* Implementation of symbolic matrices */
/*
- * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2003 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
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*/
+#include <string>
+#include <iostream>
+#include <sstream>
#include <algorithm>
#include <map>
#include <stdexcept>
#include "matrix.h"
-#include "archive.h"
#include "numeric.h"
#include "lst.h"
-#include "utils.h"
-#include "debugmsg.h"
+#include "idx.h"
+#include "indexed.h"
+#include "add.h"
#include "power.h"
#include "symbol.h"
+#include "operators.h"
#include "normal.h"
+#include "archive.h"
+#include "utils.h"
-#ifndef NO_NAMESPACE_GINAC
namespace GiNaC {
-#endif // ndef NO_NAMESPACE_GINAC
-GINAC_IMPLEMENT_REGISTERED_CLASS(matrix, basic)
+GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(matrix, basic,
+ print_func<print_context>(&matrix::do_print).
+ print_func<print_latex>(&matrix::do_print_latex).
+ print_func<print_tree>(&basic::do_print_tree).
+ print_func<print_python_repr>(&matrix::do_print_python_repr))
//////////
-// default constructor, destructor, copy constructor, assignment operator
-// and helpers:
+// default constructor
//////////
-// public
-
/** Default ctor. Initializes to 1 x 1-dimensional zero-matrix. */
-matrix::matrix()
- : inherited(TINFO_matrix), row(1), col(1)
-{
- debugmsg("matrix default constructor",LOGLEVEL_CONSTRUCT);
- m.push_back(_ex0());
-}
-
-matrix::~matrix()
-{
- debugmsg("matrix destructor",LOGLEVEL_DESTRUCT);
-}
-
-matrix::matrix(const matrix & other)
-{
- debugmsg("matrix copy constructor",LOGLEVEL_CONSTRUCT);
- copy(other);
-}
-
-const matrix & matrix::operator=(const matrix & other)
+matrix::matrix() : inherited(TINFO_matrix), row(1), col(1), m(1, _ex0)
{
- debugmsg("matrix operator=",LOGLEVEL_ASSIGNMENT);
- if (this != &other) {
- destroy(1);
- copy(other);
- }
- return *this;
-}
-
-// protected
-
-void matrix::copy(const matrix & other)
-{
- inherited::copy(other);
- row = other.row;
- col = other.col;
- m = other.m; // STL's vector copying invoked here
-}
-
-void matrix::destroy(bool call_parent)
-{
- if (call_parent) inherited::destroy(call_parent);
+ setflag(status_flags::not_shareable);
}
//////////
* @param r number of rows
* @param c number of cols */
matrix::matrix(unsigned r, unsigned c)
- : inherited(TINFO_matrix), row(r), col(c)
+ : inherited(TINFO_matrix), row(r), col(c), m(r*c, _ex0)
{
- debugmsg("matrix constructor from unsigned,unsigned",LOGLEVEL_CONSTRUCT);
- m.resize(r*c, _ex0());
+ setflag(status_flags::not_shareable);
}
- // protected
+// protected
/** Ctor from representation, for internal use only. */
matrix::matrix(unsigned r, unsigned c, const exvector & m2)
- : inherited(TINFO_matrix), row(r), col(c), m(m2)
+ : inherited(TINFO_matrix), row(r), col(c), m(m2)
+{
+ setflag(status_flags::not_shareable);
+}
+
+/** Construct matrix from (flat) list of elements. If the list has fewer
+ * elements than the matrix, the remaining matrix elements are set to zero.
+ * If the list has more elements than the matrix, the excessive elements are
+ * thrown away. */
+matrix::matrix(unsigned r, unsigned c, const lst & l)
+ : inherited(TINFO_matrix), row(r), col(c), m(r*c, _ex0)
{
- debugmsg("matrix constructor from unsigned,unsigned,exvector",LOGLEVEL_CONSTRUCT);
+ setflag(status_flags::not_shareable);
+
+ size_t i = 0;
+ for (lst::const_iterator it = l.begin(); it != l.end(); ++it, ++i) {
+ size_t x = i % c;
+ size_t y = i / c;
+ if (y >= r)
+ break; // matrix smaller than list: throw away excessive elements
+ m[y*c+x] = *it;
+ }
}
//////////
// 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)
+matrix::matrix(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
{
- return (new matrix(n, sym_lst))->setflag(status_flags::dynallocated);
+ setflag(status_flags::not_shareable);
+
+ 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;
+ }
}
-/** 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++;
- }
+ 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;
+ }
}
+DEFAULT_UNARCHIVE(matrix)
+
//////////
-// functions overriding virtual functions from bases classes
+// functions overriding virtual functions from base classes
//////////
// public
-basic * matrix::duplicate() const
+void matrix::print_elements(const print_context & c, const char *row_start, const char *row_end, const char *row_sep, const char *col_sep) const
+{
+ for (unsigned ro=0; ro<row; ++ro) {
+ c.s << row_start;
+ for (unsigned co=0; co<col; ++co) {
+ m[ro*col+co].print(c);
+ if (co < col-1)
+ c.s << col_sep;
+ else
+ c.s << row_end;
+ }
+ if (ro < row-1)
+ c.s << row_sep;
+ }
+}
+
+void matrix::do_print(const print_context & c, unsigned level) const
{
- debugmsg("matrix duplicate",LOGLEVEL_DUPLICATE);
- return new matrix(*this);
+ c.s << "[";
+ print_elements(c, "[", "]", ",", ",");
+ c.s << "]";
}
-void matrix::print(ostream & os, unsigned upper_precedence) const
+void matrix::do_print_latex(const print_latex & c, unsigned level) const
{
- debugmsg("matrix print",LOGLEVEL_PRINT);
- os << "[[ ";
- for (unsigned r=0; r<row-1; ++r) {
- os << "[[";
- for (unsigned c=0; c<col-1; ++c) {
- os << m[r*col+c] << ",";
- }
- os << m[col*(r+1)-1] << "]], ";
- }
- os << "[[";
- for (unsigned c=0; c<col-1; ++c) {
- os << m[(row-1)*col+c] << ",";
- }
- os << m[row*col-1] << "]] ]]";
+ c.s << "\\left(\\begin{array}{" << std::string(col,'c') << "}";
+ print_elements(c, "", "", "\\\\", "&");
+ c.s << "\\end{array}\\right)";
}
-void matrix::printraw(ostream & os) const
+void matrix::do_print_python_repr(const print_python_repr & c, unsigned level) const
{
- debugmsg("matrix printraw",LOGLEVEL_PRINT);
- os << "matrix(" << row << "," << col <<",";
- for (unsigned r=0; r<row-1; ++r) {
- os << "(";
- for (unsigned c=0; c<col-1; ++c) {
- os << m[r*col+c] << ",";
- }
- os << m[col*(r-1)-1] << "),";
- }
- os << "(";
- for (unsigned c=0; c<col-1; ++c) {
- os << m[(row-1)*col+c] << ",";
- }
- os << m[row*col-1] << "))";
+ c.s << class_name() << '(';
+ print_elements(c, "[", "]", ",", ",");
+ c.s << ')';
}
/** nops is defined to be rows x columns. */
-unsigned matrix::nops() const
+size_t matrix::nops() const
{
- return row*col;
+ return static_cast<size_t>(row) * static_cast<size_t>(col);
}
/** returns matrix entry at position (i/col, i%col). */
-ex matrix::op(int i) const
+ex matrix::op(size_t i) const
{
- return m[i];
+ GINAC_ASSERT(i<nops());
+
+ return m[i];
}
-/** returns matrix entry at position (i/col, i%col). */
-ex & matrix::let_op(int i)
+/** returns writable matrix entry at position (i/col, i%col). */
+ex & matrix::let_op(size_t i)
{
- return m[i];
+ GINAC_ASSERT(i<nops());
+
+ ensure_if_modifiable();
+ return m[i];
}
-/** expands the elements of a matrix entry by entry. */
-ex matrix::expand(unsigned options) const
+/** Evaluate matrix entry by entry. */
+ex matrix::eval(int level) const
{
- exvector tmp(row*col);
- for (unsigned i=0; i<row*col; ++i) {
- tmp[i]=m[i].expand(options);
- }
- return matrix(row, col, tmp);
+ // check if we have to do anything at all
+ if ((level==1)&&(flags & status_flags::evaluated))
+ return *this;
+
+ // emergency break
+ if (level == -max_recursion_level)
+ throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
+
+ // eval() entry by entry
+ exvector m2(row*col);
+ --level;
+ for (unsigned r=0; r<row; ++r)
+ for (unsigned c=0; c<col; ++c)
+ m2[r*col+c] = m[r*col+c].eval(level);
+
+ return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
+ status_flags::evaluated);
}
-/** Search ocurrences. A matrix 'has' an expression if it is the expression
- * itself or one of the elements 'has' it. */
-bool matrix::has(const ex & other) const
+ex matrix::subs(const exmap & mp, unsigned options) const
{
- GINAC_ASSERT(other.bp!=0);
-
- // tautology: it is the expression itself
- if (is_equal(*other.bp)) return true;
-
- // search all the elements
- for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
- if ((*r).has(other)) return true;
- }
- return false;
+ exvector m2(row * col);
+ for (unsigned r=0; r<row; ++r)
+ for (unsigned c=0; c<col; ++c)
+ m2[r*col+c] = m[r*col+c].subs(mp, options);
+
+ return matrix(row, col, m2).subs_one_level(mp, options);
}
-/** evaluate matrix entry by entry. */
-ex matrix::eval(int level) const
+// protected
+
+int matrix::compare_same_type(const basic & other) const
{
- debugmsg("matrix eval",LOGLEVEL_MEMBER_FUNCTION);
-
- // check if we have to do anything at all
- if ((level==1)&&(flags & status_flags::evaluated))
- return *this;
-
- // emergency break
- if (level == -max_recursion_level)
- throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
-
- // eval() entry by entry
- exvector m2(row*col);
- --level;
- for (unsigned r=0; r<row; ++r) {
- for (unsigned c=0; c<col; ++c) {
- m2[r*col+c] = m[r*col+c].eval(level);
- }
- }
-
- return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
- status_flags::evaluated );
+ GINAC_ASSERT(is_exactly_a<matrix>(other));
+ const matrix &o = static_cast<const matrix &>(other);
+
+ // compare number of rows
+ if (row != o.rows())
+ return row < o.rows() ? -1 : 1;
+
+ // compare number of columns
+ if (col != o.cols())
+ return col < o.cols() ? -1 : 1;
+
+ // equal number of rows and columns, compare individual elements
+ int cmpval;
+ 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;
+ }
+ }
+ // all elements are equal => matrices are equal;
+ return 0;
}
-/** evaluate matrix numerically entry by entry. */
-ex matrix::evalf(int level) const
+bool matrix::match_same_type(const basic & other) const
{
- debugmsg("matrix evalf",LOGLEVEL_MEMBER_FUNCTION);
-
- // check if we have to do anything at all
- if (level==1)
- return *this;
-
- // emergency break
- if (level == -max_recursion_level) {
- throw (std::runtime_error("matrix::evalf(): recursion limit exceeded"));
- }
-
- // evalf() entry by entry
- exvector m2(row*col);
- --level;
- for (unsigned r=0; r<row; ++r) {
- for (unsigned c=0; c<col; ++c) {
- m2[r*col+c] = m[r*col+c].evalf(level);
- }
- }
- return matrix(row, col, m2);
+ GINAC_ASSERT(is_exactly_a<matrix>(other));
+ const matrix & o = static_cast<const matrix &>(other);
+
+ // The number of rows and columns must be the same. This is necessary to
+ // prevent a 2x3 matrix from matching a 3x2 one.
