* Implementation of symbolic matrices */
/*
- * GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
*
* 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
*/
#include <algorithm>
+#include <map>
#include <stdexcept>
#include "matrix.h"
+#include "archive.h"
+#include "utils.h"
#include "debugmsg.h"
+#include "numeric.h"
-#ifndef NO_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC
namespace GiNaC {
-#endif // ndef NO_GINAC_NAMESPACE
+#endif // ndef NO_NAMESPACE_GINAC
+
+GINAC_IMPLEMENT_REGISTERED_CLASS(matrix, basic)
//////////
// default constructor, destructor, copy constructor, assignment operator
/** Default ctor. Initializes to 1 x 1-dimensional zero-matrix. */
matrix::matrix()
- : basic(TINFO_matrix), row(1), col(1)
+ : inherited(TINFO_matrix), row(1), col(1)
{
debugmsg("matrix default constructor",LOGLEVEL_CONSTRUCT);
- m.push_back(exZERO());
+ m.push_back(_ex0());
}
matrix::~matrix()
debugmsg("matrix destructor",LOGLEVEL_DESTRUCT);
}
-matrix::matrix(matrix const & other)
+matrix::matrix(const matrix & other)
{
debugmsg("matrix copy constructor",LOGLEVEL_CONSTRUCT);
copy(other);
}
-matrix const & matrix::operator=(matrix const & other)
+const matrix & matrix::operator=(const matrix & other)
{
debugmsg("matrix operator=",LOGLEVEL_ASSIGNMENT);
if (this != &other) {
// protected
-void matrix::copy(matrix const & other)
+void matrix::copy(const matrix & other)
{
- basic::copy(other);
+ inherited::copy(other);
row=other.row;
col=other.col;
m=other.m; // use STL's vector copying
void matrix::destroy(bool call_parent)
{
- if (call_parent) basic::destroy(call_parent);
+ if (call_parent) inherited::destroy(call_parent);
}
//////////
*
* @param r number of rows
* @param c number of cols */
-matrix::matrix(int r, int c)
- : basic(TINFO_matrix), row(r), col(c)
+matrix::matrix(unsigned r, unsigned c)
+ : inherited(TINFO_matrix), row(r), col(c)
{
- debugmsg("matrix constructor from int,int",LOGLEVEL_CONSTRUCT);
- m.resize(r*c, exZERO());
+ debugmsg("matrix constructor from unsigned,unsigned",LOGLEVEL_CONSTRUCT);
+ m.resize(r*c, _ex0());
}
// protected
/** Ctor from representation, for internal use only. */
-matrix::matrix(int r, int c, vector<ex> const & m2)
- : basic(TINFO_matrix), row(r), col(c), m(m2)
+matrix::matrix(unsigned r, unsigned c, const exvector & m2)
+ : inherited(TINFO_matrix), row(r), col(c), m(m2)
+{
+ debugmsg("matrix constructor from unsigned,unsigned,exvector",LOGLEVEL_CONSTRUCT);
+}
+
+//////////
+// archiving
+//////////
+
+/** Construct object from archive_node. */
+matrix::matrix(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
+{
+ debugmsg("matrix constructor from archive_node", LOGLEVEL_CONSTRUCT);
+ if (!(n.find_unsigned("row", row)) || !(n.find_unsigned("col", col)))
+ throw (std::runtime_error("unknown matrix dimensions in archive"));
+ m.reserve(row * col);
+ for (unsigned int i=0; true; i++) {
+ ex e;
+ if (n.find_ex("m", e, sym_lst, i))
+ m.push_back(e);
+ else
+ break;
+ }
+}
+
+/** Unarchive the object. */
+ex matrix::unarchive(const archive_node &n, const lst &sym_lst)
{
- debugmsg("matrix constructor from int,int,vector<ex>",LOGLEVEL_CONSTRUCT);
+ return (new matrix(n, sym_lst))->setflag(status_flags::dynallocated);
+}
+
+/** Archive the object. */
+void matrix::archive(archive_node &n) const
+{
+ inherited::archive(n);
+ n.add_unsigned("row", row);
+ n.add_unsigned("col", col);
+ exvector::const_iterator i = m.begin(), iend = m.end();
+ while (i != iend) {
+ n.add_ex("m", *i);
+ i++;
+ }
}
//////////
{
debugmsg("matrix print",LOGLEVEL_PRINT);
os << "[[ ";
- for (int r=0; r<row-1; ++r) {
+ for (unsigned r=0; r<row-1; ++r) {
os << "[[";
- for (int c=0; c<col-1; ++c) {
+ for (unsigned c=0; c<col-1; ++c) {
os << m[r*col+c] << ",";
}
os << m[col*(r+1)-1] << "]], ";
}
os << "[[";
- for (int c=0; c<col-1; ++c) {
+ for (unsigned c=0; c<col-1; ++c) {
os << m[(row-1)*col+c] << ",";
}
os << m[row*col-1] << "]] ]]";
{
debugmsg("matrix printraw",LOGLEVEL_PRINT);
os << "matrix(" << row << "," << col <<",";
- for (int r=0; r<row-1; ++r) {
+ for (unsigned r=0; r<row-1; ++r) {
os << "(";
- for (int c=0; c<col-1; ++c) {
+ for (unsigned c=0; c<col-1; ++c) {
os << m[r*col+c] << ",";
}
os << m[col*(r-1)-1] << "),";
}
os << "(";
- for (int c=0; c<col-1; ++c) {
+ for (unsigned c=0; c<col-1; ++c) {
os << m[(row-1)*col+c] << ",";
}
os << m[row*col-1] << "))";
}
/** nops is defined to be rows x columns. */
-int matrix::nops() const
+unsigned matrix::nops() const
{
return row*col;
}
/** returns matrix entry at position (i/col, i%col). */
-ex & matrix::let_op(int const i)
+ex matrix::op(int i) const
+{
+ return m[i];
+}
+
+/** returns matrix entry at position (i/col, i%col). */
+ex & matrix::let_op(int i)
{
return m[i];
}
/** expands the elements of a matrix entry by entry. */
ex matrix::expand(unsigned options) const
{
- vector<ex> tmp(row*col);
- for (int i=0; i<row*col; ++i) {
+ exvector tmp(row*col);
