-1) = a/b - ceil((a-b)/b)
359 * = (a - b cdiv_q(a-b,b) - b + b)/b
360 * = (cdiv_r(a,b)+b)/b
361 * Finally, we compute v + lambda * U
362 * The denominator of lambda can be d1*d2, that of lambda2 = lambda*U
363 * can be at most d1, since it is integer if v = 0.
364 * The denominator of v + lambda2 is 1.
366 void lattice_point(Value
* values
, const mat_ZZ
& rays
, mat_ZZ
& vertex
,
367 unsigned long det
, int *closed
)
369 unsigned dim
= rays
.NumRows();
370 vertex
.SetDims(det
, dim
);
372 lattice_point(values
, rays
, vertex
[0], closed
);
375 Matrix
* Rays
= rays2matrix2(rays
);
377 Smith(Rays
, &U
, &W
, &D
);
382 unsigned long det2
= 1;
383 for (int i
= 0 ; i
< D
->NbRows
; ++i
)
384 det2
*= mpz_get_ui(D
->p
[i
][i
]);
387 Matrix
*T
= Matrix_Alloc(W
->NbRows
+1, W
->NbColumns
+1);
388 for (int i
= 0; i
< W
->NbRows
; ++i
)
389 Vector_Scale(W
->p
[i
], T
->p
[i
], values
[dim
], W
->NbColumns
);
393 value_set_si(tmp
, -1);
394 Vector_Scale(values
, T
->p
[dim
], tmp
, dim
);
396 value_assign(T
->p
[dim
][dim
], values
[dim
]);
398 Rays
= rays2matrix(rays
);
399 Matrix
*inv
= Matrix_Alloc(Rays
->NbRows
, Rays
->NbColumns
);
400 int ok
= Matrix_Inverse(Rays
, inv
);
404 Matrix
*T2
= Matrix_Alloc(dim
+1, dim
+1);
405 Matrix_Product(T
, inv
, T2
);
408 Rays
= rays2matrix(rays
);
410 Vector
*lambda
= Vector_Alloc(dim
+1);
411 Vector
*lambda2
= Vector_Alloc(dim
+1);
412 FORALL_COSETS(det
, D
, i
, k
)
413 Vector_Matrix_Product(k
->p
, T2
, lambda
->p
);
414 for (int j
= 0; j
< dim
; ++j
)
415 if (!closed
|| closed
[j
])
416 mpz_fdiv_r(lambda
->p
[j
], lambda
->p
[j
], lambda
->p
[dim
]);
418 mpz_cdiv_r(lambda
->p
[j
], lambda
->p
[j
], lambda
->p
[dim
]);
419 value_addto(lambda
->p
[j
], lambda
->p
[j
], lambda
->p
[dim
]);
421 Vector_Matrix_Product(lambda
->p
, Rays
, lambda2
->p
);
422 for (int j
= 0; j
< dim
; ++j
)
423 assert(mpz_divisible_p(lambda2
->p
[j
], inv
->p
[dim
][dim
]));
424 Vector_AntiScale(lambda2
->p
, lambda2
->p
, inv
->p
[dim
][dim
], dim
+1);
425 Vector_Add(lambda2
->p
, values
, lambda2
->p
, dim
);
426 for (int j
= 0; j
< dim
; ++j
)
427 assert(mpz_divisible_p(lambda2
->p
[j
], values
[dim
]));
428 Vector_AntiScale(lambda2
->p
, lambda2
->p
, values
[dim
], dim
+1);
429 values2zz(lambda2
->p
, vertex
[i
], dim
);
432 Vector_Free(lambda2
);
440 /* Returns the power of (t+1) in the term of a rational generating function,
441 * i.e., the scalar product of the actual lattice point and lambda.
442 * The lattice point is the unique lattice point in the fundamental parallelepiped
443 * of the unimodual cone i shifted to the parametric vertex W/lcm.
445 * The rows of W refer to the coordinates of the vertex
446 * The first nparam columns are the coefficients of the parameters
447 * and the final column is the constant term.
448 * lcm is the common denominator of all coefficients.