+ return row == o.rows() && col == o.cols();
}
-// protected
+/** Automatic symbolic evaluation of an indexed matrix. */
+ex matrix::eval_indexed(const basic & i) const
+{
+ GINAC_ASSERT(is_a<indexed>(i));
+ GINAC_ASSERT(is_a<matrix>(i.op(0)));
+
+ bool all_indices_unsigned = static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint);
+
+ // Check indices
+ if (i.nops() == 2) {
+
+ // One index, must be one-dimensional vector
+ if (row != 1 && col != 1)
+ throw (std::runtime_error("matrix::eval_indexed(): vector must have exactly 1 index"));
+
+ const idx & i1 = ex_to<idx>(i.op(1));
+
+ if (col == 1) {
+
+ // Column vector
+ if (!i1.get_dim().is_equal(row))
+ throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
+
+ // Index numeric -> return vector element
+ if (all_indices_unsigned) {
+ unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
+ if (n1 >= row)
+ throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
+ return (*this)(n1, 0);
+ }
+
+ } else {
+
+ // Row vector
+ if (!i1.get_dim().is_equal(col))
+ throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
+
+ // Index numeric -> return vector element
+ if (all_indices_unsigned) {
+ unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
+ if (n1 >= col)
+ throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
+ return (*this)(0, n1);
+ }
+ }
+
+ } else if (i.nops() == 3) {
+
+ // Two indices
+ const idx & i1 = ex_to<idx>(i.op(1));
+ const idx & i2 = ex_to<idx>(i.op(2));
+
+ if (!i1.get_dim().is_equal(row))
+ throw (std::runtime_error("matrix::eval_indexed(): dimension of first index must match number of rows"));
+ if (!i2.get_dim().is_equal(col))
+ throw (std::runtime_error("matrix::eval_indexed(): dimension of second index must match number of columns"));
+
+ // Pair of dummy indices -> compute trace
+ if (is_dummy_pair(i1, i2))
+ return trace();
+
+ // Both indices numeric -> return matrix element
+ if (all_indices_unsigned) {
+ unsigned n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
+ if (n1 >= row)
+ throw (std::runtime_error("matrix::eval_indexed(): value of first index exceeds number of rows"));
+ if (n2 >= col)
+ throw (std::runtime_error("matrix::eval_indexed(): value of second index exceeds number of columns"));
+ return (*this)(n1, n2);
+ }
+
+ } else
+ throw (std::runtime_error("matrix::eval_indexed(): matrix must have exactly 2 indices"));
+
+ return i.hold();
+}
-int matrix::compare_same_type(const basic & other) const
+/** Sum of two indexed matrices. */
+ex matrix::add_indexed(const ex & self, const ex & other) const
+{
+ GINAC_ASSERT(is_a<indexed>(self));
+ GINAC_ASSERT(is_a<matrix>(self.op(0)));
+ GINAC_ASSERT(is_a<indexed>(other));
+ GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
+
+ // Only add two matrices
+ if (is_a<matrix>(other.op(0))) {
+ GINAC_ASSERT(other.nops() == 2 || other.nops() == 3);
+
+ const matrix &self_matrix = ex_to<matrix>(self.op(0));
+ const matrix &other_matrix = ex_to<matrix>(other.op(0));
+
+ if (self.nops() == 2 && other.nops() == 2) { // vector + vector
+
+ if (self_matrix.row == other_matrix.row)
+ return indexed(self_matrix.add(other_matrix), self.op(1));
+ else if (self_matrix.row == other_matrix.col)
+ return indexed(self_matrix.add(other_matrix.transpose()), self.op(1));
+
+ } else if (self.nops() == 3 && other.nops() == 3) { // matrix + matrix
+
+ if (self.op(1).is_equal(other.op(1)) && self.op(2).is_equal(other.op(2)))
+ return indexed(self_matrix.add(other_matrix), self.op(1), self.op(2));
+ else if (self.op(1).is_equal(other.op(2)) && self.op(2).is_equal(other.op(1)))
+ return indexed(self_matrix.add(other_matrix.transpose()), self.op(1), self.op(2));
+
+ }
+ }
+
+ // Don't know what to do, return unevaluated sum
+ return self + other;
+}
+
+/** Product of an indexed matrix with a number. */
+ex matrix::scalar_mul_indexed(const ex & self, const numeric & other) const
+{
+ GINAC_ASSERT(is_a<indexed>(self));
+ GINAC_ASSERT(is_a<matrix>(self.op(0)));
+ GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
+
+ const matrix &self_matrix = ex_to<matrix>(self.op(0));
+
+ if (self.nops() == 2)
+ return indexed(self_matrix.mul(other), self.op(1));
+ else // self.nops() == 3
+ return indexed(self_matrix.mul(other), self.op(1), self.op(2));
+}
+
+/** Contraction of an indexed matrix with something else. */
+bool matrix::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
{
- GINAC_ASSERT(is_exactly_of_type(other, matrix));
- const matrix & o = static_cast<matrix &>(const_cast<basic &>(other));
-
- // compare number of rows
- if (row != o.rows())
- return row < o.rows() ? -1 : 1;
-
- // compare number of columns
- if (col != o.cols())
- return col < o.cols() ? -1 : 1;
-
- // equal number of rows and columns, compare individual elements
- int cmpval;
- 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;
- }
- }
- // all elements are equal => matrices are equal;
- return 0;
+ GINAC_ASSERT(is_a<indexed>(*self));
+ GINAC_ASSERT(is_a<indexed>(*other));
+ GINAC_ASSERT(self->nops() == 2 || self->nops() == 3);
+ GINAC_ASSERT(is_a<matrix>(self->op(0)));
+
+ // Only contract with other matrices
+ if (!is_a<matrix>(other->op(0)))
+ return false;
+
+ GINAC_ASSERT(other->nops() == 2 || other->nops() == 3);
+
+ const matrix &self_matrix = ex_to<matrix>(self->op(0));
+ const matrix &other_matrix = ex_to<matrix>(other->op(0));
+
+ if (self->nops() == 2) {
+
+ if (other->nops() == 2) { // vector * vector (scalar product)
+
+ if (self_matrix.col == 1) {
+ if (other_matrix.col == 1) {
+ // Column vector * column vector, transpose first vector
+ *self = self_matrix.transpose().mul(other_matrix)(0, 0);
+ } else {
+ // Column vector * row vector, swap factors
+ *self = other_matrix.mul(self_matrix)(0, 0);
+ }
+ } else {
+ if (other_matrix.col == 1) {
+ // Row vector * column vector, perfect
+ *self = self_matrix.mul(other_matrix)(0, 0);
+ } else {
+ // Row vector * row vector, transpose second vector
+ *self = self_matrix.mul(other_matrix.transpose())(0, 0);
+ }
+ }
+ *other = _ex1;
+ return true;
+
+ } else { // vector * matrix
+
+ // B_i * A_ij = (B*A)_j (B is row vector)
+ if (is_dummy_pair(self->op(1), other->op(1))) {
+ if (self_matrix.row == 1)
+ *self = indexed(self_matrix.mul(other_matrix), other->op(2));
+ else
+ *self = indexed(self_matrix.transpose().mul(other_matrix), other->op(2));
+ *other = _ex1;
+ return true;
+ }
+
+ // B_j * A_ij = (A*B)_i (B is column vector)
+ if (is_dummy_pair(self->op(1), other->op(2))) {
+ if (self_matrix.col == 1)
+ *self = indexed(other_matrix.mul(self_matrix), other->op(1));
+ else
+ *self = indexed(other_matrix.mul(self_matrix.transpose()), other->op(1));
+ *other = _ex1;
+ return true;
+ }
+ }
+
+ } else if (other->nops() == 3) { // matrix * matrix
+
+ // A_ij * B_jk = (A*B)_ik
+ if (is_dummy_pair(self->op(2), other->op(1))) {
+ *self = indexed(self_matrix.mul(other_matrix), self->op(1), other->op(2));
+ *other = _ex1;
+ return true;
+ }
+
+ // A_ij * B_kj = (A*Btrans)_ik
