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|
/**************************************************************************
* *
* Regina - A Normal Surface Theory Calculator *
* Computational Engine *
* *
* Copyright (c) 1999-2008, Ben Burton *
* For further details contact Ben Burton (bab@debian.org). *
* *
* This program is free software; you can redistribute it and/or *
* modify it under the terms of the GNU General Public License as *
* published by the Free Software Foundation; either version 2 of the *
* License, or (at your option) any later version. *
* *
* This program is distributed in the hope that it will be useful, but *
* WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU *
* General Public License for more details. *
* *
* You should have received a copy of the GNU General Public *
* License along with this program; if not, write to the Free *
* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, *
* MA 02110-1301, USA. *
* *
**************************************************************************/
/* end stub */
#include "maths/matrixops.h"
#include "maths/numbertheory.h"
namespace regina {
void smithNormalForm(NMatrixInt& matrix) {
unsigned long currStage = 0;
unsigned long nonEmptyRows = matrix.rows();
unsigned long nonEmptyCols = matrix.columns();
bool flag;
unsigned long i, j, k;
NLargeInteger d, u, v, a, b;
NLargeInteger tmp;
while ((currStage < nonEmptyRows) && (currStage < nonEmptyCols)) {
loopStart:
// Have we got an empty row?
flag = true;
for (i=currStage; i<nonEmptyCols; i++)
if (matrix.entry(currStage, i) != 0) {
flag = false;
break;
}
if (flag) {
// Empty row!
if (currStage == nonEmptyRows-1) {
nonEmptyRows--;
continue;
}
// Switch it with a row at the bottom.
for (i=currStage; i<nonEmptyCols; i++)
matrix.entry(currStage, i).swap(
matrix.entry(nonEmptyRows-1, i));
nonEmptyRows--;
continue;
}
// Have we got an empty column?
flag = true;
for (i=currStage; i<nonEmptyRows; i++)
if (matrix.entry(i, currStage) != 0) {
flag = false;
break;
}
if (flag) {
// Empty column!
if (currStage == nonEmptyCols-1) {
nonEmptyCols--;
continue;
}
// Switch it with a column on the end.
for (i=currStage; i<nonEmptyRows; i++)
matrix.entry(i, currStage).swap(
matrix.entry(i, nonEmptyCols-1));
nonEmptyCols--;
continue;
}
// Get zeros in the current row.
for (i=currStage+1; i<nonEmptyCols; i++) {
if (matrix.entry(currStage, i) == 0)
continue;
// Put a zero in (currStage, i).
a = matrix.entry(currStage, currStage);
b = matrix.entry(currStage, i);
d = a.gcdWithCoeffs(b, u, v);
a.divByExact(d);
b.divByExact(d);
// Do a modification to columns currStage and i.
for (j=currStage; j<nonEmptyRows; j++) {
tmp = u * matrix.entry(j, currStage) +
v * matrix.entry(j, i);
matrix.entry(j, i) = a * matrix.entry(j, i) -
b * matrix.entry(j, currStage);
matrix.entry(j, currStage) = tmp;
}
}
// Get zeros in the current column.
// Check to see if we change anything and thus muck up the row.
flag = false;
for (i=currStage+1; i<nonEmptyRows; i++) {
if (matrix.entry(i, currStage) == 0)
continue;
// Put a zero in (i, currStage).
flag = true;
a = matrix.entry(currStage, currStage);
b = matrix.entry(i, currStage);
d = a.gcdWithCoeffs(b, u, v);
a.divByExact(d);
b.divByExact(d);
// Do a modification to rows currStage and i.
for (j=currStage; j<nonEmptyCols; j++) {
tmp = u * matrix.entry(currStage, j) +
v * matrix.entry(i, j);
matrix.entry(i, j) = a * matrix.entry(i, j) -
b * matrix.entry(currStage, j);
matrix.entry(currStage, j) = tmp;
}
}
if (flag) {
// The clean row was mucked up.
continue;
}
// Check that entry (currStage, currStage) divides everything
// else.
for (i=currStage+1; i<nonEmptyRows; i++)
for (j=currStage+1; j<nonEmptyCols; j++)
if ((matrix.entry(i, j) % matrix.entry(currStage, currStage))
!= 0) {
// Add row i to the current stage row and start this
// stage over.
for (k=currStage+1; k<nonEmptyCols; k++)
matrix.entry(currStage, k) +=
matrix.entry(i, k);
goto loopStart;
}
// This stage is complete!