+ for (unsigned i=0; i<row*col; ++i) {
tmp[i]=m[i].expand(options);
}
return matrix(row, col, tmp);
/** Search ocurrences. A matrix 'has' an expression if it is the expression
* itself or one of the elements 'has' it. */
-bool matrix::has(ex const & other) const
+bool matrix::has(const ex & other) const
{
GINAC_ASSERT(other.bp!=0);
if (is_equal(*other.bp)) return true;
// search all the elements
- for (vector<ex>::const_iterator r=m.begin(); r!=m.end(); ++r) {
+ for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
if ((*r).has(other)) return true;
}
return false;
debugmsg("matrix eval",LOGLEVEL_MEMBER_FUNCTION);
// check if we have to do anything at all
- if ((level==1)&&(flags & status_flags::evaluated)) {
+ if ((level==1)&&(flags & status_flags::evaluated))
return *this;
- }
// emergency break
- if (level == -max_recursion_level) {
+ if (level == -max_recursion_level)
throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
- }
// eval() entry by entry
- vector<ex> m2(row*col);
+ exvector m2(row*col);
--level;
- for (int r=0; r<row; ++r) {
- for (int c=0; c<col; ++c) {
+ for (unsigned r=0; r<row; ++r) {
+ for (unsigned c=0; c<col; ++c) {
m2[r*col+c] = m[r*col+c].eval(level);
}
}
debugmsg("matrix evalf",LOGLEVEL_MEMBER_FUNCTION);
// check if we have to do anything at all
- if (level==1) {
+ if (level==1)
return *this;
- }
// emergency break
if (level == -max_recursion_level) {
}
// evalf() entry by entry
- vector<ex> m2(row*col);
+ exvector m2(row*col);
--level;
- for (int r=0; r<row; ++r) {
- for (int c=0; c<col; ++c) {
+ for (unsigned r=0; r<row; ++r) {
+ for (unsigned c=0; c<col; ++c) {
m2[r*col+c] = m[r*col+c].evalf(level);
}
}
// protected
-int matrix::compare_same_type(basic const & other) const
+int matrix::compare_same_type(const basic & other) const
{
GINAC_ASSERT(is_exactly_of_type(other, matrix));
- matrix const & o=static_cast<matrix &>(const_cast<basic &>(other));
+ const matrix & o = static_cast<matrix &>(const_cast<basic &>(other));
// compare number of rows
- if (row != o.rows()) {
+ if (row != o.rows())
return row < o.rows() ? -1 : 1;
- }
// compare number of columns
- if (col != o.cols()) {
+ if (col != o.cols())
return col < o.cols() ? -1 : 1;
- }
// equal number of rows and columns, compare individual elements
int cmpval;
- for (int r=0; r<row; ++r) {
- for (int c=0; c<col; ++c) {
- cmpval=((*this)(r,c)).compare(o(r,c));
+ for (unsigned r=0; r<row; ++r) {
+ for (unsigned c=0; c<col; ++c) {
+ cmpval = ((*this)(r,c)).compare(o(r,c));
if (cmpval!=0) return cmpval;
}
}
/** Sum of matrices.
*
* @exception logic_error (incompatible matrices) */
-matrix matrix::add(matrix const & other) const
+matrix matrix::add(const matrix & other) const
{
- if (col != other.col || row != other.row) {
+ if (col != other.col || row != other.row)
throw (std::logic_error("matrix::add(): incompatible matrices"));
- }
- vector<ex> sum(this->m);
- vector<ex>::iterator i;
- vector<ex>::const_iterator ci;
+ exvector sum(this->m);
+ exvector::iterator i;
+ exvector::const_iterator ci;
for (i=sum.begin(), ci=other.m.begin();
i!=sum.end();
++i, ++ci) {
return matrix(row,col,sum);
}
+
/** Difference of matrices.
*
* @exception logic_error (incompatible matrices) */
-matrix matrix::sub(matrix const & other) const
+matrix matrix::sub(const matrix & other) const
{
- if (col != other.col || row != other.row) {
+ if (col != other.col || row != other.row)
throw (std::logic_error("matrix::sub(): incompatible matrices"));
- }
- vector<ex> dif(this->m);
- vector<ex>::iterator i;
- vector<ex>::const_iterator ci;
+ exvector dif(this->m);
+ exvector::iterator i;
+ exvector::const_iterator ci;
for (i=dif.begin(), ci=other.m.begin();
i!=dif.end();
++i, ++ci) {
return matrix(row,col,dif);
}
+
/** Product of matrices.
*
* @exception logic_error (incompatible matrices) */
-matrix matrix::mul(matrix const & other) const
+matrix matrix::mul(const matrix & other) const
{
- if (col != other.row) {
+ if (col != other.row)
throw (std::logic_error("matrix::mul(): incompatible matrices"));
- }
- vector<ex> prod(row*other.col);
- for (int i=0; i<row; ++i) {
- for (int j=0; j<other.col; ++j) {
- for (int l=0; l<col; ++l) {
+ exvector prod(row*other.col);
+ for (unsigned i=0; i<row; ++i) {
+ for (unsigned j=0; j<other.col; ++j) {
+ for (unsigned l=0; l<col; ++l) {
prod[i*other.col+j] += m[i*col+l] * other.m[l*other.col+j];
}
}
return matrix(row, other.col, prod);
}
+
/** operator() to access elements.
*
* @param ro row of element
* @param co column of element
* @exception range_error (index out of range) */
-ex const & matrix::operator() (int ro, int co) const
+const ex & matrix::operator() (unsigned ro, unsigned co) const
{
- if (ro<0 || ro>=row || co<0 || co>=col) {
+ if (ro<0 || ro>=row || co<0 || co>=col)
throw (std::range_error("matrix::operator(): index out of range"));
- }
return m[ro*col+co];
}
+
/** Set individual elements manually.