450 static evalue
**lattice_point_fractional(const mat_ZZ
& rays
, vec_ZZ
& lambda
,
452 unsigned long det
, int *closed
)
454 unsigned nparam
= V
->NbColumns
-2;
455 evalue
**E
= new evalue
*[det
];
457 Matrix
* Rays
= rays2matrix2(rays
);
458 Matrix
*T
= Transpose(Rays
);
459 Matrix
*T2
= Matrix_Copy(T
);
460 Matrix
*inv
= Matrix_Alloc(T2
->NbRows
, T2
->NbColumns
);
461 int ok
= Matrix_Inverse(T2
, inv
);
465 matrix2zz(V
, vertex
, V
->NbRows
, V
->NbColumns
-1);
468 num
= lambda
* vertex
;
470 evalue
*EP
= multi_monom(num
);
472 evalue_div(EP
, V
->p
[0][nparam
+1]);
474 Matrix
*L
= Matrix_Alloc(inv
->NbRows
, V
->NbColumns
);
475 Matrix_Product(inv
, V
, L
);
478 matrix2zz(T
, RT
, T
->NbRows
, T
->NbColumns
);
481 vec_ZZ p
= lambda
* RT
;
484 for (int i
= 0; i
< L
->NbRows
; ++i
) {
485 Vector_Oppose(L
->p
[i
], L
->p
[i
], nparam
+1);
486 fractional_part(L
->p
[i
], nparam
+1, V
->p
[i
][nparam
+1], p
[i
],
487 EP
, NULL
, closed
&& !closed
[i
]);
491 for (int i
= 0; i
< L
->NbRows
; ++i
)
492 value_assign(L
->p
[i
][nparam
+1], V
->p
[i
][nparam
+1]);
496 mpz_set_ui(denom
, det
);
497 value_multiply(denom
, L
->p
[0][nparam
+1], denom
);
500 Smith(Rays
, &U
, &W
, &D
);
504 unsigned long det2
= 1;
505 for (int i
= 0 ; i
< D
->NbRows
; ++i
)
506 det2
*= mpz_get_ui(D
->p
[i
][i
]);
509 Matrix_Transposition(inv
);
510 Matrix
*T2
= Matrix_Alloc(W
->NbRows
, inv
->NbColumns
);
511 Matrix_Product(W
, inv
, T2
);
514 unsigned dim
= D
->NbRows
;
515 Vector
*lambda
= Vector_Alloc(dim
);
517 Vector
*row
= Vector_Alloc(nparam
+1);
518 FORALL_COSETS(det
, D
, i
, k
)
519 Vector_Matrix_Product(k
->p
, T2
, lambda
->p
);
522 evalue_copy(E
[i
], EP
);
523 for (int j
= 0; j
< L
->NbRows
; ++j
) {
524 Vector_Oppose(L
->p
[j
], row
->p
, nparam
+1);
525 value_addmul(row
->p
[nparam
], L
->p
[j
][nparam
+1], lambda
->p
[j
]);
526 fractional_part(row
->p
, nparam
+1, denom
, p
[j
],
527 E
[i
], NULL
, closed
&& !closed
[j
]);
537 free_evalue_refs(EP
);
548 static evalue
**lattice_point(const mat_ZZ
& rays
, vec_ZZ
& lambda
,
550 unsigned long det
, int *closed
,
551 barvinok_options
*options
)
553 evalue
**lp
= lattice_point_fractional(rays
, lambda
, V
->Vertex
, det
, closed
);
554 if (options
->lookup_table
) {
555 for (int i
= 0; i
< det
; ++i
)
556 evalue_mod2table(lp
[i
], V
->Vertex
->NbColumns
-2);
561 /* returns the unique lattice point in the fundamental parallelepiped
562 * of the unimodual cone C shifted to the parametric vertex V.
564 * The return values num and E_vertex are such that
565 * coordinate i of this lattice point is equal to
567 * num[i] + E_vertex[i]
569 void lattice_point(Param_Vertices
*V
, const mat_ZZ
& rays
, vec_ZZ
& num
,
570 evalue
**E_vertex
, barvinok_options
*options
)
572 unsigned nparam
= V
->Vertex
->NbColumns
- 2;
573 unsigned dim
= rays
.NumCols();
575 vertex
.SetLength(nparam
+1);
580 value_set_si(lcm
, 1);
582 for (int j
= 0; j
< V
->Vertex
->NbRows
; ++j
) {
583 value_lcm(lcm
, V
->Vertex
->p
[j
][nparam
+1], &lcm
);
586 if (value_notone_p(lcm
)) {
587 Matrix
* mv
= Matrix_Alloc(dim
, nparam
+1);
588 for (int j
= 0 ; j
< dim
; ++j
) {
589 value_division(tmp
, lcm
, V
->Vertex
->p
[j
][nparam
+1]);
590 Vector_Scale(V
->Vertex
->p
[j
], mv
->p
[j
], tmp
, nparam
+1);
593 Matrix
* Rays
= rays2matrix2(rays
);