// Make sure the diagonal entry is positive before leaving it.
if (matrix.entry(currStage, currStage) < 0)
matrix.entry(currStage, currStage).negate();
currStage++;
}
}
void smithNormalForm(NMatrixInt& matrix,
NMatrixInt& rowSpaceBasis, NMatrixInt& rowSpaceBasisInv,
NMatrixInt& colSpaceBasis, NMatrixInt& colSpaceBasisInv) {
unsigned long currStage = 0;
unsigned long nonEmptyRows = matrix.rows();
unsigned long nonEmptyCols = matrix.columns();
bool flag;
unsigned long i, j, k;
NLargeInteger d, u, v, a, b;
NLargeInteger tmp;
rowSpaceBasis.makeIdentity();
rowSpaceBasisInv.makeIdentity();
colSpaceBasis.makeIdentity();
colSpaceBasisInv.makeIdentity();
while ((currStage < nonEmptyRows) && (currStage < nonEmptyCols)) {
loopStart:
// Have we got an empty row?
flag = true;
for (i=currStage; i<nonEmptyCols; i++)
if (matrix.entry(currStage, i) != 0) {
flag = false;
break;
}
if (flag) {
// Empty row!
if (currStage == nonEmptyRows-1) {
nonEmptyRows--;
continue;
}
// Switch it with a row at the bottom.
for (i=currStage; i<nonEmptyCols; i++) {
matrix.entry(currStage, i).swap(
matrix.entry(nonEmptyRows-1, i));
}
colSpaceBasis.swapRows(currStage, nonEmptyRows-1);
colSpaceBasisInv.swapColumns(currStage, nonEmptyRows-1);
nonEmptyRows--;
continue;
}
// Have we got an empty column?
flag = true;
for (i=currStage; i<nonEmptyRows; i++)
if (matrix.entry(i, currStage) != 0) {
flag = false;
break;
}
if (flag) {
// Empty column!
if (currStage == nonEmptyCols-1) {
nonEmptyCols--;
continue;
}
// Switch it with a column on the end.
for (i=currStage; i<nonEmptyRows; i++) {
matrix.entry(i, currStage).swap(
matrix.entry(i, nonEmptyCols-1));
}
rowSpaceBasis.swapColumns(currStage, nonEmptyCols-1);
rowSpaceBasisInv.swapRows(currStage, nonEmptyCols-1);
nonEmptyCols--;
continue;
}
// Get zeros in the current row.
for (i=currStage+1; i<nonEmptyCols; i++) {
if (matrix.entry(currStage, i) == 0)
continue;
// Put a zero in (currStage, i).
a = matrix.entry(currStage, currStage);
b = matrix.entry(currStage, i);
d = a.gcdWithCoeffs(b, u, v);
a.divByExact(d);
b.divByExact(d);
// Do a modification to columns currStage and i.
for (j=currStage; j<nonEmptyRows; j++) {
tmp = u * matrix.entry(j, currStage) +
v * matrix.entry(j, i);
matrix.entry(j, i) = a * matrix.entry(j, i) -
b * matrix.entry(j, currStage);
matrix.entry(j, currStage) = tmp;
}
for (j=0; j<matrix.columns(); j++) {
tmp = u * rowSpaceBasis.entry(j, currStage) +
v * rowSpaceBasis.entry(j, i);
rowSpaceBasis.entry(j, i) = a * rowSpaceBasis.entry(j, i) -
b * rowSpaceBasis.entry(j, currStage);
rowSpaceBasis.entry(j, currStage) = tmp;
tmp = a * rowSpaceBasisInv.entry(currStage, j) +
b * rowSpaceBasisInv.entry(i, j);
rowSpaceBasisInv.entry(i, j) =
u * rowSpaceBasisInv.entry(i, j) -
v * rowSpaceBasisInv.entry(currStage, j);
rowSpaceBasisInv.entry(currStage, j) = tmp;
}
}
// Get zeros in the current column.