*
* @exception range_error (index out of range) */
-matrix & matrix::set(int ro, int co, ex value)
+matrix & matrix::set(unsigned ro, unsigned co, ex value)
{
- if (ro<0 || ro>=row || co<0 || co>=col) {
+ if (ro<0 || ro>=row || co<0 || co>=col)
throw (std::range_error("matrix::set(): index out of range"));
- }
ensure_if_modifiable();
- m[ro*col+co]=value;
+ m[ro*col+co] = value;
return *this;
}
+
/** Transposed of an m x n matrix, producing a new n x m matrix object that
* represents the transposed. */
matrix matrix::transpose(void) const
{
- vector<ex> trans(col*row);
+ exvector trans(col*row);
- for (int r=0; r<col; ++r) {
- for (int c=0; c<row; ++c) {
+ for (unsigned r=0; r<col; ++r)
+ for (unsigned c=0; c<row; ++c)
trans[r*row+c] = m[c*col+r];
- }
- }
- return matrix(col,row,trans);
-}
-
-/* Determiant of purely numeric matrix, using pivoting. This routine is only
- * called internally by matrix::determinant(). */
-ex determinant_numeric(const matrix & M)
-{
- GINAC_ASSERT(M.rows()==M.cols()); // cannot happen, just in case...
- matrix tmp(M);
- ex det=exONE();
- ex piv;
-
- for (int r1=0; r1<M.rows(); ++r1) {
- int indx = tmp.pivot(r1);
- if (indx == -1) {
- return exZERO();
- }
- if (indx != 0) {
- det *= exMINUSONE();
- }
- det = det * tmp.m[r1*M.cols()+r1];
- for (int r2=r1+1; r2<M.rows(); ++r2) {
- piv = tmp.m[r2*M.cols()+r1] / tmp.m[r1*M.cols()+r1];
- for (int c=r1+1; c<M.cols(); c++) {
- tmp.m[r2*M.cols()+c] -= piv * tmp.m[r1*M.cols()+c];
- }
- }
- }
- return det;
-}
-
-// Compute the sign of a permutation of a vector of things, used internally
-// by determinant_symbolic_perm() where it is instantiated for int.
-template <class T>
-int permutation_sign(vector<T> s)
-{
- if (s.size() < 2)
- return 0;
- int sigma=1;
- for (typename vector<T>::iterator i=s.begin(); i!=s.end()-1; ++i) {
- for (typename vector<T>::iterator j=i+1; j!=s.end(); ++j) {
- if (*i == *j)
- return 0;
- if (*i > *j) {
- iter_swap(i,j);
- sigma = -sigma;
- }
- }
- }
- return sigma;
-}
-
-/** Determinant built by application of the full permutation group. This
- * routine is only called internally by matrix::determinant(). */
-ex determinant_symbolic_perm(const matrix & M)
-{
- GINAC_ASSERT(M.rows()==M.cols()); // cannot happen, just in case...
-
- if (M.rows()==1) { // speed things up
- return M(0,0);
- }
-
- ex det;
- ex term;
- vector<int> sigma(M.cols());
- for (int i=0; i<M.cols(); ++i) sigma[i]=i;
-
- do {
- term = M(sigma[0],0);
- for (int i=1; i<M.cols(); ++i) term *= M(sigma[i],i);
- det += permutation_sign(sigma)*term;
- } while (next_permutation(sigma.begin(), sigma.end()));
- return det;
+ return matrix(col,row,trans);
}
-/** Recursive determiant for small matrices having at least one symbolic entry.
- * This algorithm is also known as Laplace-expansion. This routine is only
- * called internally by matrix::determinant(). */
-ex determinant_symbolic_minor(const matrix & M)
-{
- GINAC_ASSERT(M.rows()==M.cols()); // cannot happen, just in case...
-
- if (M.rows()==1) { // end of recursion
- return M(0,0);
- }
- if (M.rows()==2) { // speed things up
- return (M(0,0)*M(1,1)-
- M(1,0)*M(0,1));
- }
- if (M.rows()==3) { // speed things up even a little more
- return ((M(2,1)*M(0,2)-M(2,2)*M(0,1))*M(1,0)+
- (M(1,2)*M(0,1)-M(1,1)*M(0,2))*M(2,0)+
- (M(2,2)*M(1,1)-M(2,1)*M(1,2))*M(0,0));
- }
-
- ex det;
- matrix minorM(M.rows()-1,M.cols()-1);
- for (int r1=0; r1<M.rows(); ++r1) {
- // assemble the minor matrix
- for (int r=0; r<minorM.rows(); ++r) {
- for (int c=0; c<minorM.cols(); ++c) {
- if (r<r1) {
- minorM.set(r,c,M(r,c+1));
- } else {
- minorM.set(r,c,M(r+1,c+1));
- }
- }
- }
- // recurse down
- if (r1%2) {
- det -= M(r1,0) * determinant_symbolic_minor(minorM);
- } else {
- det += M(r1,0) * determinant_symbolic_minor(minorM);
- }
- }
- return det;
-}
-
-/* Leverrier algorithm for large matrices having at least one symbolic entry.
- * This routine is only called internally by matrix::determinant(). The
- * algorithm is deemed bad for symbolic matrices since it returns expressions
- * that are very hard to canonicalize. */
-/*ex determinant_symbolic_leverrier(const matrix & M)
- *{
- * GINAC_ASSERT(M.rows()==M.cols()); // cannot happen, just in case...
- *
- * matrix B(M);
- * matrix I(M.row, M.col);
- * ex c=B.trace();
- * for (int i=1; i<M.row; ++i) {
- * for (int j=0; j<M.row; ++j)
- * I.m[j*M.col+j] = c;
- * B = M.mul(B.sub(I));
- * c = B.trace()/ex(i+1);
- * }
- * if (M.row%2) {
- * return c;
- * } else {
- * return -c;
- * }
- *}*/
/** Determinant of square matrix. This routine doesn't actually calculate the
* determinant, it only implements some heuristics about which algorithm to
- * call. When the parameter for normalization is explicitly turned off this
- * method does not normalize its result at the end, which might imply that
- * the symbolic 2x2 matrix [[a/(a-b),1],[b/(a-b),1]] is not immediatly
- * recognized to be unity. (This is Mathematica's default behaviour, it
- * should be used with care.)
