void
ppl_min_for_le_pointset (ppl_Pointset_Powerset_C_Polyhedron_t ps,
ppl_Linear_Expression_t le, Value res)
{
ppl_Coefficient_t num, denom;
Value dv, nv;
int minimum, err;
value_init (nv);
value_init (dv);
ppl_new_Coefficient (&num);
ppl_new_Coefficient (&denom);
err = ppl_Pointset_Powerset_C_Polyhedron_minimize (ps, le, num, denom, &minimum);
if (err > 0)
{
ppl_Coefficient_to_mpz_t (num, nv);
ppl_Coefficient_to_mpz_t (denom, dv);
gcc_assert (value_notzero_p (dv));
value_division (res, nv, dv);
}
value_clear (nv);
value_clear (dv);
ppl_delete_Coefficient (num);
ppl_delete_Coefficient (denom);
}
/*
* Return the number of ways to choose 'b' items from 'a' items, that is,
* return a!/(b!(a-b)!)
*/
void CNP(int a,int b, Value *result) {
int i;
Value tmp;
value_init(tmp);
value_set_si(*result,1);
/* If number of items is less than the number to be choosen, return 1 */
if(a <= b) {
value_clear(tmp);
return;
}
for(i=a;i>b;--i) {
value_set_si(tmp,i);
value_multiply(*result,*result,tmp);
}
for(i=1;i<=(a-b);++i) {
value_set_si(tmp,i);
value_division(*result,*result,tmp);
}
value_clear(tmp);
} /* CNP */
/*
* Compute n!/(p!(n-p)!)
*/
void Binomial(int n, int p, Value *result) {
int i;
Value tmp;
Value f;
value_init(tmp);value_init(f);
if (n-p<p)
p=n-p;
if (p!=0) {
value_set_si(*result,(n-p+1));
for (i=n-p+2;i<=n;i++) {
value_set_si(tmp,i);
value_multiply(*result,*result,tmp);
}
Factorial(p,&f);
value_division(*result,*result,f);
}
else
value_set_si(*result,1);
value_clear(f);value_clear(tmp);
} /* Binomial */
/*
* Given a rational matrix 'Mat'(k x k), compute its inverse rational matrix
* 'MatInv' k x k.
* The output is 1,
* if 'Mat' is non-singular (invertible), otherwise the output is 0. Note::
* (1) Matrix 'Mat' is modified during the inverse operation.
* (2) Matrix 'MatInv' must be preallocated before passing into this function.
*/
int Matrix_Inverse(Matrix *Mat,Matrix *MatInv ) {
int i, k, j, c;
Value x, gcd, piv;
Value m1,m2;
Value *den;
if(Mat->NbRows != Mat->NbColumns) {
fprintf(stderr,"Trying to invert a non-square matrix !\n");
return 0;
}
/* Initialize all the 'Value' variables */
value_init(x); value_init(gcd); value_init(piv);
value_init(m1); value_init(m2);
k = Mat->NbRows;
/* Initialise MatInv */
Vector_Set(MatInv->p[0],0,k*k);
/* Initialize 'MatInv' to Identity matrix form. Each diagonal entry is set*/
/* to 1. Last column of each row (denominator of each entry in a row) is */
/* also set to 1. */
for(i=0;i<k;++i) {
value_set_si(MatInv->p[i][i],1);
/* value_set_si(MatInv->p[i][k],1); /* denum */
}
/* Apply Gauss-Jordan elimination method on the two matrices 'Mat' and */
/* 'MatInv' in parallel. */
for(i=0;i<k;++i) {
/* Check if the diagonal entry (new pivot) is non-zero or not */
if(value_zero_p(Mat->p[i][i])) {
/* Search for a non-zero pivot down the column(i) */
for(j=i;j<k;++j)
if(value_notzero_p(Mat->p[j][i]))
break;
/* If no non-zero pivot is found, the matrix 'Mat' is non-invertible */
