* Implementation of symbolic matrices */
/*
- * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2001 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 "numeric.h"
+#include "lst.h"
+#include "idx.h"
+#include "indexed.h"
+#include "power.h"
+#include "symbol.h"
+#include "normal.h"
+#include "print.h"
#include "archive.h"
#include "utils.h"
#include "debugmsg.h"
-#ifndef NO_NAMESPACE_GINAC
namespace GiNaC {
-#endif // ndef NO_NAMESPACE_GINAC
GINAC_IMPLEMENT_REGISTERED_CLASS(matrix, basic)
//////////
-// default constructor, destructor, copy constructor, assignment operator
-// and helpers:
+// default ctor, dtor, copy ctor, assignment operator and helpers:
//////////
-// 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()
+matrix::matrix() : inherited(TINFO_matrix), row(1), col(1)
{
- debugmsg("matrix destructor",LOGLEVEL_DESTRUCT);
+ debugmsg("matrix default ctor",LOGLEVEL_CONSTRUCT);
+ m.push_back(_ex0());
}
-matrix::matrix(const matrix & other)
-{
- debugmsg("matrix copy constructor",LOGLEVEL_CONSTRUCT);
- copy(other);
-}
-
-const matrix & matrix::operator=(const matrix & other)
-{
- 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; // use STL's vector copying
+ 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);
-}
+DEFAULT_DESTROY(matrix)
//////////
-// other constructors
+// other ctors
//////////
// public
* @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)
{
- debugmsg("matrix constructor from unsigned,unsigned",LOGLEVEL_CONSTRUCT);
- m.resize(r*c, _ex0());
+ debugmsg("matrix ctor from unsigned,unsigned",LOGLEVEL_CONSTRUCT);
+ m.resize(r*c, _ex0());
}
// 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)
{
- debugmsg("matrix constructor from unsigned,unsigned,exvector",LOGLEVEL_CONSTRUCT);
+ debugmsg("matrix ctor from unsigned,unsigned,exvector",LOGLEVEL_CONSTRUCT);
+}
+
+/** 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)
+{
+ debugmsg("matrix ctor from unsigned,unsigned,lst",LOGLEVEL_CONSTRUCT);
+ m.resize(r*c, _ex0());
+
+ for (unsigned i=0; i<l.nops(); i++) {
+ unsigned x = i % c;
+ unsigned y = i / c;
+ if (y >= r)
+ break; // matrix smaller than list: throw away excessive elements
+ m[y*c+x] = l.op(i);
+ }
}
//////////
// 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)
-{
- return (new matrix(n, sym_lst))->setflag(status_flags::dynallocated);
+ debugmsg("matrix ctor 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;
+ }
}
-/** 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
-{
- debugmsg("matrix duplicate",LOGLEVEL_DUPLICATE);
- return new matrix(*this);
-}
-
-void matrix::print(ostream & os, unsigned upper_precedence) const
+void matrix::print(const print_context & 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] << "]] ]]";
-}
-
-void matrix::printraw(ostream & os) 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] << "))";
+ debugmsg("matrix print", LOGLEVEL_PRINT);
+
+ if (is_of_type(c, print_tree)) {
+
+ inherited::print(c, level);
+
+ } else {
+
+ c.s << "[";
+ for (unsigned y=0; y<row-1; ++y) {
+ c.s << "[";
+ for (unsigned x=0; x<col-1; ++x) {
+ m[y*col+x].print(c);
+ c.s << ",";
+ }
+ m[col*(y+1)-1].print(c);
+ c.s << "],";
+ }
+ c.s << "[";
+ for (unsigned x=0; x<col-1; ++x) {
+ m[(row-1)*col+x].print(c);
+ c.s << ",";
+ }
+ m[row*col-1].print(c);
+ c.s << "]]";
+
+ }
}
/** nops is defined to be rows x columns. */
unsigned matrix::nops() const
{
- return row*col;
+ return row*col;
}
/** returns matrix entry at position (i/col, i%col). */
ex matrix::op(int i) const
{
- return m[i];
+ return m[i];
}
/** returns matrix entry at position (i/col, i%col). */
ex & matrix::let_op(int i)
{
- return m[i];
+ GINAC_ASSERT(i>=0);
+ GINAC_ASSERT(i<nops());
+
+ 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);
+ 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 );
}
-/** 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 lst & ls, const lst & lr, bool no_pattern) 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(ls, lr, no_pattern);
-/** evaluate matrix entry by entry. */
-ex matrix::eval(int level) 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 );
-}
-
-/** evaluate matrix numerically entry by entry. */
-ex matrix::evalf(int level) 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);
+ return ex(matrix(row, col, m2)).bp->basic::subs(ls, lr, no_pattern);
}
// protected
int matrix::compare_same_type(const basic & other) 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_exactly_of_type(other, matrix));
+ 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;
+}
+
+bool matrix::match_same_type(const basic & other) const
+{
+ GINAC_ASSERT(is_exactly_of_type(other, matrix));
+ 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();
+}
+
+/** Automatic symbolic evaluation of an indexed matrix. */
+ex matrix::eval_indexed(const basic & i) const
+{
+ GINAC_ASSERT(is_of_type(i, indexed));
+ GINAC_ASSERT(is_ex_of_type(i.op(0), matrix));
+
+ 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();
}
+/** Sum of two indexed matrices. */
+ex matrix::add_indexed(const ex & self, const ex & other) const
+{
+ GINAC_ASSERT(is_ex_of_type(self, indexed));
+ GINAC_ASSERT(is_ex_of_type(self.op(0), matrix));
+ GINAC_ASSERT(is_ex_of_type(other, indexed));
+ GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
+
+ // Only add two matrices
+ if (is_ex_of_type(other.op(0), matrix)) {
+ 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_ex_of_type(self, indexed));
+ GINAC_ASSERT(is_ex_of_type(self.op(0), matrix));
+ 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_ex_of_type(*self, indexed));
+ GINAC_ASSERT(is_ex_of_type(*other, indexed));
+ GINAC_ASSERT(self->nops() == 2 || self->nops() == 3);
+ GINAC_ASSERT(is_ex_of_type(self->op(0), matrix));
+
+ // Only contract with other matrices
+ if (!is_ex_of_type(other->op(0), matrix))
+ 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);
}
+
/** Difference of matrices.