+ * call. If all the elements of the matrix are elements of an integral domain
+ * the determinant is also in that integral domain and the result is expanded
+ * only. If one or more elements are from a quotient field the determinant is
+ * usually also in that quotient field and the result is normalized before it
+ * is returned. This implies that the determinant of the symbolic 2x2 matrix
+ * [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
+ * behaves like MapleV and unlike Mathematica.)
*
- * @param normalized may be set to false if no normalization of the
- * result is desired (i.e. to force Mathematica behavior, Maple
- * does normalize the result).
* @return the determinant as a new expression
* @exception logic_error (matrix not square) */
-ex matrix::determinant(bool normalized) const
+ex matrix::determinant(void) const
{
- if (row != col) {
+ if (row != col)
throw (std::logic_error("matrix::determinant(): matrix not square"));
+ GINAC_ASSERT(row*col==m.capacity());
+
+ ex det;
+ bool numeric_flag = true;
+ bool normal_flag = false;
+ for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
+ if (!(*r).info(info_flags::numeric))
+ numeric_flag = false;
+ if ((*r).info(info_flags::rational_function) &&
+ !(*r).info(info_flags::crational_polynomial))
+ normal_flag = true;
}
-
- // check, if there are non-numeric entries in the matrix:
- for (vector<ex>::const_iterator r=m.begin(); r!=m.end(); ++r) {
- if (!(*r).info(info_flags::numeric)) {
- if (normalized) {
- return determinant_symbolic_minor(*this).normal();
- } else {
- return determinant_symbolic_perm(*this);
- }
- }
- }
- // if it turns out that all elements are numeric
- return determinant_numeric(*this);
+
+ if (numeric_flag)
+ det = determinant_numeric();
+ else
+ if (normal_flag)
+ det = determinant_minor().normal();
+ else
+ det = determinant_minor(); // is already expanded!
+
+ return det;
}
-/** Trace of a matrix.
+
+/** Trace of a matrix. The result is normalized if it is in some quotient
+ * field and expanded only otherwise. This implies that the trace of the
+ * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
*
* @return the sum of diagonal elements
* @exception logic_error (matrix not square) */
ex matrix::trace(void) const
{
- if (row != col) {
+ if (row != col)
throw (std::logic_error("matrix::trace(): matrix not square"));
- }
+ GINAC_ASSERT(row*col==m.capacity());
ex tr;
- for (int r=0; r<col; ++r) {
+ for (unsigned r=0; r<col; ++r)
tr += m[r*col+r];
- }
- return tr;
+
+ if (tr.info(info_flags::rational_function) &&
+ !tr.info(info_flags::crational_polynomial))
+ return tr.normal();
+ else
+ return tr.expand();
}
+
/** Characteristic Polynomial. The characteristic polynomial of a matrix M is
* defined as the determiant of (M - lambda * 1) where 1 stands for the unit
* matrix of the same dimension as M. This method returns the characteristic
* @return characteristic polynomial as new expression
* @exception logic_error (matrix not square)
* @see matrix::determinant() */
-ex matrix::charpoly(ex const & lambda) const
+ex matrix::charpoly(const ex & lambda) const
{
- if (row != col) {
+ if (row != col)
throw (std::logic_error("matrix::charpoly(): matrix not square"));
- }
matrix M(*this);
- for (int r=0; r<col; ++r) {
+ for (unsigned r=0; r<col; ++r)
M.m[r*col+r] -= lambda;
- }
+
return (M.determinant());
}
+
/** Inverse of this matrix.
*
* @return the inverted matrix
* @exception runtime_error (singular matrix) */
matrix matrix::inverse(void) const
{
- if (row != col) {
+ if (row != col)
throw (std::logic_error("matrix::inverse(): matrix not square"));
- }
matrix tmp(row,col);
// set tmp to the unit matrix
- for (int i=0; i<col; ++i) {
- tmp.m[i*col+i] = exONE();
- }
+ for (unsigned i=0; i<col; ++i)
+ tmp.m[i*col+i] = _ex1();
+
// create a copy of this matrix
matrix cpy(*this);
- for (int r1=0; r1<row; ++r1) {
+ for (unsigned r1=0; r1<row; ++r1) {
int indx = cpy.pivot(r1);
if (indx == -1) {
throw (std::runtime_error("matrix::inverse(): singular matrix"));
}
if (indx != 0) { // swap rows r and indx of matrix tmp
- for (int i=0; i<col; ++i) {
+ for (unsigned i=0; i<col; ++i) {
tmp.m[r1*col+i].swap(tmp.m[indx*col+i]);
}
}
ex a1 = cpy.m[r1*col+r1];
- for (int c=0; c<col; ++c) {
+ for (unsigned c=0; c<col; ++c) {
cpy.m[r1*col+c] /= a1;
tmp.m[r1*col+c] /= a1;
}
- for (int r2=0; r2<row; ++r2) {
+ for (unsigned r2=0; r2<row; ++r2) {
if (r2 != r1) {
ex a2 = cpy.m[r2*col+r1];
- for (int c=0; c<col; ++c) {
+ for (unsigned c=0; c<col; ++c) {
cpy.m[r2*col+c] -= a2 * cpy.m[r1*col+c];
tmp.m[r2*col+c] -= a2 * tmp.m[r1*col+c];
}
return tmp;
}
-void matrix::ffe_swap(int r1, int c1, int r2 ,int c2)
+
+// superfluous helper function
+void matrix::ffe_swap(unsigned r1, unsigned c1, unsigned r2 ,unsigned c2)
{
ensure_if_modifiable();
- ex tmp=ffe_get(r1,c1);
+ ex tmp = ffe_get(r1,c1);
ffe_set(r1,c1,ffe_get(r2,c2));
ffe_set(r2,c2,tmp);
}
-void matrix::ffe_set(int r, int c, ex e)
+// superfluous helper function
+void matrix::ffe_set(unsigned r, unsigned c, ex e)
{
set(r-1,c-1,e);
}
-ex matrix::ffe_get(int r, int c) const
+// superfluous helper function
+ex matrix::ffe_get(unsigned r, unsigned c) const
{
return operator()(r-1,c-1);
}
/** Solve a set of equations for an m x n matrix by fraction-free Gaussian
- * elimination. Based on algorithm 9.1 from 'Algorithms for Computer Algebra'
+ * elimination. Based on algorithm 9.1 from 'Algorithms for Computer Algebra'
* by Keith O. Geddes et al.