+ if (is_dummy_pair(self->op(2), other->op(2))) {
+ *self = indexed(self_matrix.mul(other_matrix.transpose()), self->op(1), other->op(1));
+ *other = _ex1;
+ return true;
+ }
+
+ // A_ji * B_jk = (Atrans*B)_ik
+ if (is_dummy_pair(self->op(1), other->op(1))) {
+ *self = indexed(self_matrix.transpose().mul(other_matrix), self->op(2), other->op(2));
+ *other = _ex1;
+ return true;
+ }
+
+ // A_ji * B_kj = (B*A)_ki
+ if (is_dummy_pair(self->op(1), other->op(2))) {
+ *self = indexed(other_matrix.mul(self_matrix), other->op(1), self->op(2));
+ *other = _ex1;
+ return true;
+ }
+ }
+
+ return false;
}
+
//////////
// non-virtual functions in this class
//////////
* @exception logic_error (incompatible matrices) */
matrix matrix::add(const matrix & other) const
{
- if (col != other.col || row != other.row)
- throw (std::logic_error("matrix::add(): incompatible matrices"));
-
- exvector sum(this->m);
- exvector::iterator i;
- exvector::const_iterator ci;
- for (i=sum.begin(), ci=other.m.begin();
- i!=sum.end();
- ++i, ++ci) {
- (*i) += (*ci);
- }
- return matrix(row,col,sum);
+ if (col != other.col || row != other.row)
+ throw std::logic_error("matrix::add(): incompatible matrices");
+
+ exvector sum(this->m);
+ exvector::iterator i = sum.begin(), end = sum.end();
+ exvector::const_iterator ci = other.m.begin();
+ while (i != end)
+ *i++ += *ci++;
+
+ return matrix(row,col,sum);
}
* @exception logic_error (incompatible matrices) */
matrix matrix::sub(const matrix & other) const
{
- if (col != other.col || row != other.row)
- throw (std::logic_error("matrix::sub(): incompatible matrices"));
-
- exvector dif(this->m);
- exvector::iterator i;
- exvector::const_iterator ci;
- for (i=dif.begin(), ci=other.m.begin();
- i!=dif.end();
- ++i, ++ci) {
- (*i) -= (*ci);
- }
- return matrix(row,col,dif);
+ if (col != other.col || row != other.row)
+ throw std::logic_error("matrix::sub(): incompatible matrices");
+
+ exvector dif(this->m);
+ exvector::iterator i = dif.begin(), end = dif.end();
+ exvector::const_iterator ci = other.m.begin();
+ while (i != end)
+ *i++ -= *ci++;
+
+ return matrix(row,col,dif);
}
* @exception logic_error (incompatible matrices) */
matrix matrix::mul(const matrix & other) const
{
- if (col != other.row)
- throw (std::logic_error("matrix::mul(): incompatible matrices"));
-
- exvector prod(row*other.col);
-
- for (unsigned r1=0; r1<row; ++r1) {
- for (unsigned c=0; c<col; ++c) {
- if (m[r1*col+c].is_zero())
- continue;
- for (unsigned r2=0; r2<other.col; ++r2)
- prod[r1*other.col+r2] += m[r1*col+c] * other.m[c*other.col+r2];
- }
- }
- return matrix(row, other.col, prod);
+ if (this->cols() != other.rows())
+ throw std::logic_error("matrix::mul(): incompatible matrices");
+
+ exvector prod(this->rows()*other.cols());
+
+ for (unsigned r1=0; r1<this->rows(); ++r1) {
+ for (unsigned c=0; c<this->cols(); ++c) {
+ if (m[r1*col+c].is_zero())
+ continue;
+ for (unsigned r2=0; r2<other.cols(); ++r2)
+ prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]).expand();
+ }
+ }
+ return matrix(row, other.col, prod);
+}
+
+
+/** Product of matrix and scalar. */
+matrix matrix::mul(const numeric & other) const
+{
+ exvector prod(row * col);
+
+ for (unsigned r=0; r<row; ++r)
+ for (unsigned c=0; c<col; ++c)
+ prod[r*col+c] = m[r*col+c] * other;
+
+ return matrix(row, col, prod);
+}
+
+
+/** Product of matrix and scalar expression. */
+matrix matrix::mul_scalar(const ex & other) const
+{
+ if (other.return_type() != return_types::commutative)
+ throw std::runtime_error("matrix::mul_scalar(): non-commutative scalar");
+
+ exvector prod(row * col);
+
+ for (unsigned r=0; r<row; ++r)
+ for (unsigned c=0; c<col; ++c)
+ prod[r*col+c] = m[r*col+c] * other;
+
+ return matrix(row, col, prod);
+}
+
+
+/** Power of a matrix. Currently handles integer exponents only. */
+matrix matrix::pow(const ex & expn) const
+{
+ if (col!=row)
+ throw (std::logic_error("matrix::pow(): matrix not square"));
+
+ if (is_exactly_a<numeric>(expn)) {
+ // Integer cases are computed by successive multiplication, using the
+ // obvious shortcut of storing temporaries, like A^4 == (A*A)*(A*A).
+ if (expn.info(info_flags::integer)) {
+ numeric b = ex_to<numeric>(expn);
+ matrix A(row,col);
+ if (expn.info(info_flags::negative)) {
+ b *= -1;
+ A = this->inverse();
+ } else {
+ A = *this;
+ }
+ matrix C(row,col);
+ for (unsigned r=0; r<row; ++r)
+ C(r,r) = _ex1;
+ if (b.is_zero())
+ return C;
+ // This loop computes the representation of b in base 2 from right
+ // to left and multiplies the factors whenever needed. Note
+ // that this is not entirely optimal but close to optimal and
+ // "better" algorithms are much harder to implement. (See Knuth,
+ // TAoCP2, section "Evaluation of Powers" for a good discussion.)
+ while (b!=_num1) {
+ if (b.is_odd()) {
+ C = C.mul(A);
+ --b;
+ }
+ b /= _num2; // still integer.
+ A = A.mul(A);
+ }
+ return A.mul(C);
+ }
+ }
+ throw (std::runtime_error("matrix::pow(): don't know how to handle exponent"));
}
-/** operator() to access elements.
+/** operator() to access elements for reading.
*
* @param ro row of element
- * @param co column of element
+ * @param co column of element
* @exception range_error (index out of range) */
const ex & matrix::operator() (unsigned ro, unsigned co) const
{
- if (ro<0 || ro>=row || co<0 || co>=col)
- throw (std::range_error("matrix::operator(): index out of range"));
-
- return m[ro*col+co];
+ if (ro>=row || co>=col)
+ throw (std::range_error("matrix::operator(): index out of range"));
+
+ return m[ro*col+co];
}
-/** Set individual elements manually.
+/** operator() to access elements for writing.
*
+ * @param ro row of element
+ * @param co column of element
* @exception range_error (index out of range) */
-matrix & matrix::set(unsigned ro, unsigned co, ex value)
+ex & matrix::operator() (unsigned ro, unsigned co)
{
- 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;
- return *this;
+ if (ro>=row || co>=col)
+ throw (std::range_error("matrix::operator(): index out of range"));
+
+ ensure_if_modifiable();
+ return m[ro*col+co];
}
/** Transposed of an m x n matrix, producing a new n x m matrix object that
* represents the transposed. */
-matrix matrix::transpose(void) const
+matrix matrix::transpose() const
{
- exvector trans(col*row);
-
- 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);
+ exvector trans(this->cols()*this->rows());
+
+ for (unsigned r=0; r<this->cols(); ++r)
+ for (unsigned c=0; c<this->rows(); ++c)
+ trans[r*this->rows()+c] = m[c*this->cols()+r];
+
+ return matrix(this->cols(),this->rows(),trans);
}
-
/** Determinant of square matrix. This routine doesn't actually calculate the
* determinant, it only implements some heuristics about which algorithm to
- * call. If all the elements of the matrix are elements of an integral domain
+ * run. 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
* [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
* behaves like MapleV and unlike Mathematica.)