/* Return 0. */
if(j==k) {
/* Clear all the 'Value' variables */
value_clear(x); value_clear(gcd); value_clear(piv);
value_clear(m1); value_clear(m2);
return 0;
}
/* Exchange the rows, row(i) and row(j) so that the diagonal element */
/* Mat->p[i][i] (pivot) is non-zero. Repeat the same operations on */
/* matrix 'MatInv'. */
for(c=0;c<k;++c) {
/* Interchange rows, row(i) and row(j) of matrix 'Mat' */
value_assign(x,Mat->p[j][c]);
value_assign(Mat->p[j][c],Mat->p[i][c]);
value_assign(Mat->p[i][c],x);
/* Interchange rows, row(i) and row(j) of matrix 'MatInv' */
value_assign(x,MatInv->p[j][c]);
value_assign(MatInv->p[j][c],MatInv->p[i][c]);
value_assign(MatInv->p[i][c],x);
}
}
/* Make all the entries in column(i) of matrix 'Mat' zero except the */
/* diagonal entry. Repeat the same sequence of operations on matrix */
/* 'MatInv'. */
for(j=0;j<k;++j) {
if (j==i) continue; /* Skip the pivot */
value_assign(x,Mat->p[j][i]);
if(value_notzero_p(x)) {
value_assign(piv,Mat->p[i][i]);
value_gcd(gcd, x, piv);
if (value_notone_p(gcd) ) {
value_divexact(x, x, gcd);
value_divexact(piv, piv, gcd);
}
for(c=((j>i)?i:0);c<k;++c) {
value_multiply(m1,piv,Mat->p[j][c]);
value_multiply(m2,x,Mat->p[i][c]);
value_subtract(Mat->p[j][c],m1,m2);
}
for(c=0;c<k;++c) {
value_multiply(m1,piv,MatInv->p[j][c]);
value_multiply(m2,x,MatInv->p[i][c]);
value_subtract(MatInv->p[j][c],m1,m2);
}
/* Simplify row(j) of the two matrices 'Mat' and 'MatInv' by */
/* dividing the rows with the common GCD. */
Vector_Gcd(&MatInv->p[j][0],k,&m1);
Vector_Gcd(&Mat->p[j][0],k,&m2);
value_gcd(gcd, m1, m2);
if(value_notone_p(gcd)) {
for(c=0;c<k;++c) {
value_divexact(Mat->p[j][c], Mat->p[j][c], gcd);
value_divexact(MatInv->p[j][c], MatInv->p[j][c], gcd);
}
}
}
}
}
/* Find common denom for each row */
den = (Value *)malloc(k*sizeof(Value));
value_set_si(x,1);
for(j=0 ; j<k ; ++j) {
value_init(den[j]);
value_assign(den[j],Mat->p[j][j]);
/* gcd is always positive */
Vector_Gcd(&MatInv->p[j][0],k,&gcd);
value_gcd(gcd, gcd, den[j]);
if (value_neg_p(den[j]))
value_oppose(gcd,gcd); /* make denominator positive */
if (value_notone_p(gcd)) {
for (c=0; c<k; c++)
value_divexact(MatInv->p[j][c], MatInv->p[j][c], gcd); /* normalize */
value_divexact(den[j], den[j], gcd);
}
value_gcd(gcd, x, den[j]);
value_divexact(m1, den[j], gcd);
value_multiply(x,x,m1);
}
if (value_notone_p(x))
for(j=0 ; j<k ; ++j) {
for (c=0; c<k; c++) {
value_division(m1,x,den[j]);
value_multiply(MatInv->p[j][c],MatInv->p[j][c],m1); /* normalize */
}
}
/* Clear all the 'Value' variables */
for(j=0 ; j<k ; ++j) {
value_clear(den[j]);
}
value_clear(x); value_clear(gcd); value_clear(piv);
value_clear(m1); value_clear(m2);
free(den);
return 1;
} /* Matrix_Inverse */
/*
* Given (m x n) integer matrix 'X' and n x (k+1) rational matrix 'P', compute
* the rational m x (k+1) rational matrix 'S'. The last column in each row of
* the rational matrices is used to store the common denominator of elements
* in a row.