*
* @param vars n x p matrix
* @param rhs m x p matrix
* @exception logic_error (incompatible matrices)
* @exception runtime_error (singular matrix) */
-matrix matrix::fraction_free_elim(matrix const & vars,
- matrix const & rhs) const
+matrix matrix::fraction_free_elim(const matrix & vars,
+ const matrix & rhs) const
{
- if ((row != rhs.row) || (col != vars.row) || (rhs.col != vars.col)) {
- throw (std::logic_error("matrix::solve(): incompatible matrices"));
- }
+ // FIXME: implement a Sasaki-Murao scheme which avoids division at all!
+ if ((row != rhs.row) || (col != vars.row) || (rhs.col != vars.col))
+ throw (std::logic_error("matrix::fraction_free_elim(): incompatible matrices"));
- matrix a(*this); // make a copy of the matrix
- matrix b(rhs); // make a copy of the rhs vector
+ matrix a(*this); // make a copy of the matrix
+ matrix b(rhs); // make a copy of the rhs vector
// given an m x n matrix a, reduce it to upper echelon form
- int m=a.row;
- int n=a.col;
- int sign=1;
- ex divisor=1;
- int r=1;
+ unsigned m = a.row;
+ unsigned n = a.col;
+ int sign = 1;
+ ex divisor = 1;
+ unsigned r = 1;
// eliminate below row r, with pivot in column k
- for (int k=1; (k<=n)&&(r<=m); ++k) {
+ for (unsigned k=1; (k<=n)&&(r<=m); ++k) {
// find a nonzero pivot
- int p;
- for (p=r; (p<=m)&&(a.ffe_get(p,k).is_equal(exZERO())); ++p) {}
+ unsigned p;
+ for (p=r; (p<=m)&&(a.ffe_get(p,k).is_equal(_ex0())); ++p) {}
// pivot is in row p
if (p<=m) {
if (p!=r) {
// switch rows p and r
- for (int j=k; j<=n; ++j) {
+ for (unsigned j=k; j<=n; ++j)
a.ffe_swap(p,j,r,j);
- }
b.ffe_swap(p,1,r,1);
// keep track of sign changes due to row exchange
- sign=-sign;
+ sign = -sign;
}
- for (int i=r+1; i<=m; ++i) {
- for (int j=k+1; j<=n; ++j) {
+ for (unsigned i=r+1; i<=m; ++i) {
+ for (unsigned j=k+1; j<=n; ++j) {
a.ffe_set(i,j,(a.ffe_get(r,k)*a.ffe_get(i,j)
-a.ffe_get(r,j)*a.ffe_get(i,k))/divisor);
a.ffe_set(i,j,a.ffe_get(i,j).normal() /*.normal() */ );
b.ffe_set(i,1,b.ffe_get(i,1).normal() /*.normal() */ );
a.ffe_set(i,k,0);
}
- divisor=a.ffe_get(r,k);
+ divisor = a.ffe_get(r,k);
r++;
}
}
// optionally compute the determinant for square or augmented matrices
- // if (r==m+1) { det=sign*divisor; } else { det=0; }
+ // if (r==m+1) { det = sign*divisor; } else { det = 0; }
/*
- for (int r=1; r<=m; ++r) {
- for (int c=1; c<=n; ++c) {
+ for (unsigned r=1; r<=m; ++r) {
+ for (unsigned c=1; c<=n; ++c) {
cout << a.ffe_get(r,c) << "\t";
}
cout << " | " << b.ffe_get(r,1) << endl;
#ifdef DO_GINAC_ASSERT
// test if we really have an upper echelon matrix
- int zero_in_last_row=-1;
- for (int r=1; r<=m; ++r) {
+ int zero_in_last_row = -1;
+ for (unsigned r=1; r<=m; ++r) {
int zero_in_this_row=0;
- for (int c=1; c<=n; ++c) {
- if (a.ffe_get(r,c).is_equal(exZERO())) {
+ for (unsigned c=1; c<=n; ++c) {
+ if (a.ffe_get(r,c).is_equal(_ex0()))
zero_in_this_row++;
- } else {
+ else
break;
- }
}
GINAC_ASSERT((zero_in_this_row>zero_in_last_row)||(zero_in_this_row=n));
- zero_in_last_row=zero_in_this_row;
+ zero_in_last_row = zero_in_this_row;
}
#endif // def DO_GINAC_ASSERT
+ /*
+ cout << "after" << endl;
+ cout << "a=" << a << endl;
+ cout << "b=" << b << endl;
+ */
+
// assemble solution
matrix sol(n,1);
- int last_assigned_sol=n+1;
- for (int r=m; r>0; --r) {
- int first_non_zero=1;
- while ((first_non_zero<=n)&&(a.ffe_get(r,first_non_zero).is_zero())) {
+ unsigned last_assigned_sol = n+1;
+ for (unsigned r=m; r>0; --r) {
+ unsigned first_non_zero = 1;
+ while ((first_non_zero<=n)&&(a.ffe_get(r,first_non_zero).is_zero()))
first_non_zero++;
- }
if (first_non_zero>n) {
// row consists only of zeroes, corresponding rhs must be 0 as well
if (!b.ffe_get(r,1).is_zero()) {
} else {
// assign solutions for vars between first_non_zero+1 and
// last_assigned_sol-1: free parameters
- for (int c=first_non_zero+1; c<=last_assigned_sol-1; ++c) {
+ for (unsigned c=first_non_zero+1; c<=last_assigned_sol-1; ++c) {
sol.ffe_set(c,1,vars.ffe_get(c,1));
}
- ex e=b.ffe_get(r,1);
- for (int c=first_non_zero+1; c<=n; ++c) {
+ ex e = b.ffe_get(r,1);
+ for (unsigned c=first_non_zero+1; c<=n; ++c) {