*
+ * @param algo allows to chose an algorithm
* @return the determinant as a new expression
- * @exception logic_error (matrix not square) */
-ex matrix::determinant(void) const
+ * @exception logic_error (matrix not square)
+ * @see determinant_algo */
+ex matrix::determinant(unsigned algo) const
{
- if (row!=col)
- throw (std::logic_error("matrix::determinant(): matrix not square"));
- GINAC_ASSERT(row*col==m.capacity());
- if (this->row==1) // continuation would be pointless
- return m[0];
-
- // Gather some information about the matrix:
- bool numeric_flag = true;
- bool normal_flag = false;
- unsigned sparse_count = 0; // count non-zero elements
- for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
- if (!(*r).is_zero())
- ++sparse_count;
- 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;
- }
-
- // Purely numeric matrix handled by Gauss elimination
- if (numeric_flag) {
- ex det = 1;
- matrix tmp(*this);
- int sign = tmp.gauss_elimination();
- for (int d=0; d<row; ++d)
- det *= tmp.m[d*col+d];
- return (sign*det);
- }
-
- // Does anybody know when a matrix is really sparse?
- // Maybe <~row/2.2 nonzero elements average in a row?
- if (5*sparse_count<=row*col) {
- // copy *this:
- matrix tmp(*this);
- int sign;
- sign = tmp.fraction_free_elimination(true);
- if (normal_flag)
- return (sign*tmp.m[row*col-1]).normal();
- else
- return (sign*tmp.m[row*col-1]).expand();
- }
-
- // Now come the minor expansion schemes. We always develop such that the
- // smallest minors (i.e, the trivial 1x1 ones) are on the rightmost column.
- // For this to be efficient it turns out that the emptiest columns (i.e.
- // the ones with most zeros) should be the ones on the right hand side.
- // Therefore we presort the columns of the matrix:
- typedef pair<unsigned,unsigned> uintpair; // # of zeros, column
- vector<uintpair> c_zeros; // number of zeros in column
- for (unsigned c=0; c<col; ++c) {
- unsigned acc = 0;
- for (unsigned r=0; r<row; ++r)
- if (m[r*col+c].is_zero())
- ++acc;
- c_zeros.push_back(uintpair(acc,c));
- }
- sort(c_zeros.begin(),c_zeros.end());
- vector<unsigned> pre_sort; // unfortunately vector<uintpair> can't be used
- // for permutation_sign.
- for (vector<uintpair>::iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
- pre_sort.push_back(i->second);
- int sign = permutation_sign(pre_sort);
- exvector result(row*col); // represents sorted matrix
- unsigned c = 0;
- for (vector<unsigned>::iterator i=pre_sort.begin();
- i!=pre_sort.end();
- ++i,++c) {
- for (unsigned r=0; r<row; ++r)
- result[r*col+c] = m[r*col+(*i)];
- }
-
- if (normal_flag)
- return sign*matrix(row,col,result).determinant_minor().normal();
- return sign*matrix(row,col,result).determinant_minor();
+ if (row!=col)
+ throw (std::logic_error("matrix::determinant(): matrix not square"));
+ GINAC_ASSERT(row*col==m.capacity());
+
+ // Gather some statistical information about this matrix:
+ bool numeric_flag = true;
+ bool normal_flag = false;
+ unsigned sparse_count = 0; // counts non-zero elements
+ exvector::const_iterator r = m.begin(), rend = m.end();
+ while (r != rend) {
+ exmap srl; // symbol replacement list
+ ex rtest = r->to_rational(srl);
+ if (!rtest.is_zero())
+ ++sparse_count;
+ if (!rtest.info(info_flags::numeric))
+ numeric_flag = false;
+ if (!rtest.info(info_flags::crational_polynomial) &&
+ rtest.info(info_flags::rational_function))
+ normal_flag = true;
+ ++r;
+ }
+
+ // Here is the heuristics in case this routine has to decide:
+ if (algo == determinant_algo::automatic) {
+ // Minor expansion is generally a good guess:
+ algo = determinant_algo::laplace;
+ // Does anybody know when a matrix is really sparse?
+ // Maybe <~row/2.236 nonzero elements average in a row?
+ if (row>3 && 5*sparse_count<=row*col)
+ algo = determinant_algo::bareiss;
+ // Purely numeric matrix can be handled by Gauss elimination.
+ // This overrides any prior decisions.
+ if (numeric_flag)
+ algo = determinant_algo::gauss;
+ }
+
+ // Trap the trivial case here, since some algorithms don't like it
+ if (this->row==1) {
+ // for consistency with non-trivial determinants...
+ if (normal_flag)
+ return m[0].normal();
+ else
+ return m[0].expand();
+ }
+
+ // Compute the determinant
+ switch(algo) {
+ case determinant_algo::gauss: {
+ ex det = 1;
+ matrix tmp(*this);
+ int sign = tmp.gauss_elimination(true);
+ for (unsigned d=0; d<row; ++d)
+ det *= tmp.m[d*col+d];
+ if (normal_flag)
+ return (sign*det).normal();
+ else
+ return (sign*det).normal().expand();
+ }
+ case determinant_algo::bareiss: {
+ matrix tmp(*this);
+ int sign;
+ sign = tmp.fraction_free_elimination(true);
+ if (normal_flag)
+ return (sign*tmp.m[row*col-1]).normal();
+ else
+ return (sign*tmp.m[row*col-1]).expand();
+ }
+ case determinant_algo::divfree: {
+ matrix tmp(*this);
+ int sign;
+ sign = tmp.division_free_elimination(true);
+ if (sign==0)
+ return _ex0;
+ ex det = tmp.m[row*col-1];
+ // factor out accumulated bogus slag
+ for (unsigned d=0; d<row-2; ++d)
+ for (unsigned j=0; j<row-d-2; ++j)
+ det = (det/tmp.m[d*col+d]).normal();
+ return (sign*det);
+ }
+ case determinant_algo::laplace:
+ default: {
+ // This is the minor expansion scheme. We always develop such
+ // that the smallest minors (i.e, the trivial 1x1 ones) are on the
+ // rightmost column. For this to be efficient, empirical tests
+ // have shown that the emptiest columns (i.e. the ones with most
+ // zeros) should be the ones on the right hand side -- although
+ // this might seem counter-intuitive (and in contradiction to some
+ // literature like the FORM manual). Please go ahead and test it
+ // if you don't believe me! Therefore we presort the columns of
+ // the matrix:
+ typedef std::pair<unsigned,unsigned> uintpair;
+ std::vector<uintpair> c_zeros; // number of zeros in column
+ for (unsigned c=0; c<col; ++c) {
+ unsigned acc = 0;
+ for (unsigned r=0; r<row; ++r)
+ if (m[r*col+c].is_zero())
+ ++acc;
+ c_zeros.push_back(uintpair(acc,c));
+ }
+ std::sort(c_zeros.begin(),c_zeros.end());
+ std::vector<unsigned> pre_sort;
+ for (std::vector<uintpair>::const_iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
+ pre_sort.push_back(i->second);
+ std::vector<unsigned> pre_sort_test(pre_sort); // permutation_sign() modifies the vector so we make a copy here
+ int sign = permutation_sign(pre_sort_test.begin(), pre_sort_test.end());
+ exvector result(row*col); // represents sorted matrix
+ unsigned c = 0;
+ for (std::vector<unsigned>::const_iterator i=pre_sort.begin();
+ i!=pre_sort.end();
+ ++i,++c) {
+ for (unsigned r=0; r<row; ++r)
+ result[r*col+c] = m[r*col+(*i)];
+ }
+
+ if (normal_flag)
+ return (sign*matrix(row,col,result).determinant_minor()).normal();
+ else
+ return sign*matrix(row,col,result).determinant_minor();
+ }
+ }
}
*
* @return the sum of diagonal elements
* @exception logic_error (matrix not square) */
-ex matrix::trace(void) const
+ex matrix::trace() const
{
- if (row != col)
- throw (std::logic_error("matrix::trace(): matrix not square"));
- GINAC_ASSERT(row*col==m.capacity());
-
- ex tr;
- for (unsigned r=0; r<col; ++r)
- tr += m[r*col+r];
-
- if (tr.info(info_flags::rational_function) &&
- !tr.info(info_flags::crational_polynomial))
- return tr.normal();
- else
- return tr.expand();
+ if (row != col)
+ throw (std::logic_error("matrix::trace(): matrix not square"));
+
+ ex tr;
+ for (unsigned r=0; r<col; ++r)
+ tr += m[r*col+r];
+
+ if (tr.info(info_flags::rational_function) &&
+ !tr.info(info_flags::crational_polynomial))
+ return tr.normal();
+ else
+ return tr.expand();
}
* @return characteristic polynomial as new expression
* @exception logic_error (matrix not square)
* @see matrix::determinant() */
-ex matrix::charpoly(const symbol & lambda) const
+ex matrix::charpoly(const ex & lambda) const
{
- if (row != col)
- throw (std::logic_error("matrix::charpoly(): matrix not square"));
-
- bool numeric_flag = true;
- for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
- if (!(*r).info(info_flags::numeric)) {
- numeric_flag = false;
- }
- }
-
- // The pure numeric case is traditionally rather common. Hence, it is
- // trapped and we use Leverrier's algorithm which goes as row^3 for
- // every coefficient. The expensive part is the matrix multiplication.