594 Matrix
*T
= Transpose(Rays
);
595 Matrix
*T2
= Matrix_Copy(T
);
596 Matrix
*inv
= Matrix_Alloc(T2
->NbRows
, T2
->NbColumns
);
597 int ok
= Matrix_Inverse(T2
, inv
);
601 Matrix
*L
= Matrix_Alloc(inv
->NbRows
, mv
->NbColumns
);
602 Matrix_Product(inv
, mv
, L
);
611 evalue
*remainders
[dim
];
612 for (int i
= 0; i
< dim
; ++i
) {
613 remainders
[i
] = evalue_zero();
615 ceil(L
->p
[i
], nparam
+1, lcm
, one
, remainders
[i
], options
);
620 for (int i
= 0; i
< V
->Vertex
->NbRows
; ++i
) {
621 values2zz(mv
->p
[i
], vertex
, nparam
+1);
622 E_vertex
[i
] = multi_monom(vertex
);
625 value_set_si(f
.x
.n
, 1);
626 value_assign(f
.d
, lcm
);
628 emul(&f
, E_vertex
[i
]);
630 for (int j
= 0; j
< dim
; ++j
) {
631 if (value_zero_p(T
->p
[i
][j
]))
635 evalue_copy(&cp
, remainders
[j
]);
636 if (value_notone_p(T
->p
[i
][j
])) {
637 value_set_si(f
.d
, 1);
638 value_assign(f
.x
.n
, T
->p
[i
][j
]);
641 eadd(&cp
, E_vertex
[i
]);
642 free_evalue_refs(&cp
);
645 for (int i
= 0; i
< dim
; ++i
) {
646 free_evalue_refs(remainders
[i
]);
650 free_evalue_refs(&f
);
661 for (int i
= 0; i
< V
->Vertex
->NbRows
; ++i
) {
663 if (First_Non_Zero(V
->Vertex
->p
[i
], nparam
) == -1) {
665 value2zz(V
->Vertex
->p
[i
][nparam
], num
[i
]);
667 values2zz(V
->Vertex
->p
[i
], vertex
, nparam
+1);
668 E_vertex
[i
] = multi_monom(vertex
);
674 static int lattice_point_fixed(Param_Vertices
* V
, const mat_ZZ
& rays
,
675 vec_ZZ
& lambda
, term_info
* term
, unsigned long det
, int *closed
)
677 unsigned nparam
= V
->Vertex
->NbColumns
- 2;
678 unsigned dim
= rays
.NumCols();
680 for (int i
= 0; i
< dim
; ++i
)
681 if (First_Non_Zero(V
->Vertex
->p
[i
], nparam
) != -1)
684 Vector
*fixed
= Vector_Alloc(dim
+1);
685 for (int i
= 0; i
< dim
; ++i
)
686 value_assign(fixed
->p
[i
], V
->Vertex
->p
[i
][nparam
]);
687 value_assign(fixed
->p
[dim
], V
->Vertex
->p
[0][nparam
+1]);
690 lattice_point(fixed
->p
, rays
, vertex
, det
, closed
);
692 term
->constant
= vertex
* lambda
;
698 /* Returns the power of (t+1) in the term of a rational generating function,
699 * i.e., the scalar product of the actual lattice point and lambda.
700 * The lattice point is the unique lattice point in the fundamental parallelepiped
701 * of the unimodual cone i shifted to the parametric vertex V.
703 * The result is returned in term.
705 void lattice_point(Param_Vertices
* V
, const mat_ZZ
& rays
, vec_ZZ
& lambda
,
706 term_info
* term
, unsigned long det
, int *closed
,
707 barvinok_options
*options
)
709 unsigned nparam
= V
->Vertex
->NbColumns
- 2;
710 unsigned dim
= rays
.NumCols();
712 vertex
.SetDims(V
->Vertex
->NbRows
, nparam
+1);
714 Param_Vertex_Common_Denominator(V
);
716 if (lattice_point_fixed(V
, rays
, lambda
, term
, det
, closed
))
719 if (det
!= 1 || closed
|| value_notone_p(V
->Vertex
->p
[0][nparam
+1])) {
720 term
->E
= lattice_point(rays
, lambda
, V
, det
, closed
, options
);
723 for (int i
= 0; i
< V
->Vertex
->NbRows
; ++i
) {
724 assert(value_one_p(V
->Vertex
->p
[i
][nparam
+1])); // for now
725 values2zz(V
->Vertex
->p
[i
], vertex
[i
], nparam
+1);
729 num
= lambda
* vertex
;
732 for (int j
= 0; j
< nparam
; ++j
)
736 term
->E
= new evalue
*[1];
737 term
->E
[0] = multi_monom(num
);
740 term
->constant
.SetLength(1);
741 term
->constant
[0] = num
[nparam
];