e=e-a.ffe_get(r,c)*sol.ffe_get(c,1);
}
sol.ffe_set(first_non_zero,1,
(e/a.ffe_get(r,first_non_zero)).normal());
- last_assigned_sol=first_non_zero;
+ last_assigned_sol = first_non_zero;
}
}
// assign solutions for vars between 1 and
// last_assigned_sol-1: free parameters
- for (int c=1; c<=last_assigned_sol-1; ++c) {
+ for (unsigned c=1; c<=last_assigned_sol-1; ++c)
sol.ffe_set(c,1,vars.ffe_get(c,1));
- }
-
- /*
- for (int c=1; c<=n; ++c) {
- cout << vars.ffe_get(c,1) << "->" << sol.ffe_get(c,1) << endl;
- }
- */
#ifdef DO_GINAC_ASSERT
// test solution with echelon matrix
- for (int r=1; r<=m; ++r) {
- ex e=0;
- for (int c=1; c<=n; ++c) {
- e=e+a.ffe_get(r,c)*sol.ffe_get(c,1);
- }
+ for (unsigned r=1; r<=m; ++r) {
+ ex e = 0;
+ for (unsigned c=1; c<=n; ++c)
+ e = e+a.ffe_get(r,c)*sol.ffe_get(c,1);
if (!(e-b.ffe_get(r,1)).normal().is_zero()) {
cout << "e=" << e;
cout << "b.ffe_get(" << r<<",1)=" << b.ffe_get(r,1) << endl;
}
GINAC_ASSERT((e-b.ffe_get(r,1)).normal().is_zero());
}
-
+
// test solution with original matrix
- for (int r=1; r<=m; ++r) {
- ex e=0;
- for (int c=1; c<=n; ++c) {
- e=e+ffe_get(r,c)*sol.ffe_get(c,1);
- }
+ for (unsigned r=1; r<=m; ++r) {
+ ex e = 0;
+ for (unsigned c=1; c<=n; ++c)
+ e = e+ffe_get(r,c)*sol.ffe_get(c,1);
try {
- if (!(e-rhs.ffe_get(r,1)).normal().is_zero()) {
- cout << "e=" << e << endl;
- e.printtree(cout);
- ex en=e.normal();
- cout << "e.normal()=" << en << endl;
- en.printtree(cout);
- cout << "rhs.ffe_get(" << r<<",1)=" << rhs.ffe_get(r,1) << endl;
- cout << "diff=" << (e-rhs.ffe_get(r,1)).normal() << endl;
- }
+ if (!(e-rhs.ffe_get(r,1)).normal().is_zero()) {
+ cout << "e=" << e << endl;
+ e.printtree(cout);
+ ex en = e.normal();
+ cout << "e.normal()=" << en << endl;
+ en.printtree(cout);
+ cout << "rhs.ffe_get(" << r<<",1)=" << rhs.ffe_get(r,1) << endl;
+ cout << "diff=" << (e-rhs.ffe_get(r,1)).normal() << endl;
+ }
} catch (...) {
- ex xxx=e-rhs.ffe_get(r,1);
+ ex xxx = e - rhs.ffe_get(r,1);
cerr << "xxx=" << xxx << endl << endl;
}
GINAC_ASSERT((e-rhs.ffe_get(r,1)).normal().is_zero());
#endif // def DO_GINAC_ASSERT
return sol;
-}
+}
+
+/** Solve a set of equations for an m x n matrix.
+ *
+ * @param vars n x p matrix
+ * @param rhs m x p matrix
+ * @exception logic_error (incompatible matrices)
+ * @exception runtime_error (singular matrix) */
+matrix matrix::solve(const matrix & vars,
+ const matrix & rhs) const
+{
+ if ((row != rhs.row) || (col != vars.row) || (rhs.col != vars.col))
+ throw (std::logic_error("matrix::solve(): incompatible matrices"));
-/** Solve simultaneous set of equations. */
-matrix matrix::solve(matrix const & v) const
+ throw (std::runtime_error("FIXME: need implementation."));
+}
+
+/** Old and obsolete interface: */
+matrix matrix::old_solve(const matrix & v) const
{
- if (!(row == col && col == v.row)) {
+ if ((v.row != col) || (col != v.row))
throw (std::logic_error("matrix::solve(): incompatible matrices"));
- }
- // build the extended matrix of *this with v attached to the right
+ // build the augmented matrix of *this with v attached to the right
matrix tmp(row,col+v.col);
- for (int r=0; r<row; ++r) {
- for (int c=0; c<col; ++c) {
- tmp.m[r*tmp.col+c] = m[r*col+c];
- }
- for (int c=0; c<v.col; ++c) {
+ for (unsigned r=0; r<row; ++r) {
+ for (unsigned c=0; c<col; ++c)
+ tmp.m[r*tmp.col+c] = this->m[r*col+c];
+ for (unsigned c=0; c<v.col; ++c)
tmp.m[r*tmp.col+c+col] = v.m[r*v.col+c];
+ }
+ // cout << "augmented: " << tmp << endl;
+ tmp.gauss_elimination();
+ // cout << "degaussed: " << tmp << endl;
+ // assemble the solution matrix
+ exvector sol(v.row*v.col);
+ for (unsigned c=0; c<v.col; ++c) {
+ for (unsigned r=row; r>0; --r) {
+ for (unsigned i=r; i<col; ++i)
+ sol[(r-1)*v.col+c] -= tmp.m[(r-1)*tmp.col+i]*sol[i*v.col+c];
+ sol[(r-1)*v.col+c] += tmp.m[(r-1)*tmp.col+col+c];
+ sol[(r-1)*v.col+c] = (sol[(r-1)*v.col+c]/tmp.m[(r-1)*tmp.col+(r-1)]).normal();
}
}
- for (int r1=0; r1<row; ++r1) {
+ return matrix(v.row, v.col, sol);
+}
+
+
+// protected
+
+/** Determinant of purely numeric matrix, using pivoting.