// Check to see if we change anything and thus muck up the row.
flag = false;
for (i=currStage+1; i<nonEmptyRows; i++) {
if (matrix.entry(i, currStage) == 0)
continue;
// Put a zero in (i, currStage).
flag = true;
a = matrix.entry(currStage, currStage);
b = matrix.entry(i, currStage);
d = a.gcdWithCoeffs(b, u, v);
a.divByExact(d);
b.divByExact(d);
// Do a modification to rows currStage and i.
for (j=currStage; j<nonEmptyCols; j++) {
tmp = u * matrix.entry(currStage, j) +
v * matrix.entry(i, j);
matrix.entry(i, j) = a * matrix.entry(i, j) -
b * matrix.entry(currStage, j);
matrix.entry(currStage, j) = tmp;
}
for (j=0; j<matrix.rows(); j++) {
tmp = u * colSpaceBasis.entry(currStage, j) +
v * colSpaceBasis.entry(i, j);
colSpaceBasis.entry(i, j) = a * colSpaceBasis.entry(i, j) -
b * colSpaceBasis.entry(currStage, j);
colSpaceBasis.entry(currStage, j) = tmp;
tmp = a * colSpaceBasisInv.entry(j, currStage) +
b * colSpaceBasisInv.entry(j, i);
colSpaceBasisInv.entry(j, i) =
u * colSpaceBasisInv.entry(j, i) -
v * colSpaceBasisInv.entry(j, currStage);
colSpaceBasisInv.entry(j, currStage) = tmp;
}
}
if (flag) {
// The clean row was mucked up.
continue;
}
// Check that entry (currStage, currStage) divides everything
// else.
for (i=currStage+1; i<nonEmptyRows; i++)
for (j=currStage+1; j<nonEmptyCols; j++)
if ((matrix.entry(i, j) % matrix.entry(currStage, currStage))
!= 0) {
// Add row i to the current stage row and start this
// stage over.
for (k=currStage+1; k<nonEmptyCols; k++)
matrix.entry(currStage, k) += matrix.entry(i, k);
colSpaceBasis.addRow(i, currStage);
colSpaceBasisInv.addCol(currStage, i, -1);
goto loopStart;
}
// This stage is complete!
// Make sure the diagonal entry is positive before leaving it.
if (matrix.entry(currStage, currStage) < 0) {
matrix.entry(currStage, currStage).negate();
for (j=0; j<matrix.rows(); j++) {
// we're thinking of this as a row op
colSpaceBasis.entry( currStage, j ).negate();
colSpaceBasisInv.entry( j,currStage ).negate();
}
}
currStage++;
}
}
void columnEchelonForm(NMatrixInt &M, NMatrixInt &R, NMatrixInt &Ri,
const std::vector<unsigned> &rowList) {
unsigned i,j;
unsigned CR=0;
unsigned CC=0;
// these are the indices of the current WORKING rows and columns
// respectively.
// thus the entries of M above CR will not change, and to the left of
// CC all that can happen is some reduction.
std::vector<unsigned> rowNZlist; // in the current row, this is the
// list of column coordinates
// for the non-zero entries.
NLargeInteger d,r; // given two NLargeIntegers a and b, we will represent
// a/b by d and a % b by r in the algorithm.
NLargeInteger u,v,gcd, a,b; // for column operations u,v,a,b represent
// a 2x2 matrix.