- if (numeric_flag) {
- matrix B(*this);
- ex c = B.trace();
- ex poly = power(lambda,row)-c*power(lambda,row-1);
- for (unsigned i=1; i<row; ++i) {
- for (unsigned j=0; j<row; ++j)
- B.m[j*col+j] -= c;
- B = this->mul(B);
- c = B.trace()/ex(i+1);
- poly -= c*power(lambda,row-i-1);
- }
- if (row%2)
- return -poly;
- else
- return poly;
- }
-
- matrix M(*this);
- for (unsigned r=0; r<col; ++r)
- M.m[r*col+r] -= lambda;
-
- return M.determinant().collect(lambda);
+ if (row != col)
+ throw (std::logic_error("matrix::charpoly(): matrix not square"));
+
+ bool numeric_flag = true;
+ exvector::const_iterator r = m.begin(), rend = m.end();
+ while (r!=rend && numeric_flag==true) {
+ if (!r->info(info_flags::numeric))
+ numeric_flag = false;
+ ++r;
+ }
+
+ // The pure numeric case is traditionally rather common. Hence, it is
+ // trapped and we use Leverrier's algorithm which goes as row^3 for
+ // every coefficient. The expensive part is the matrix multiplication.
+ if (numeric_flag) {
+
+ matrix B(*this);
+ ex c = B.trace();
+ ex poly = power(lambda, row) - c*power(lambda, row-1);
+ for (unsigned i=1; i<row; ++i) {
+ for (unsigned j=0; j<row; ++j)
+ B.m[j*col+j] -= c;
+ B = this->mul(B);
+ c = B.trace() / ex(i+1);
+ poly -= c*power(lambda, row-i-1);
+ }
+ if (row%2)
+ return -poly;
+ else
+ return poly;
+
+ } else {
+
+ matrix M(*this);
+ for (unsigned r=0; r<col; ++r)
+ M.m[r*col+r] -= lambda;
+
+ return M.determinant().collect(lambda);
+ }
}
* @return the inverted matrix
* @exception logic_error (matrix not square)
* @exception runtime_error (singular matrix) */
-matrix matrix::inverse(void) const
+matrix matrix::inverse() const
{
- if (row != col)
- throw (std::logic_error("matrix::inverse(): matrix not square"));
-
- matrix tmp(row,col);
- // set tmp to the unit matrix
- for (unsigned i=0; i<col; ++i)
- tmp.m[i*col+i] = _ex1();
-
- // create a copy of this matrix
- matrix cpy(*this);
- 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 (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 (unsigned c=0; c<col; ++c) {
- cpy.m[r1*col+c] /= a1;
- tmp.m[r1*col+c] /= a1;
- }
- for (unsigned r2=0; r2<row; ++r2) {
- if (r2 != r1) {
- ex a2 = cpy.m[r2*col+r1];
- 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;
+ if (row != col)
+ throw (std::logic_error("matrix::inverse(): matrix not square"));
+
+ // This routine actually doesn't do anything fancy at all. We compute the
+ // inverse of the matrix A by solving the system A * A^{-1} == Id.
+
+ // First populate the identity matrix supposed to become the right hand side.
+ matrix identity(row,col);
+ for (unsigned i=0; i<row; ++i)
+ identity(i,i) = _ex1;
+
+ // Populate a dummy matrix of variables, just because of compatibility with
+ // matrix::solve() which wants this (for compatibility with under-determined
+ // systems of equations).
+ matrix vars(row,col);
+ for (unsigned r=0; r<row; ++r)
+ for (unsigned c=0; c<col; ++c)
+ vars(r,c) = symbol();
+
+ matrix sol(row,col);
+ try {
+ sol = this->solve(vars,identity);
+ } catch (const std::runtime_error & e) {
+ if (e.what()==std::string("matrix::solve(): inconsistent linear system"))
+ throw (std::runtime_error("matrix::inverse(): singular matrix"));
+ else
+ throw;
+ }
+ return sol;
}
-/** 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'
- * 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(const matrix & vars,
- const matrix & rhs) const
-{
- // FIXME: use implementation of matrix::fraction_free_elimination
- 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
-
- // given an m x n matrix a, reduce it to upper echelon form
- unsigned m = a.row;
- unsigned n = a.col;
- int sign = 1;
- ex divisor = 1;
- unsigned r = 0;
-
- // eliminate below row r, with pivot in column k
- for (unsigned k=0; (k<n)&&(r<m); ++k) {
- // find a nonzero pivot
- unsigned p;
- for (p=r; (p<m)&&(a.m[p*a.cols()+k].is_zero()); ++p) {}
- // pivot is in row p
- if (p<m) {
- if (p!=r) {
- // swap rows p and r
- for (unsigned j=k; j<n; ++j)
- a.m[p*a.cols()+j].swap(a.m[r*a.cols()+j]);
- a.m[p*a.cols()].swap(a.m[r*a.cols()]);
- // keep track of sign changes due to row exchange
- sign = -sign;
- }
- for (unsigned i=r+1; i<m; ++i) {
- for (unsigned j=k+1; j<n; ++j) {
- a.set(i,j,(a.m[r*a.cols()+k]*a.m[i*a.cols()+j]
- -a.m[r*a.cols()+j]*a.m[i*a.cols()+k])/divisor);
- a.set(i,j,a.m[i*a.cols()+j].normal());
- }
- b.set(i,0,(a.m[r*a.cols()+k]*b.m[i*b.cols()]
- -b.m[r*b.cols()]*a.m[i*a.cols()+k])/divisor);
- b.set(i,0,b.m[i*b.cols()].normal());
- a.set(i,k,0);
- }
- divisor = a.m[r*a.cols()+k];
- r++;
- }
- }
-
-#ifdef DO_GINAC_ASSERT
- // test if we really have an upper echelon matrix
- int zero_in_last_row = -1;
- for (unsigned r=0; r<m; ++r) {
- int zero_in_this_row=0;
- for (unsigned c=0; c<n; ++c) {
- if (a.m[r*a.cols()+c].is_zero())
- zero_in_this_row++;
- 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;
- }
-#endif // def DO_GINAC_ASSERT
-
- // assemble solution
- matrix sol(n,1);
- unsigned last_assigned_sol = n+1;
- for (int r=m-1; r>=0; --r) {
- unsigned first_non_zero = 1;
- while ((first_non_zero<=n)&&(a.m[r*a.cols()+(first_non_zero-1)].is_zero()))
- first_non_zero++;
- if (first_non_zero>n) {
- // row consists only of zeroes, corresponding rhs must be 0 as well
- if (!b.m[r*b.cols()].is_zero()) {
- throw (std::runtime_error("matrix::fraction_free_elim(): singular matrix"));
- }
- } else {
- // assign solutions for vars between first_non_zero+1 and
- // last_assigned_sol-1: free parameters
- for (unsigned c=first_non_zero; c<last_assigned_sol-1; ++c)
- sol.set(c,0,vars.m[c*vars.cols()]);
- ex e = b.m[r*b.cols()];
- for (unsigned c=first_non_zero; c<n; ++c)
- e -= a.m[r*a.cols()+c]*sol.m[c*sol.cols()];
- sol.set(first_non_zero-1,0,
- (e/a.m[r*a.cols()+(first_non_zero-1)]).normal());
- last_assigned_sol = first_non_zero;
- }
- }
- // assign solutions for vars between 1 and
- // last_assigned_sol-1: free parameters
- for (unsigned c=0; c<last_assigned_sol-1; ++c)
- sol.set(c,0,vars.m[c*vars.cols()]);
-
-#ifdef DO_GINAC_ASSERT
- // test solution with echelon matrix
- for (unsigned r=0; r<m; ++r) {
- ex e = 0;
- for (unsigned c=0; c<n; ++c)
- e += a.m[r*a.cols()+c]*sol.m[c*sol.cols()];
- if (!(e-b.m[r*b.cols()]).normal().is_zero()) {
- cout << "e=" << e;
- cout << "b(" << r <<",0)=" << b.m[r*b.cols()] << endl;
- cout << "diff=" << (e-b.m[r*b.cols()]).normal() << endl;
- }
- GINAC_ASSERT((e-b.m[r*b.cols()]).normal().is_zero());
- }
-
- // test solution with original matrix
- for (unsigned r=0; r<m; ++r) {
- ex e = 0;
- for (unsigned c=0; c<n; ++c)
- e += this->m[r*cols()+c]*sol.m[c*sol.cols()];
- try {
- if (!(e-rhs.m[r*rhs.cols()]).normal().is_zero()) {
- cout << "e==" << e << endl;
- e.printtree(cout);
- ex en = e.normal();
- cout << "e.normal()=" << en << endl;
- en.printtree(cout);
- cout << "rhs(" << r <<",0)=" << rhs.m[r*rhs.cols()] << endl;
- cout << "diff=" << (e-rhs.m[r*rhs.cols()]).normal() << endl;
- }
- } catch (...) {
- ex xxx = e - rhs.m[r*rhs.cols()];
- cerr << "xxx=" << xxx << endl << endl;
- }
- GINAC_ASSERT((e-rhs.m[r*rhs.cols()]).normal().is_zero());
- }
-#endif // def DO_GINAC_ASSERT
-
- return sol;
-}
-
-/** Solve a set of equations for an m x n matrix.