+ *
+ * @see matrix::determinant() */
+ex matrix::determinant_numeric(void) const
+{
+ matrix tmp(*this);
+ ex det = _ex1();
+ ex piv;
+
+ for (unsigned r1=0; r1<row; ++r1) {
int indx = tmp.pivot(r1);
- if (indx == -1) {
- throw (std::runtime_error("matrix::solve(): singular matrix"));
- }
- for (int c=r1; c<tmp.col; ++c) {
- tmp.m[r1*tmp.col+c] /= tmp.m[r1*tmp.col+r1];
- }
- for (int r2=r1+1; r2<row; ++r2) {
- for (int c=r1; c<tmp.col; ++c) {
- tmp.m[r2*tmp.col+c]
- -= tmp.m[r2*tmp.col+r1] * tmp.m[r1*tmp.col+c];
+ if (indx == -1)
+ return _ex0();
+ if (indx != 0)
+ det *= _ex_1();
+ det = det * tmp.m[r1*col+r1];
+ for (unsigned r2=r1+1; r2<row; ++r2) {
+ piv = tmp.m[r2*col+r1] / tmp.m[r1*col+r1];
+ for (unsigned c=r1+1; c<col; c++) {
+ tmp.m[r2*col+c] -= piv * tmp.m[r1*col+c];
}
}
}
- // assemble the solution matrix
- vector<ex> sol(v.row*v.col);
- for (int c=0; c<v.col; ++c) {
- for (int r=col-1; r>=0; --r) {
- sol[r*v.col+c] = tmp[r*tmp.col+c];
- for (int i=r+1; i<col; ++i) {
- sol[r*v.col+c]
- -= tmp[r*tmp.col+i] * sol[i*v.col+c];
+ return det;
+}
+
+
+/* Leverrier algorithm for large matrices having at least one symbolic entry.
+ * This routine is only called internally by matrix::determinant(). The
+ * algorithm is very bad for symbolic matrices since it returns expressions
+ * that are quite hard to expand. */
+/*ex matrix::determinant_leverrier(const matrix & M)
+ *{
+ * GINAC_ASSERT(M.rows()==M.cols()); // cannot happen, just in case...
+ *
+ * matrix B(M);
+ * matrix I(M.row, M.col);
+ * ex c=B.trace();
+ * for (unsigned i=1; i<M.row; ++i) {
+ * for (unsigned j=0; j<M.row; ++j)
+ * I.m[j*M.col+j] = c;
+ * B = M.mul(B.sub(I));
+ * c = B.trace()/ex(i+1);
+ * }
+ * if (M.row%2) {
+ * return c;
+ * } else {
+ * return -c;
+ * }
+ *}*/
+
+
+/** Recursive determinant for small matrices having at least one symbolic
+ * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
+ * some bookkeeping to avoid calculation of the same submatrices ("minors")
+ * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
+ * is better than elimination schemes for matrices of sparse multivariate
+ * polynomials and also for matrices of dense univariate polynomials if the
+ * matrix' dimesion is larger than 7.
+ *
+ * @return the determinant as a new expression (in expanded form)
+ * @see matrix::determinant() */
+ex matrix::determinant_minor(void) const
+{
+ // for small matrices the algorithm does not make any sense:
+ if (this->row==1)
+ return m[0];
+ if (this->row==2)
+ return (m[0]*m[3]-m[2]*m[1]).expand();
+ if (this->row==3)
+ return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
+ m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
+ m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
+
+ // This algorithm can best be understood by looking at a naive
+ // implementation of Laplace-expansion, like this one:
+ // ex det;
+ // matrix minorM(this->row-1,this->col-1);
+ // for (unsigned r1=0; r1<this->row; ++r1) {
+ // // shortcut if element(r1,0) vanishes
+ // if (m[r1*col].is_zero())
+ // continue;
+ // // assemble the minor matrix
+ // for (unsigned r=0; r<minorM.rows(); ++r) {
+ // for (unsigned c=0; c<minorM.cols(); ++c) {
+ // if (r<r1)
+ // minorM.set(r,c,m[r*col+c+1]);
+ // else
+ // minorM.set(r,c,m[(r+1)*col+c+1]);
+ // }
+ // }
+ // // recurse down and care for sign:
+ // if (r1%2)
+ // det -= m[r1*col] * minorM.determinant_minor();
+ // else
+ // det += m[r1*col] * minorM.determinant_minor();
+ // }
+ // return det.expand();
+ // What happens is that while proceeding down many of the minors are
+ // computed more than once. In particular, there are binomial(n,k)
+ // kxk minors and each one is computed factorial(n-k) times. Therefore
+ // it is reasonable to store the results of the minors. We proceed from
+ // right to left. At each column c we only need to retrieve the minors
+ // calculated in step c-1. We therefore only have to store at most
+ // 2*binomial(n,n/2) minors.
+
+ // Unique flipper counter for partitioning into minors
+ vector<unsigned> Pkey;
+ Pkey.reserve(this->col);
+ // key for minor determinant (a subpartition of Pkey)
+ vector<unsigned> Mkey;
+ Mkey.reserve(this->col-1);
+ // we store our subminors in maps, keys being the rows they arise from
+ typedef map<vector<unsigned>,class ex> Rmap;
+ typedef map<vector<unsigned>,class ex>::value_type Rmap_value;
+ Rmap A;
+ Rmap B;
+ ex det;
+ // initialize A with last column:
+ for (unsigned r=0; r<this->col; ++r) {
+ Pkey.erase(Pkey.begin(),Pkey.end());
+ Pkey.push_back(r);
+ A.insert(Rmap_value(Pkey,m[this->col*r+this->col-1]));
+ }
+ // proceed from right to left through matrix
+ for (int c=this->col-2; c>=0; --c) {
+ Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
+ Mkey.erase(Mkey.begin(),Mkey.end());
+ for (unsigned i=0; i<this->col-c; ++i)
+ Pkey.push_back(i);
+ unsigned fc = 0; // controls logic for our strange flipper counter
+ do {
+ A.insert(Rmap_value(Pkey,_ex0()));
+ det = _ex0();
+ for (unsigned r=0; r<this->col-c; ++r) {
+ // maybe there is nothing to do?