*/
void rat_prodmat(Matrix *S,Matrix *X,Matrix *P) {
int i,j,k;
int last_column_index = P->NbColumns - 1;
Value lcm, old_lcm,gcd,last_column_entry,s1;
Value m1,m2;
/* Initialize all the 'Value' variables */
value_init(lcm); value_init(old_lcm); value_init(gcd);
value_init(last_column_entry); value_init(s1);
value_init(m1); value_init(m2);
/* Compute the LCM of last column entries (denominators) of rows */
value_assign(lcm,P->p[0][last_column_index]);
for(k=1;k<P->NbRows;++k) {
value_assign(old_lcm,lcm);
value_assign(last_column_entry,P->p[k][last_column_index]);
value_gcd(gcd, lcm, last_column_entry);
value_divexact(m1, last_column_entry, gcd);
value_multiply(lcm,lcm,m1);
}
/* S[i][j] = Sum(X[i][k] * P[k][j] where Sum is extended over k = 1..nbrows*/
for(i=0;i<X->NbRows;++i)
for(j=0;j<P->NbColumns-1;++j) {
/* Initialize s1 to zero. */
value_set_si(s1,0);
for(k=0;k<P->NbRows;++k) {
/* If the LCM of last column entries is one, simply add the products */
if(value_one_p(lcm)) {
value_addmul(s1, X->p[i][k], P->p[k][j]);
}
/* Numerator (num) and denominator (denom) of S[i][j] is given by :- */
/* numerator = Sum(X[i][k]*P[k][j]*lcm/P[k][last_column_index]) and */
/* denominator= lcm where Sum is extended over k = 1..nbrows. */
else {
value_multiply(m1,X->p[i][k],P->p[k][j]);
value_division(m2,lcm,P->p[k][last_column_index]);
value_addmul(s1, m1, m2);
}
}
value_assign(S->p[i][j],s1);
}
for(i=0;i<S->NbRows;++i) {
value_assign(S->p[i][last_column_index],lcm);
/* Normalize the rows so that last element >=0 */
Vector_Normalize_Positive(&S->p[i][0],S->NbColumns,S->NbColumns-1);
}
/* Clear all the 'Value' variables */
value_clear(lcm); value_clear(old_lcm); value_clear(gcd);
value_clear(last_column_entry); value_clear(s1);
value_clear(m1); value_clear(m2);
return;
} /* rat_prodmat */
/*
* Basic hermite engine
*/
static int hermite(Matrix *H,Matrix *U,Matrix *Q) {
int nc, nr, i, j, k, rank, reduced, pivotrow;
Value pivot,x,aux;
Value *temp1, *temp2;
/* T -1 T */
/* Computes form: A = Q H and U A = H and U = Q */
if (!H) {
errormsg1("Domlib", "nullH", "hermite: ? Null H");
return -1;
}
nc = H->NbColumns;
nr = H->NbRows;
temp1 = (Value *) malloc(nc * sizeof(Value));
temp2 = (Value *) malloc(nr * sizeof(Value));
if (!temp1 ||!temp2) {
errormsg1("Domlib", "outofmem", "out of memory space");
return -1;
}
/* Initialize all the 'Value' variables */
value_init(pivot); value_init(x);
value_init(aux);
for(i=0;i<nc;i++)
value_init(temp1[i]);
for(i=0;i<nr;i++)
value_init(temp2[i]);
#ifdef DEBUG
fprintf(stderr,"Start -----------\n");
Matrix_Print(stderr,0,H);
#endif
for (k=0, rank=0; k<nc && rank<nr; k=k+1) {
reduced = 1; /* go through loop the first time */
#ifdef DEBUG
fprintf(stderr, "Working on col %d. Rank=%d ----------\n", k+1, rank+1);
#endif
while (reduced) {
reduced=0;
/* 1. find pivot row */
value_absolute(pivot,H->p[rank][k]);
/* the kth-diagonal element */
pivotrow = rank;
/* find the row i>rank with smallest nonzero element in col k */
for (i=rank+1; i<nr; i++) {
value_absolute(x,H->p[i][k]);
if (value_notzero_p(x) &&
(value_lt(x,pivot) || value_zero_p(pivot))) {
value_assign(pivot,x);
pivotrow = i;
}
}
/* 2. Bring pivot to diagonal (exchange rows pivotrow and rank) */
if (pivotrow != rank) {
Vector_Exchange(H->p[pivotrow],H->p[rank],nc);
if (U)
Vector_Exchange(U->p[pivotrow],U->p[rank],nr);