+/** Solve a linear system consisting of a m x n matrix and a m x p right hand
+ * side by applying an elimination scheme to the augmented matrix.
*
- * @param vars n x p matrix
+ * @param vars n x p matrix, all elements must be symbols
* @param rhs m x p matrix
+ * @param algo selects the solving algorithm
+ * @return n x p solution matrix
* @exception logic_error (incompatible matrices)
- * @exception runtime_error (singular matrix) */
+ * @exception invalid_argument (1st argument must be matrix of symbols)
+ * @exception runtime_error (inconsistent linear system)
+ * @see solve_algo */
matrix matrix::solve(const matrix & vars,
- const matrix & rhs) const
+ const matrix & rhs,
+ unsigned algo) const
{
- if ((row != rhs.row) || (col != vars.row) || (rhs.col != vars.col))
- throw (std::logic_error("matrix::solve(): incompatible matrices"));
-
- throw (std::runtime_error("FIXME: need implementation."));
-}
-
-/** Old and obsolete interface: */
-matrix matrix::old_solve(const matrix & v) const
-{
- if ((v.row != col) || (col != v.row))
- throw (std::logic_error("matrix::solve(): incompatible matrices"));
-
- // build the augmented matrix of *this with v attached to the right
- matrix tmp(row,col+v.col);
- 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();
- }
- }
- return matrix(v.row, v.col, sol);
+ const unsigned m = this->rows();
+ const unsigned n = this->cols();
+ const unsigned p = rhs.cols();
+
+ // syntax checks
+ if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
+ throw (std::logic_error("matrix::solve(): incompatible matrices"));
+ for (unsigned ro=0; ro<n; ++ro)
+ for (unsigned co=0; co<p; ++co)
+ if (!vars(ro,co).info(info_flags::symbol))
+ throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
+
+ // build the augmented matrix of *this with rhs attached to the right
+ matrix aug(m,n+p);
+ for (unsigned r=0; r<m; ++r) {
+ for (unsigned c=0; c<n; ++c)
+ aug.m[r*(n+p)+c] = this->m[r*n+c];
+ for (unsigned c=0; c<p; ++c)
+ aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
+ }
+
+ // Gather some statistical information about the augmented matrix:
+ bool numeric_flag = true;
+ exvector::const_iterator r = aug.m.begin(), rend = aug.m.end();
+ while (r!=rend && numeric_flag==true) {
+ if (!r->info(info_flags::numeric))
+ numeric_flag = false;
+ ++r;
+ }
+
+ // Here is the heuristics in case this routine has to decide:
+ if (algo == solve_algo::automatic) {
+ // Bareiss (fraction-free) elimination is generally a good guess:
+ algo = solve_algo::bareiss;
+ // For m<3, Bareiss elimination is equivalent to division free
+ // elimination but has more logistic overhead
+ if (m<3)
+ algo = solve_algo::divfree;
+ // This overrides any prior decisions.
+ if (numeric_flag)
+ algo = solve_algo::gauss;
+ }
+
+ // Eliminate the augmented matrix:
+ switch(algo) {
+ case solve_algo::gauss:
+ aug.gauss_elimination();
+ break;
+ case solve_algo::divfree:
+ aug.division_free_elimination();
+ break;
+ case solve_algo::bareiss:
+ default:
+ aug.fraction_free_elimination();
+ }
+
+ // assemble the solution matrix:
+ matrix sol(n,p);
+ for (unsigned co=0; co<p; ++co) {
+ unsigned last_assigned_sol = n+1;
+ for (int r=m-1; r>=0; --r) {
+ unsigned fnz = 1; // first non-zero in row
+ while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].is_zero()))
+ ++fnz;
+ if (fnz>n) {
+ // row consists only of zeros, corresponding rhs must be 0, too
+ if (!aug.m[r*(n+p)+n+co].is_zero()) {
+ throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
+ }
+ } else {
+ // assign solutions for vars between fnz+1 and
+ // last_assigned_sol-1: free parameters
+ for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
+ sol(c,co) = vars.m[c*p+co];
+ ex e = aug.m[r*(n+p)+n+co];
+ for (unsigned c=fnz; c<n; ++c)
+ e -= aug.m[r*(n+p)+c]*sol.m[c*p+co];
+ sol(fnz-1,co) = (e/(aug.m[r*(n+p)+(fnz-1)])).normal();
+ last_assigned_sol = fnz;
+ }
+ }
+ // assign solutions for vars between 1 and
+ // last_assigned_sol-1: free parameters
+ for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
+ sol(ro,co) = vars(ro,co);
+ }
+
+ return sol;
}
*
* @return the determinant as a new expression (in expanded form)
* @see matrix::determinant() */
-ex matrix::determinant_minor(void) const
+ex matrix::determinant_minor() 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 {
- 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 time:
- det = det.expand();
- // store the new determinant at its place in B:
- if (!det.is_zero())
- 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;
+ // for small matrices the algorithm does not make any sense:
+ const unsigned n = this->cols();
+ if (n==1)
+ return m[0].expand();
+ if (n==2)
+ return (m[0]*m[3]-m[2]*m[1]).expand();
+ if (n==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->rows()-1,this->cols()-1);
+ // for (unsigned r1=0; r1<this->rows(); ++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(r,c) = m[r*col+c+1];
+ // else
+ // minorM(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
+ std::vector<unsigned> Pkey;
+ Pkey.reserve(n);
+ // key for minor determinant (a subpartition of Pkey)
+ std::vector<unsigned> Mkey;
+ Mkey.reserve(n-1);
+ // we store our subminors in maps, keys being the rows they arise from
+ typedef std::map<std::vector<unsigned>,class ex> Rmap;
+ typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
+ Rmap A;
+ Rmap B;
+ ex det;
+ // initialize A with last column:
+ for (unsigned r=0; r<n; ++r) {
+ Pkey.erase(Pkey.begin(),Pkey.end());
+ Pkey.push_back(r);
+ A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
+ }
+ // proceed from right to left through matrix
+ for (int c=n-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<n-c; ++i)
+ Pkey.push_back(i);
+ unsigned fc = 0; // controls logic for our strange flipper counter
+ do {
+ det = _ex0;
+ for (unsigned r=0; r<n-c; ++r) {
+ // maybe there is nothing to do?
+ if (m[Pkey[r]*n+c].is_zero())
+ continue;
+ // create the sorted key for all possible minors
+ Mkey.erase(Mkey.begin(),Mkey.end());
+ for (unsigned i=0; i<n-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]*n+c]*A[Mkey];
+ else
+ det += m[Pkey[r]*n+c]*A[Mkey];
+ }
+ // prevent build-up of deep nesting of expressions saves time:
+ det = det.expand();
+ // store the new determinant at its place in B:
+ if (!det.is_zero())
+ B.insert(Rmap_value(Pkey,det));
+ // increment our strange flipper counter
+ for (fc=n-c; fc>0; --fc) {
+ ++Pkey[fc-1];
+ if (Pkey[fc-1]<fc+c)
+ break;
+ }
+ if (fc<n-c && fc>0)
+ for (unsigned j=fc; j<n-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;
}
-/** Perform the steps of an ordinary Gaussian elimination to bring the matrix
- * into an upper echelon form.
+/** Perform the steps of an ordinary Gaussian elimination to bring the m x n
+ * matrix into an upper echelon form. The algorithm is ok for matrices
+ * with numeric coefficients but quite unsuited for symbolic matrices.
*
+ * @param det may be set to true to save a lot of space if one is only
+ * interested in the diagonal elements (i.e. for calculating determinants).
+ * The others are set to zero in this case.