NLargeInteger tmp;
// the algorithm will think of itself as working top to bottom.
while ( (CR<rowList.size()) && (CC<M.columns())) {
// build rowNZlist
rowNZlist.clear();
for (i=CC;i<M.columns();i++)
if (M.entry(rowList[CR],i) != 0)
rowNZlist.push_back(i);
// now the column operations
if (rowNZlist.size() == 0) {
// nothing to do.
CR++;
continue;
} else if (rowNZlist.size() == 1) {
// let's move this entry to be the leading entry.
if (rowNZlist[0]==CC) {
// step 1: ensure entry(CR,CC) is positive.
if (M.entry(rowList[CR],CC)<0) {
// step 1: negate column CC.
for (i=0;i<M.rows();i++)
M.entry(i,CC)=(-M.entry(i,CC));
// step 2: modify R, this is a right multiplication so a
// column op.
for (i=0;i<R.rows();i++)
R.entry(i,CC)=(-R.entry(i,CC));
// step 3: modify Ri, so this is a row op.
for (i=0;i<Ri.columns();i++)
Ri.entry(CC,i)=(-Ri.entry(CC,i));
// done.
}
// step 2: reduce entries(CR,i) for i<CC
for (i=0;i<CC;i++) { // write entry(CR,i) as d*entry(CR,CC) + r.
d = M.entry(rowList[CR],i).divisionAlg(
M.entry(rowList[CR],CC), r );
// perform reduction on column i. this is subtracting
// d times column CC from column i.
for (j=0;j<M.rows();j++)
M.entry(j,i) = M.entry(j,i) - d*M.entry(j,CC);
// modify R
for (j=0;j<R.rows();j++)
R.entry(j,i) = R.entry(j,i) - d*R.entry(j,CC);
// modify Ri -- the corresponding row op is addition of
// d times row i to row CC
for (j=0;j<Ri.columns();j++)
Ri.entry(CC,j) = Ri.entry(CC,j) + d*Ri.entry(i,j);
// done.
}
// done, move on.
CC++;
CR++;
continue;
} else {
// permute column rowNZlist[0] with CC.
for (i=0;i<M.rows();i++)
M.entry(i,CC).swap(M.entry(i,rowNZlist[0]));
// modify R
for (i=0;i<R.rows();i++)
R.entry(i,CC).swap(R.entry(i,rowNZlist[0]));
// modify Ri
for (i=0;i<Ri.columns();i++)
Ri.entry(CC,i).swap(Ri.entry(rowNZlist[0],i));
// done.
continue;
}
} else {
// there is at least 2 non-zero entries to deal with. we go
// through them, one by one, a pair at a time.
while (rowNZlist.size()>1) {
// do column reduction on columns rowNZlist[0] and rowNZlist[1]
// first we need to find the approp modification matrix.
// This will be the matrix ( u -b ) where ua+vb = 1. We get
// ( v a )
// a and b from entry(CR, r[0]) and entry(CR, r[1])
// by dividing by their GCD, found with
// rowNZlist[0].gcdWithCoeffs(rowNZlist[1],u,v)
gcd = M.entry(rowList[CR], rowNZlist[0]).gcdWithCoeffs(
M.entry(rowList[CR], rowNZlist[1]), u,v);
a = M.entry(rowList[CR], rowNZlist[0]).divExact(gcd);
b = M.entry(rowList[CR], rowNZlist[1]).divExact(gcd);
// so multiplication on the right by the above matrix
// corresponds to replacing column r[0] by u r[0] + v r[1]
// and column r[1] by -b r[0] + a r[1].