if (Q)
Vector_Exchange(Q->p[pivotrow],Q->p[rank],nr);
#ifdef DEBUG
fprintf(stderr,"Exchange rows %d and %d -----------\n", rank+1, pivotrow+1);
Matrix_Print(stderr,0,H);
#endif
}
value_assign(pivot,H->p[rank][k]); /* actual ( no abs() ) pivot */
/* 3. Invert the row 'rank' if pivot is negative */
if (value_neg_p(pivot)) {
value_oppose(pivot,pivot); /* pivot = -pivot */
for (j=0; j<nc; j++)
value_oppose(H->p[rank][j],H->p[rank][j]);
/* H->p[rank][j] = -(H->p[rank][j]); */
if (U)
for (j=0; j<nr; j++)
value_oppose(U->p[rank][j],U->p[rank][j]);
/* U->p[rank][j] = -(U->p[rank][j]); */
if (Q)
for (j=0; j<nr; j++)
value_oppose(Q->p[rank][j],Q->p[rank][j]);
/* Q->p[rank][j] = -(Q->p[rank][j]); */
#ifdef DEBUG
fprintf(stderr,"Negate row %d -----------\n", rank+1);
Matrix_Print(stderr,0,H);
#endif
}
if (value_notzero_p(pivot)) {
/* 4. Reduce the column modulo the pivot */
/* This eventually zeros out everything below the */
/* diagonal and produces an upper triangular matrix */
for (i=rank+1;i<nr;i++) {
value_assign(x,H->p[i][k]);
if (value_notzero_p(x)) {
value_modulus(aux,x,pivot);
/* floor[integer division] (corrected for neg x) */
if (value_neg_p(x) && value_notzero_p(aux)) {
/* x=(x/pivot)-1; */
value_division(x,x,pivot);
value_decrement(x,x);
}
else
value_division(x,x,pivot);
for (j=0; j<nc; j++) {
value_multiply(aux,x,H->p[rank][j]);
value_subtract(H->p[i][j],H->p[i][j],aux);
}
/* U->p[i][j] -= (x * U->p[rank][j]); */
if (U)
for (j=0; j<nr; j++) {
value_multiply(aux,x,U->p[rank][j]);
value_subtract(U->p[i][j],U->p[i][j],aux);
}
/* Q->p[rank][j] += (x * Q->p[i][j]); */
if (Q)
for(j=0;j<nr;j++) {
value_addmul(Q->p[rank][j], x, Q->p[i][j]);
}
reduced = 1;
#ifdef DEBUG
fprintf(stderr,
"row %d = row %d - %d row %d -----------\n", i+1, i+1, x, rank+1);
Matrix_Print(stderr,0,H);
#endif
} /* if (x) */
} /* for (i) */
} /* if (pivot != 0) */
} /* while (reduced) */
/* Last finish up this column */
/* 5. Make pivot column positive (above pivot row) */
/* x should be zero for i>k */
if (value_notzero_p(pivot)) {
for (i=0; i<rank; i++) {
value_assign(x,H->p[i][k]);
if (value_notzero_p(x)) {
value_modulus(aux,x,pivot);
/* floor[integer division] (corrected for neg x) */
if (value_neg_p(x) && value_notzero_p(aux)) {
value_division(x,x,pivot);
value_decrement(x,x);
/* x=(x/pivot)-1; */
}
else
value_division(x,x,pivot);
/* H->p[i][j] -= x * H->p[rank][j]; */
for (j=0; j<nc; j++) {
value_multiply(aux,x,H->p[rank][j]);
value_subtract(H->p[i][j],H->p[i][j],aux);
}
/* U->p[i][j] -= x * U->p[rank][j]; */
if (U)
for (j=0; j<nr; j++) {
value_multiply(aux,x,U->p[rank][j]);
value_subtract(U->p[i][j],U->p[i][j],aux);
}
/* Q->p[rank][j] += x * Q->p[i][j]; */
if (Q)
for (j=0; j<nr; j++) {
value_addmul(Q->p[rank][j], x, Q->p[i][j]);
}
#ifdef DEBUG
fprintf(stderr,
"row %d = row %d - %d row %d -----------\n", i+1, i+1, x, rank+1);
Matrix_Print(stderr,0,H);
#endif
} /* if (x) */
} /* for (i) */
rank++;
} /* if (pivot!=0) */
} /* for (k) */
/* Clear all the 'Value' variables */
value_clear(pivot); value_clear(x);
value_clear(aux);
for(i=0;i<nc;i++)
value_clear(temp1[i]);
for(i=0;i<nr;i++)
value_clear(temp2[i]);
free(temp2);
free(temp1);
return rank;
} /* Hermite */
/* GaussSimplify --
Given Mat1, a matrix of equalities, performs Gaussian elimination.
Find a minimum basis, Returns the rank.