* @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 matrix::gauss_elimination(const bool det)
{
- ensure_if_modifiable();
- int sign = 1;
- ex piv;
- 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) {
- piv = this->m[r2*col+r1] / this->m[r1*col+r1];
- for (unsigned c=r1+1; c<col; ++c)
- this->m[r2*col+c] -= piv * this->m[r1*col+c];
- for (unsigned c=0; c<=r1; ++c)
- this->m[r2*col+c] = _ex0();
- }
- }
-
- return sign;
+ ensure_if_modifiable();
+ const unsigned m = this->rows();
+ const unsigned n = this->cols();
+ GINAC_ASSERT(!det || n==m);
+ int sign = 1;
+
+ unsigned r0 = 0;
+ for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
+ int indx = pivot(r0, r1, true);
+ if (indx == -1) {
+ sign = 0;
+ if (det)
+ return 0; // leaves *this in a messy state
+ }
+ if (indx>=0) {
+ if (indx > 0)
+ sign = -sign;
+ for (unsigned r2=r0+1; r2<m; ++r2) {
+ if (!this->m[r2*n+r1].is_zero()) {
+ // yes, there is something to do in this row
+ ex piv = this->m[r2*n+r1] / this->m[r0*n+r1];
+ for (unsigned c=r1+1; c<n; ++c) {
+ this->m[r2*n+c] -= piv * this->m[r0*n+c];
+ if (!this->m[r2*n+c].info(info_flags::numeric))
+ this->m[r2*n+c] = this->m[r2*n+c].normal();
+ }
+ }
+ // fill up left hand side with zeros
+ for (unsigned c=0; c<=r1; ++c)
+ this->m[r2*n+c] = _ex0;
+ }
+ if (det) {
+ // save space by deleting no longer needed elements
+ for (unsigned c=r0+1; c<n; ++c)
+ this->m[r0*n+c] = _ex0;
+ }
+ ++r0;
+ }
+ }
+
+ return sign;
}
-/** Perform the steps of division free elimination to bring the matrix
+/** Perform the steps of division free elimination to bring the m x n matrix
* into an upper echelon form.
*
+ * @param det may be set to true to save a lot of space if one is only
+ * interested in the diagonal elements (i.e. for calculating determinants).
+ * The others are set to zero in this case.
* @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::division_free_elimination(void)
+int matrix::division_free_elimination(const bool det)
{
- 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[r1*col+r1]*this->m[r2*col+c] - this->m[r2*col+r1]*this->m[r1*col+c];
- for (unsigned c=0; c<=r1; ++c)
- this->m[r2*col+c] = _ex0();
- }
- }
-
- return sign;
+ ensure_if_modifiable();
+ const unsigned m = this->rows();
+ const unsigned n = this->cols();
+ GINAC_ASSERT(!det || n==m);
+ int sign = 1;
+
+ unsigned r0 = 0;
+ for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
+ int indx = pivot(r0, r1, true);
+ if (indx==-1) {
+ sign = 0;
+ if (det)
+ return 0; // leaves *this in a messy state
+ }
+ if (indx>=0) {
+ if (indx>0)
+ sign = -sign;
+ for (unsigned r2=r0+1; r2<m; ++r2) {
+ for (unsigned c=r1+1; c<n; ++c)
+ this->m[r2*n+c] = (this->m[r0*n+r1]*this->m[r2*n+c] - this->m[r2*n+r1]*this->m[r0*n+c]).expand();
+ // fill up left hand side with zeros
+ for (unsigned c=0; c<=r1; ++c)
+ this->m[r2*n+c] = _ex0;
+ }
+ if (det) {
+ // save space by deleting no longer needed elements
+ for (unsigned c=r0+1; c<n; ++c)
+ this->m[r0*n+c] = _ex0;
+ }
+ ++r0;
+ }
+ }
+
+ return sign;
}
* is possible, since we know the divisor at each step.
*
* @param det may be set to true to save a lot of space if one is only
- * interested in the last element (i.e. for calculating determinants), the
+ * interested in the last element (i.e. for calculating determinants). The
* others are set to zero in this case.
* @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::fraction_free_elimination(bool det)
+int matrix::fraction_free_elimination(const bool det)
{
- // Method:
- // (single-step fraction free elimination scheme, already known to Jordan)
- //
- // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
- // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
- //
- // Bareiss (fraction-free) elimination in addition divides that element
- // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
- // Sylvester determinant that this really divides m[k+1](r,c).
- //
- // We also allow rational functions where the original prove still holds.
- // However, we must care for numerator and denominator separately and
- // "manually" work in the integral domains because of subtle cancellations
- // (see below). This blows up the bookkeeping a bit and the formula has
- // to be modified to expand like this (N{x} stands for numerator of x,
- // D{x} for denominator of x):
- // N{m[k+1](r,c)} = N{m[k](k,k)}*N{m[k](r,c)}*D{m[k](r,k)}*D{m[k](k,c)}
- // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
- // D{m[k+1](r,c)} = D{m[k](k,k)}*D{m[k](r,c)}*D{m[k](r,k)}*D{m[k](k,c)}
- // where for k>1 we now divide N{m[k+1](r,c)} by
- // N{m[k-1](k-1,k-1)}
- // and D{m[k+1](r,c)} by
- // D{m[k-1](k-1,k-1)}.
-
- GINAC_ASSERT(det || row==col);
- ensure_if_modifiable();
- if (rows()==1)
- return 1;
-
- int sign = 1;
- ex divisor_n = 1;
- ex divisor_d = 1;
- ex dividend_n;
- ex dividend_d;
-
- // We populate temporary matrices to subsequently operate on. There is
- // one holding numerators and another holding denominators of entries.
- // This is a must since the evaluator (or even earlier mul's constructor)
- // might cancel some trivial element which causes divide() to fail. The
- // elements are normalized first (yes, even though this algorithm doesn't
- // need GCDs) since the elements of *this might be unnormalized, which
- // makes things more complicated than they need to be.
- matrix tmp_n(*this);
- matrix tmp_d(row,col); // for denominators, if needed
- lst srl; // symbol replacement list
- exvector::iterator it = m.begin();
- exvector::iterator tmp_n_it = tmp_n.m.begin();
- exvector::iterator tmp_d_it = tmp_d.m.begin();
- for (; it!= m.end(); ++it, ++tmp_n_it, ++tmp_d_it) {
- (*tmp_n_it) = (*it).normal().to_rational(srl);
- (*tmp_d_it) = (*tmp_n_it).denom();
- (*tmp_n_it) = (*tmp_n_it).numer();
- }
-
- for (unsigned r1=0; r1<row-1; ++r1) {
- int indx = tmp_n.pivot(r1);
- if (det && indx==-1)
- return 0; // FIXME: what to do if det is false, some day?
- if (indx>0) {
- sign = -sign;
- // rows r1 and indx were swapped, so pivot matrix tmp_d:
- for (unsigned c=0; c<col; ++c)
- tmp_d.m[row*indx+c].swap(tmp_d.m[row*r1+c]);
- }
- if (r1>0) {
- divisor_n = tmp_n.m[(r1-1)*col+(r1-1)].expand();
- divisor_d = tmp_d.m[(r1-1)*col+(r1-1)].expand();
- // save space by deleting no longer needed elements:
- if (det) {
- for (unsigned c=0; c<col; ++c) {
- tmp_n.m[(r1-1)*col+c] = 0;
- tmp_d.m[(r1-1)*col+c] = 1;
- }
- }
- }
- for (unsigned r2=r1+1; r2<row; ++r2) {
- for (unsigned c=r1+1; c<col; ++c) {
- dividend_n = (tmp_n.m[r1*col+r1]*tmp_n.m[r2*col+c]*
- tmp_d.m[r2*col+r1]*tmp_d.m[r1*col+c]
- -tmp_n.m[r2*col+r1]*tmp_n.m[r1*col+c]*
- tmp_d.m[r1*col+r1]*tmp_d.m[r2*col+c]).expand();
- dividend_d = (tmp_d.m[r2*col+r1]*tmp_d.m[r1*col+c]*
- tmp_d.m[r1*col+r1]*tmp_d.m[r2*col+c]).expand();
- bool check = divide(dividend_n, divisor_n,
- tmp_n.m[r2*col+c],true);
- check &= divide(dividend_d, divisor_d,
- tmp_d.m[r2*col+c],true);
- GINAC_ASSERT(check);
- }
- // fill up left hand side.
- for (unsigned c=0; c<=r1; ++c)
- tmp_n.m[r2*col+c] = _ex0();
- }
- }
- // repopulate *this matrix:
- it = m.begin();
- tmp_n_it = tmp_n.m.begin();
- tmp_d_it = tmp_d.m.begin();
- for (; it!= m.end(); ++it, ++tmp_n_it, ++tmp_d_it)
- (*it) = ((*tmp_n_it)/(*tmp_d_it)).subs(srl);
-
- return sign;
+ // Method:
+ // (single-step fraction free elimination scheme, already known to Jordan)
+ //
+ // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
+ // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
+ //
+ // Bareiss (fraction-free) elimination in addition divides that element
+ // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
+ // Sylvester identity that this really divides m[k+1](r,c).
+ //
+ // We also allow rational functions where the original prove still holds.
+ // However, we must care for numerator and denominator separately and
+ // "manually" work in the integral domains because of subtle cancellations
+ // (see below). This blows up the bookkeeping a bit and the formula has
+ // to be modified to expand like this (N{x} stands for numerator of x,
+ // D{x} for denominator of x):
+ // N{m[k+1](r,c)} = N{m[k](k,k)}*N{m[k](r,c)}*D{m[k](r,k)}*D{m[k](k,c)}
+ // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
+ // D{m[k+1](r,c)} = D{m[k](k,k)}*D{m[k](r,c)}*D{m[k](r,k)}*D{m[k](k,c)}
+ // where for k>1 we now divide N{m[k+1](r,c)} by
+ // N{m[k-1](k-1,k-1)}
+ // and D{m[k+1](r,c)} by
+ // D{m[k-1](k-1,k-1)}.