+ if (m[Pkey[r]*this->col+c].is_zero())
+ continue;
+ // create the sorted key for all possible minors
+ Mkey.erase(Mkey.begin(),Mkey.end());
+ for (unsigned i=0; i<this->col-c; ++i)
+ if (i!=r)
+ Mkey.push_back(Pkey[i]);
+ // Fetch the minors and compute the new determinant
+ if (r%2)
+ det -= m[Pkey[r]*this->col+c]*A[Mkey];
+ else
+ det += m[Pkey[r]*this->col+c]*A[Mkey];
}
+ // prevent build-up of deep nesting of expressions saves some time:
+ det = det.expand();
+ // store the new determinant at its place in B:
+ B.insert(Rmap_value(Pkey,det));
+ // increment our strange flipper counter
+ for (fc=this->col-c; fc>0; --fc) {
+ ++Pkey[fc-1];
+ if (Pkey[fc-1]<fc+c)
+ break;
+ }
+ if (fc<this->col-c)
+ for (unsigned j=fc; j<this->col-c; ++j)
+ Pkey[j] = Pkey[j-1]+1;
+ } while(fc);
+ // next column, so change the role of A and B:
+ A = B;
+ B.clear();
+ }
+
+ return det;
+}
+
+
+/** Determinant built by application of the full permutation group. This
+ * routine is only called internally by matrix::determinant(). */
+ex matrix::determinant_perm(void) const
+{
+ if (rows()==1) // speed things up
+ return m[0];
+
+ ex det;
+ ex term;
+ vector<unsigned> sigma(col);
+ for (unsigned i=0; i<col; ++i)
+ sigma[i]=i;
+
+ do {
+ term = (*this)(sigma[0],0);
+ for (unsigned i=1; i<col; ++i)
+ term *= (*this)(sigma[i],i);
+ det += permutation_sign(sigma)*term;
+ } while (next_permutation(sigma.begin(), sigma.end()));
+
+ return det;
+}
+
+
+/** Perform the steps of an ordinary Gaussian elimination to bring the matrix
+ * into an upper echelon form.
+ *
+ * @return sign is 1 if an even number of rows was swapped, -1 if an odd
+ * number of rows was swapped and 0 if the matrix is singular. */
+int matrix::gauss_elimination(void)
+{
+ int sign = 1;
+ ensure_if_modifiable();
+ for (unsigned r1=0; r1<row-1; ++r1) {
+ int indx = pivot(r1);
+ if (indx == -1)
+ return 0; // Note: leaves *this in a messy state.
+ if (indx > 0)
+ sign = -sign;
+ for (unsigned r2=r1+1; r2<row; ++r2) {
+ for (unsigned c=r1+1; c<col; ++c)
+ this->m[r2*col+c] -= this->m[r2*col+r1]*this->m[r1*col+c]/this->m[r1*col+r1];
+ for (unsigned c=0; c<=r1; ++c)
+ this->m[r2*col+c] = _ex0();
}
}
- return matrix(v.row, v.col, sol);
+
+ return sign;
}
-// protected
/** Partial pivoting method.
- * Usual pivoting returns the index to the element with the largest absolute
- * value and swaps the current row with the one where the element was found.
- * Here it does the same with the first non-zero element. (This works fine,
- * but may be far from optimal for numerics.) */
-int matrix::pivot(int ro)
+ * Usual pivoting (symbolic==false) returns the index to the element with the
+ * largest absolute value in column ro and swaps the current row with the one
+ * where the element was found. With (symbolic==true) it does the same thing
+ * with the first non-zero element.
+ *
+ * @param ro is the row to be inspected
+ * @param symbolic signal if we want the first non-zero element to be pivoted
+ * (true) or the one with the largest absolute value (false).
+ * @return 0 if no interchange occured, -1 if all are zero (usually signaling
+ * a degeneracy) and positive integer k means that rows ro and k were swapped.
+ */
+int matrix::pivot(unsigned ro, bool symbolic)
{
- int k=ro;
+ unsigned k = ro;
- for (int r=ro; r<row; ++r) {
- if (!m[r*col+ro].is_zero()) {
- k = r;
- break;
+ if (symbolic) { // search first non-zero
+ for (unsigned r=ro; r<row; ++r) {
+ if (!m[r*col+ro].is_zero()) {
+ k = r;
+ break;
+ }
+ }
+ } else { // search largest
+ numeric tmp(0);
+ numeric maxn(-1);
+ for (unsigned r=ro; r<row; ++r) {
+ GINAC_ASSERT(is_ex_of_type(m[r*col+ro],numeric));
+ if ((tmp = abs(ex_to_numeric(m[r*col+ro]))) > maxn &&
+ !tmp.is_zero()) {
+ maxn = tmp;
+ k = r;
+ }
}
}
- if (m[k*col+ro].is_zero()) {
+ if (m[k*col+ro].is_zero())
return -1;
- }
if (k!=ro) { // swap rows
- for (int c=0; c<col; ++c) {
+ ensure_if_modifiable();
+ for (unsigned c=0; c<col; ++c) {
m[k*col+c].swap(m[ro*col+c]);
}
return k;
//////////
const matrix some_matrix;
-type_info const & typeid_matrix=typeid(some_matrix);
+const type_info & typeid_matrix=typeid(some_matrix);
-#ifndef NO_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC
} // namespace GiNaC
-#endif // ndef NO_GINAC_NAMESPACE
+#endif // ndef NO_NAMESPACE_GINAC
git://www.ginac.de / ginac.git / blobdiff