*
* @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);
}
+
/** Product of matrices.
*
* @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 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.
+ 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_ex_exactly_of_type(expn, numeric)) {
+ // 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();
+ // 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!=1) {
+ if (b.is_odd()) {
+ C = C.mul(A);
+ b -= 1;
+ }
+ b *= _num1_2(); // b /= 2, 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 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
{
- 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);
-}
-
-/* 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=_ex1();
- ex piv;
-
- for (unsigned r1=0; r1<M.rows(); ++r1) {
- int indx = tmp.pivot(r1);
- if (indx == -1) {
- return _ex0();
- }
- if (indx != 0) {
- det *= _ex_1();
- }
- det = det * tmp.m[r1*M.cols()+r1];
- for (unsigned r2=r1+1; r2<M.rows(); ++r2) {
- piv = tmp.m[r2*M.cols()+r1] / tmp.m[r1*M.cols()+r1];
- for (unsigned 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 <typename 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<unsigned> sigma(M.cols());
- for (unsigned i=0; i<M.cols(); ++i) sigma[i]=i;
-
- do {
- term = M(sigma[0],0);
- for (unsigned 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;
-}
-
-/** 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 (unsigned r1=0; r1<M.rows(); ++r1) {
- // 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,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 (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;
- * }
- *}*/
+ 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. 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.)
+ * 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
+ * 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).
+ * @param algo allows to chose an algorithm
* @return the determinant as a new expression
- * @exception logic_error (matrix not square) */
-ex matrix::determinant(bool normalized) const
-{
- if (row != col) {
- throw (std::logic_error("matrix::determinant(): matrix not square"));
- }
-
- // check, if there are non-numeric entries in the matrix:
- for (exvector::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);
-}
-
-/** Trace of a matrix.
+ * @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());
+
+ // 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) {
+ lst 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 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 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));
+ }
+ 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();
+ }
+ }
+}
+
+
+/** 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) {
- throw (std::logic_error("matrix::trace(): matrix not square"));
- }
-
- ex tr;
- for (unsigned r=0; r<col; ++r) {
- tr += m[r*col+r];
- }
- return tr;
-}
-
-/** 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
- * polynomial as a new expression.
+ 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();
+}
+
+
+/** Characteristic Polynomial. Following mathematica notation 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. Note that some CASs define it with a sign inside the determinant
+ * which gives rise to an overall sign if the dimension is odd. This method
+ * returns the characteristic polynomial collected in powers of lambda as a
+ * new expression.
*
* @return characteristic polynomial as new expression
* @exception logic_error (matrix not square)
* @see matrix::determinant() */
-ex matrix::charpoly(const ex & lambda) const
+ex matrix::charpoly(const symbol & lambda) const
{
- if (row != col) {
- throw (std::logic_error("matrix::charpoly(): matrix not square"));
- }
-
- matrix M(*this);
- for (unsigned r=0; r<col; ++r) {
- M.m[r*col+r] -= lambda;
- }
- return (M.determinant());
+ 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) {
+ 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;
+ }
+
+ matrix M(*this);
+ for (unsigned r=0; r<col; ++r)
+ M.m[r*col+r] -= lambda;
+
+ return M.determinant().collect(lambda);
}
+
/** Inverse of this matrix.
*
* @return the inverted matrix
* @exception runtime_error (singular matrix) */
matrix matrix::inverse(void) 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;
-}
-
-void matrix::ffe_swap(unsigned r1, unsigned c1, unsigned r2 ,unsigned c2)
-{
- ensure_if_modifiable();
-
- ex tmp=ffe_get(r1,c1);
- ffe_set(r1,c1,ffe_get(r2,c2));
- ffe_set(r2,c2,tmp);
-}
-
-void matrix::ffe_set(unsigned r, unsigned c, ex e)
-{
- set(r-1,c-1,e);
-}
-
-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'
- * by Keith O. Geddes et al.