for (i=0;i<M.rows();i++) {
tmp = u * M.entry( i, rowNZlist[0] ) +
v * M.entry(i, rowNZlist[1] );
M.entry(i,rowNZlist[1]) = a * M.entry( i, rowNZlist[1]) -
b * M.entry( i, rowNZlist[0]);
M.entry(i,rowNZlist[0]) = tmp;
}
// modify R
for (i=0;i<R.rows();i++) {
tmp = u * R.entry( i, rowNZlist[0] ) +
v * R.entry(i, rowNZlist[1] );
R.entry(i,rowNZlist[1]) = a * R.entry( i, rowNZlist[1]) -
b * R.entry( i, rowNZlist[0]);
R.entry(i,rowNZlist[0]) = tmp;
}
// modify Ri
for (i=0;i<Ri.columns();i++) {
tmp = a * Ri.entry( rowNZlist[0], i ) +
b * Ri.entry( rowNZlist[1], i );
Ri.entry( rowNZlist[1], i ) =
u * Ri.entry( rowNZlist[1], i ) -
v * Ri.entry( rowNZlist[0], i );
Ri.entry( rowNZlist[0], i ) = tmp;
}
// modify rowNZlist by deleting the entry corresponding to
// rowNZlist[1]
rowNZlist.erase( rowNZlist.begin()+1 );
// done.
}
continue;
}
}
}
std::auto_ptr<NMatrixInt> preImageOfLattice(const NMatrixInt& hom,
const std::vector<NLargeInteger>& L) {
// there are two main steps to this algorithm.
// 1) find a basis for the domain which splits into a) vectors sent to the
// complement of the primitive subspace generated by the range lattice
// and b) a basis of vectors sent to the primitive subspace generated
// by the range lattice.
// 2) modify the basis (b) by column ops to get the preimage of the lattice.
// step (1) is an application of the columnEchelonForm
// step (2) starts with another application of columnEchelonForm, but then
// it finishes with a variation on it...
unsigned i,j;
NMatrixInt basis(hom.columns(), hom.columns() );
basis.makeIdentity();
NMatrixInt basisi(hom.columns(), hom.columns() );
basisi.makeIdentity();
// and we proceed to modify it solely via column operations.
// one for every column operation performed on homModL
NMatrixInt homModL(hom);
// set up two lists: the coordinates that correspond to free generators
// of the range and coordinates corresponding to torsion generators.
// these lists need to be built from L
std::vector<unsigned> freeList;
std::vector<unsigned> torList;
for (i=0;i<L.size();i++)
if (L[i]==0)
freeList.push_back(i);
else
torList.push_back(i);
// so first put the free image part of it in column echelon form
columnEchelonForm( homModL, basis, basisi, freeList );
std::vector<unsigned> torCol(0);
bool zeroCol;
for (i=0; i<homModL.columns(); i++) {
zeroCol=true;
for (j=0; j<freeList.size(); j++)
if (homModL.entry( freeList[j], i) != 0)
zeroCol=false;
if (zeroCol)
torCol.push_back(i);
}
// set up a new matrix consisting of columns being sent to the primitive
// subspace generated by the torsion lattice.
NMatrixInt tHom( homModL.rows(), torCol.size() );
std::auto_ptr<NMatrixInt> tBasis(
new NMatrixInt( basis.rows(), torCol.size() )); // this will be the
// eventual retval.
NMatrixInt dummy( torCol.size(), 0 ); // needed when we call
// columnEchelonForm. choosing it to have 0 columns speeds up
// the algorithm.
for (i=0;i<tHom.rows();i++)
for (j=0;j<tHom.columns();j++)
tHom.entry(i,j) = homModL.entry(i, torCol[j]);
for (i=0;i<basis.rows();i++)
for (j=0;j<torCol.size();j++)
tBasis->entry(i,j) = basis.entry(i, torCol[j]);
columnEchelonForm( tHom, *tBasis, dummy, torList );
// so now we have a primitive collection of vectors being sent to the
// primitive subspace generated by the torsion lattice in the target.
// The idea is to run through the rows, for each non-zero row, through
// a basis change we can ensure there is at most one non-zero entry.