Mat1 is context, Mat2 is reduced in context of Mat1
*/
int GaussSimplify(Matrix *Mat1,Matrix *Mat2) {
int NbRows = Mat1->NbRows;
int NbCols = Mat1->NbColumns;
int *column_index;
int i, j, k, n, t, pivot, Rank;
Value gcd, tmp, *cp;
column_index=(int *)malloc(NbCols * sizeof(int));
if (!column_index) {
errormsg1("GaussSimplify", "outofmem", "out of memory space\n");
Pol_status = 1;
return 0;
}
/* Initialize all the 'Value' variables */
value_init(gcd); value_init(tmp);
Rank=0;
for (j=0; j<NbCols; j++) { /* for each column starting at */
for (i=Rank; i<NbRows; i++) /* diagonal, look down to find */
if (value_notzero_p(Mat1->p[i][j])) /* the first non-zero entry */
break;
if (i!=NbRows) { /* was one found ? */
if (i!=Rank) /* was it found below the diagonal?*/
Vector_Exchange(Mat1->p[Rank],Mat1->p[i],NbCols);
/* Normalize the pivot row */
Vector_Gcd(Mat1->p[Rank],NbCols,&gcd);
/* If (gcd >= 2) */
value_set_si(tmp,2);
if (value_ge(gcd,tmp)) {
cp = Mat1->p[Rank];
for (k=0; k<NbCols; k++,cp++)
value_division(*cp,*cp,gcd);
}
if (value_neg_p(Mat1->p[Rank][j])) {
cp = Mat1->p[Rank];
for (k=0; k<NbCols; k++,cp++)
value_oppose(*cp,*cp);
}
/* End of normalize */
pivot=i;
for (i=0;i<NbRows;i++) /* Zero out the rest of the column */
if (i!=Rank) {
if (value_notzero_p(Mat1->p[i][j])) {
Value a, a1, a2, a1abs, a2abs;
value_init(a); value_init(a1); value_init(a2);
value_init(a1abs); value_init(a2abs);
value_assign(a1,Mat1->p[i][j]);
value_absolute(a1abs,a1);
value_assign(a2,Mat1->p[Rank][j]);
value_absolute(a2abs,a2);
value_gcd(a, a1abs, a2abs);
value_divexact(a1, a1, a);
value_divexact(a2, a2, a);
value_oppose(a1,a1);
Vector_Combine(Mat1->p[i],Mat1->p[Rank],Mat1->p[i],a2,
a1,NbCols);
Vector_Normalize(Mat1->p[i],NbCols);
value_clear(a); value_clear(a1); value_clear(a2);
value_clear(a1abs); value_clear(a2abs);
}
}
column_index[Rank]=j;
Rank++;
}
} /* end of Gauss elimination */
if (Mat2) { /* Mat2 is a transformation matrix (i,j->f(i,j))....
can't scale it because can't scale both sides of -> */
/* normalizes an affine transformation */
/* priority of forms */
/* 1. i' -> i (identity) */
/* 2. i' -> i + constant (uniform) */
/* 3. i' -> constant (broadcast) */
/* 4. i' -> j (permutation) */
/* 5. i' -> j + constant ( ) */
/* 6. i' -> i + j + constant (non-uniform) */
for (k=0; k<Rank; k++) {
j = column_index[k];
for (i=0; i<(Mat2->NbRows-1);i++) { /* all but the last row 0...0 1 */
if ((i!=j) && value_notzero_p(Mat2->p[i][j])) {
/* Remove dependency of i' on j */
Value a, a1, a1abs, a2, a2abs;
value_init(a); value_init(a1); value_init(a2);
value_init(a1abs); value_init(a2abs);
value_assign(a1,Mat2->p[i][j]);
value_absolute(a1abs,a1);
value_assign(a2,Mat1->p[k][j]);
value_absolute(a2abs,a2);
value_gcd(a, a1abs, a2abs);
value_divexact(a1, a1, a);
value_divexact(a2, a2, a);
value_oppose(a1,a1);
if (value_one_p(a2)) {
Vector_Combine(Mat2->p[i],Mat1->p[k],Mat2->p[i],a2,
a1,NbCols);
/* Vector_Normalize(Mat2->p[i],NbCols); -- can't do T */
} /* otherwise, can't do it without mult lhs prod (2i,3j->...) */
value_clear(a); value_clear(a1); value_clear(a2);
value_clear(a1abs); value_clear(a2abs);
}
else if ((i==j) && value_zero_p(Mat2->p[i][j])) {
/* 'i' does not depend on j */
for (n=j+1; n < (NbCols-1); n++) {
if (value_notzero_p(Mat2->p[i][n])) { /* i' depends on some n */
value_set_si(tmp,1);
Vector_Combine(Mat2->p[i],Mat1->p[k],Mat2->p[i],tmp,
tmp,NbCols);
break;
} /* if 'i' depends on just a constant, then leave it alone.*/
}
}
}
}
/* Check last row of transformation Mat2 */
for (j=0; j<(NbCols-1); j++)
if (value_notzero_p(Mat2->p[Mat2->NbRows-1][j])) {
errormsg1("GaussSimplify", "corrtrans", "Corrupted transformation\n");
break;
}
if (value_notone_p(Mat2->p[Mat2->NbRows-1][NbCols-1])) {
errormsg1("GaussSimplify", "corrtrans", "Corrupted transformation\n");
}
}
value_clear(gcd); value_clear(tmp);
free(column_index);
return Rank;
} /* GaussSimplify */