+
+ ensure_if_modifiable();
+ const unsigned m = this->rows();
+ const unsigned n = this->cols();
+ GINAC_ASSERT(!det || n==m);
+ int sign = 1;
+ if (m==1)
+ return 1;
+ ex divisor_n = 1;
+ ex divisor_d = 1;
+ ex dividend_n;
+ ex dividend_d;
+
+ // We populate temporary matrices to subsequently operate on. There is
+ // one holding numerators and another holding denominators of entries.
+ // This is a must since the evaluator (or even earlier mul's constructor)
+ // might cancel some trivial element which causes divide() to fail. The
+ // elements are normalized first (yes, even though this algorithm doesn't
+ // need GCDs) since the elements of *this might be unnormalized, which
+ // makes things more complicated than they need to be.
+ matrix tmp_n(*this);
+ matrix tmp_d(m,n); // for denominators, if needed
+ exmap srl; // symbol replacement list
+ exvector::const_iterator cit = this->m.begin(), citend = this->m.end();
+ exvector::iterator tmp_n_it = tmp_n.m.begin(), tmp_d_it = tmp_d.m.begin();
+ while (cit != citend) {
+ ex nd = cit->normal().to_rational(srl).numer_denom();
+ ++cit;
+ *tmp_n_it++ = nd.op(0);
+ *tmp_d_it++ = nd.op(1);
+ }
+
+ unsigned r0 = 0;
+ for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
+ int indx = tmp_n.pivot(r0, r1, true);
+ if (indx==-1) {
+ sign = 0;
+ if (det)
+ return 0;
+ }
+ if (indx>=0) {
+ if (indx>0) {
+ sign = -sign;
+ // tmp_n's rows r0 and indx were swapped, do the same in tmp_d:
+ for (unsigned c=r1; c<n; ++c)
+ tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
+ }
+ for (unsigned r2=r0+1; r2<m; ++r2) {
+ for (unsigned c=r1+1; c<n; ++c) {
+ dividend_n = (tmp_n.m[r0*n+r1]*tmp_n.m[r2*n+c]*
+ tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]
+ -tmp_n.m[r2*n+r1]*tmp_n.m[r0*n+c]*
+ tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
+ dividend_d = (tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]*
+ tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
+ bool check = divide(dividend_n, divisor_n,
+ tmp_n.m[r2*n+c], true);
+ check &= divide(dividend_d, divisor_d,
+ tmp_d.m[r2*n+c], true);
+ GINAC_ASSERT(check);
+ }
+ // fill up left hand side with zeros
+ for (unsigned c=0; c<=r1; ++c)
+ tmp_n.m[r2*n+c] = _ex0;
+ }
+ if ((r1<n-1)&&(r0<m-1)) {
+ // compute next iteration's divisor
+ divisor_n = tmp_n.m[r0*n+r1].expand();
+ divisor_d = tmp_d.m[r0*n+r1].expand();
+ if (det) {
+ // save space by deleting no longer needed elements
+ for (unsigned c=0; c<n; ++c) {
+ tmp_n.m[r0*n+c] = _ex0;
+ tmp_d.m[r0*n+c] = _ex1;
+ }
+ }
+ }
+ ++r0;
+ }
+ }
+ // repopulate *this matrix:
+ exvector::iterator it = this->m.begin(), itend = this->m.end();
+ tmp_n_it = tmp_n.m.begin();
+ tmp_d_it = tmp_d.m.begin();
+ while (it != itend)
+ *it++ = ((*tmp_n_it++)/(*tmp_d_it++)).subs(srl, subs_options::no_pattern);
+
+ return sign;
}
* 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 ro is the row from where to begin
+ * @param co is the column 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 matrix::pivot(unsigned ro, unsigned co, bool symbolic)
{
- unsigned k = ro;
-
- 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())
- return -1;
- if (k!=ro) { // swap rows
- ensure_if_modifiable();
- for (unsigned c=0; c<col; ++c) {
- m[k*col+c].swap(m[ro*col+c]);
- }
- return k;
- }
- return 0;
+ unsigned k = ro;
+ if (symbolic) {
+ // search first non-zero element in column co beginning at row ro
+ while ((k<row) && (this->m[k*col+co].expand().is_zero()))
+ ++k;
+ } else {
+ // search largest element in column co beginning at row ro
+ GINAC_ASSERT(is_exactly_a<numeric>(this->m[k*col+co]));
+ unsigned kmax = k+1;
+ numeric mmax = abs(ex_to<numeric>(m[kmax*col+co]));
+ while (kmax<row) {
+ GINAC_ASSERT(is_exactly_a<numeric>(this->m[kmax*col+co]));
+ numeric tmp = ex_to<numeric>(this->m[kmax*col+co]);
+ if (abs(tmp) > mmax) {
+ mmax = tmp;
+ k = kmax;
+ }
+ ++kmax;
+ }
+ if (!mmax.is_zero())
+ k = kmax;
+ }
+ if (k==row)
+ // all elements in column co below row ro vanish
+ return -1;
+ if (k==ro)
+ // matrix needs no pivoting
+ return 0;
+ // matrix needs pivoting, so swap rows k and ro
+ ensure_if_modifiable();
+ for (unsigned c=0; c<col; ++c)
+ this->m[k*col+c].swap(this->m[ro*col+c]);
+
+ return k;
}
-/** Convert list of lists to matrix. */
-ex lst_to_matrix(const ex &l)
+ex lst_to_matrix(const lst & l)
{
- if (!is_ex_of_type(l, lst))
- throw(std::invalid_argument("argument to lst_to_matrix() must be a lst"));
+ lst::const_iterator itr, itc;
// Find number of rows and columns
- unsigned rows = l.nops(), cols = 0, i, j;
- for (i=0; i<rows; i++)
- if (l.op(i).nops() > cols)
- cols = l.op(i).nops();
+ size_t rows = l.nops(), cols = 0;
+ for (itr = l.begin(); itr != l.end(); ++itr) {
+ if (!is_a<lst>(*itr))
+ throw (std::invalid_argument("lst_to_matrix: argument must be a list of lists"));
+ if (itr->nops() > cols)
+ cols = itr->nops();
+ }
// Allocate and fill matrix
- matrix &m = *new matrix(rows, cols);
- for (i=0; i<rows; i++)
- for (j=0; j<cols; j++)
- if (l.op(i).nops() > j)
- m.set(i, j, l.op(i).op(j));
- else
- m.set(i, j, ex(0));
- return m;
+ matrix &M = *new matrix(rows, cols);
+ M.setflag(status_flags::dynallocated);
+
+ unsigned i;
+ for (itr = l.begin(), i = 0; itr != l.end(); ++itr, ++i) {
+ unsigned j;
+ for (itc = ex_to<lst>(*itr).begin(), j = 0; itc != ex_to<lst>(*itr).end(); ++itc, ++j)
+ M(i, j) = *itc;
+ }
+
+ return M;
}
-//////////
-// global constants
-//////////
+ex diag_matrix(const lst & l)
+{
+ lst::const_iterator it;
+ size_t dim = l.nops();
-const matrix some_matrix;
-const type_info & typeid_matrix=typeid(some_matrix);
+ // Allocate and fill matrix
+ matrix &M = *new matrix(dim, dim);
+ M.setflag(status_flags::dynallocated);
+
+ unsigned i;
+ for (it = l.begin(), i = 0; it != l.end(); ++it, ++i)
+ M(i, i) = *it;
+
+ return M;
+}
+
+ex unit_matrix(unsigned r, unsigned c)
+{
+ matrix &Id = *new matrix(r, c);
+ Id.setflag(status_flags::dynallocated);
+ for (unsigned i=0; i<r && i<c; i++)
+ Id(i,i) = _ex1;
+
+ return Id;
+}
+
+ex symbolic_matrix(unsigned r, unsigned c, const std::string & base_name, const std::string & tex_base_name)
+{
+ matrix &M = *new matrix(r, c);
+ M.setflag(status_flags::dynallocated | status_flags::evaluated);
+
+ bool long_format = (r > 10 || c > 10);
+ bool single_row = (r == 1 || c == 1);
+
+ for (unsigned i=0; i<r; i++) {
+ for (unsigned j=0; j<c; j++) {
+ std::ostringstream s1, s2;
+ s1 << base_name;
+ s2 << tex_base_name << "_{";
+ if (single_row) {
+ if (c == 1) {
+ s1 << i;
+ s2 << i << '}';
+ } else {
+ s1 << j;
+ s2 << j << '}';
+ }
+ } else {
+ if (long_format) {
+ s1 << '_' << i << '_' << j;
+ s2 << i << ';' << j << "}";
+ } else {
+ s1 << i << j;
+ s2 << i << j << '}';
+ }
+ }
+ M(i, j) = symbol(s1.str(), s2.str());
+ }
+ }
+
+ return M;
+}
-#ifndef NO_NAMESPACE_GINAC
} // namespace GiNaC
-#endif // ndef NO_NAMESPACE_GINAC
git://www.ginac.de / ginac.git / blobdiff