// multiply this column by the smallest factor so that it is in
// the torsion lattice, repeat. etc.
unsigned CR=0; // current row under consideration. The actual row index
// will of course be torList[CR] since all other rows are already zero.
std::vector<unsigned> rowNZlist; // in the current row, this is the list
// of column coordinates for the non-zero entries.
NLargeInteger d,r; // given two NLargeIntegers a and b, we will represent
// a/b by d and a % b by r in the algorithm.
NLargeInteger u,v,gcd, a,b; // for column operations u,v,a,b represent
// a 2x2 matrix.
NLargeInteger tmp;
while (CR<torList.size()) {
// build rowNZlist
rowNZlist.clear();
for (i=0;i<tHom.columns();i++)
if (tHom.entry(torList[CR],i) != 0)
rowNZlist.push_back(i);
// okay, so now we have a list of non-zero entries.
// case 1: rowNZlist.size()==0, increment CR, continue;
if (rowNZlist.size()==0) {
CR++;
continue;
}
// case 2: rowNZlist.size()==1 multiply column only if neccessary,
// increment CR, continue;
if (rowNZlist.size()==1) {
// check to see if tHom.entry(torList[CR], rowNZlist[0]) %
// L[torList[CR]] == 0 if not, find smallest positive integer
// s.t. when multiplied by it, above becomes true.
gcd = tHom.entry(torList[CR], rowNZlist[0]).gcd( L[torList[CR]] );
d = L[torList[CR]].divExact(gcd); // devByExact changed to divExact Sept 26th, 2007
// multiply column rowNZlist[0] of tHom by d.
for (i=0;i<torList.size();i++)
tHom.entry( torList[i], rowNZlist[0] ) *= d;
// corresponding operation on tBasis.
for (i=0;i<tBasis->rows();i++)
tBasis->entry( i, rowNZlist[0] ) *= d;
// done.
CR++;
continue;
}
// case 3: rowNZlist.size()>1.row ops to reduce rowNZlist.size().
// then continue
while (rowNZlist.size()>1) {
// do column op on columns rowNZlist[0] and rowNZlist[1]
// first we need to find the approp modification matrix. This will
// be the matrix ( u -b ) where ua+vb = 1. We get a and b from
// ( v a ) from entry(torList[CR], r[0]) and
// entry(torlist[CR], r[1]) by dividing by their GCD, found with
// rowNZlist[0].gcdWithCoeffs(rowNZlist[1],u,v)
gcd = tHom.entry(torList[CR], rowNZlist[0]).gcdWithCoeffs(
tHom.entry(torList[CR], rowNZlist[1]), u,v);
a = tHom.entry(torList[CR], rowNZlist[0]).divExact(gcd);
b = tHom.entry(torList[CR], rowNZlist[1]).divExact(gcd);
// so multiplication on the right by the above matrix corresponds
// to replacing column r[0] by u r[0] + v r[1] and column r[1] by
// -b r[0] + a r[1].
for (i=0;i<torList.size();i++) {
tmp = u * tHom.entry( torList[i], rowNZlist[0] ) +
v * tHom.entry( torList[i], rowNZlist[1] );
tHom.entry( torList[i],rowNZlist[1]) =
a * tHom.entry( torList[i], rowNZlist[1]) -
b * tHom.entry( torList[i], rowNZlist[0]);
tHom.entry( torList[i],rowNZlist[0]) = tmp;
}
// modify tBasis
for (i=0;i<tBasis->rows();i++) {
tmp = u * tBasis->entry( i, rowNZlist[0] ) +
v * tBasis->entry(i, rowNZlist[1] );
tBasis->entry(i,rowNZlist[1]) =
a * tBasis->entry( i, rowNZlist[1]) -
b * tBasis->entry( i, rowNZlist[0]);
tBasis->entry(i,rowNZlist[0]) = tmp;
}
// now rowNZlist[1] entry is zero, remove it from the list.
rowNZlist.erase( rowNZlist.begin()+1 );
}
}
return tBasis;
}
} // namespace regina
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