+ 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 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
+ * @return n x p solution matrix
* @exception logic_error (incompatible matrices)
- * @exception runtime_error (singular matrix) */
-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"));
- }
-
- 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=1;
-
- // eliminate below row r, with pivot in column k
- for (unsigned k=1; (k<=n)&&(r<=m); ++k) {
- // find a nonzero pivot
- 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 (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;
- }
- 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,(a.ffe_get(r,k)*b.ffe_get(i,1)
- -b.ffe_get(r,1)*a.ffe_get(i,k))/divisor);
- b.ffe_set(i,1,b.ffe_get(i,1).normal() /*.normal() */ );
- a.ffe_set(i,k,0);
- }
- 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; }
-
- /*
- 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 (unsigned r=1; r<=m; ++r) {
- int zero_in_this_row=0;
- for (unsigned c=1; c<=n; ++c) {
- if (a.ffe_get(r,c).is_equal(_ex0())) {
- 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 (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()) {
- 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+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 (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;
- }
- }
- // assign solutions for vars between 1 and
- // last_assigned_sol-1: free parameters
- for (unsigned c=1; c<=last_assigned_sol-1; ++c) {
- sol.ffe_set(c,1,vars.ffe_get(c,1));
- }
-
- /*
- for (unsigned 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 (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;
- cout << "diff=" << (e-b.ffe_get(r,1)).normal() << endl;
- }
- GINAC_ASSERT((e-b.ffe_get(r,1)).normal().is_zero());
- }
-
- // test solution with original matrix
- 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;
- }
- } catch (...) {
- 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 simultaneous set of equations. */
-matrix matrix::solve(const matrix & v) const
-{
- if (!(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
- 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] = 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];
- }
- }
- for (unsigned r1=0; r1<row; ++r1) {
- int indx = tmp.pivot(r1);
- if (indx == -1) {
- throw (std::runtime_error("matrix::solve(): singular matrix"));
- }
- for (unsigned c=r1; c<tmp.col; ++c) {
- tmp.m[r1*tmp.col+c] /= tmp.m[r1*tmp.col+r1];
- }
- for (unsigned r2=r1+1; r2<row; ++r2) {
- for (unsigned c=r1; c<tmp.col; ++c) {
- tmp.m[r2*tmp.col+c]
- -= tmp.m[r2*tmp.col+r1] * tmp.m[r1*tmp.col+c];
- }
- }
- }
-
- // assemble the solution matrix
- exvector sol(v.row*v.col);
- for (unsigned c=0; c<v.col; ++c) {
- for (unsigned r=col-1; r>=0; --r) {
- sol[r*v.col+c] = tmp[r*tmp.col+c];
- for (unsigned i=r+1; i<col; ++i) {
- sol[r*v.col+c]
- -= tmp[r*tmp.col+i] * sol[i*v.col+c];
- }
- }
- }
- return matrix(v.row, v.col, sol);
+ * @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,
+ unsigned algo) const
+{
+ 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) {
+ 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;
}
+
// 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(unsigned ro)
-{
- unsigned k=ro;
-
- for (unsigned r=ro; r<row; ++r) {
- if (!m[r*col+ro].is_zero()) {
- k = r;
- break;
- }
- }
- if (m[k*col+ro].is_zero()) {
- return -1;
- }
- if (k!=ro) { // swap rows
- for (unsigned c=0; c<col; ++c) {
- m[k*col+c].swap(m[ro*col+c]);
- }
- return k;
- }
- return 0;
+/** 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:
+ 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 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(const bool det)
+{
+ 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 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(const bool det)
+{
+ 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;
}
-//////////
-// global constants
-//////////
-const matrix some_matrix;
-const type_info & typeid_matrix=typeid(some_matrix);
+/** Perform the steps of Bareiss' one-step fraction free elimination to bring
+ * the matrix into an upper echelon form. Fraction free elimination means
+ * that divide is used straightforwardly, without computing GCDs first. This
+ * 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
+ * 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(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)}.
+
+ 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
+ lst 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);
+
+ return sign;
+}
+
+
+/** Partial pivoting method for matrix elimination schemes.
+ * 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 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, unsigned co, bool symbolic)
+{
+ 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_ex_of_type(this->m[k*col+co],numeric));
+ unsigned kmax = k+1;
+ numeric mmax = abs(ex_to<numeric>(m[kmax*col+co]));
+ while (kmax<row) {
+ GINAC_ASSERT(is_ex_of_type(this->m[kmax*col+co],numeric));
+ 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;
+}
+
+ex lst_to_matrix(const lst & l)
+{
+ // 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();
+
+ // Allocate and fill matrix
+ matrix &m = *new matrix(rows, cols);
+ m.setflag(status_flags::dynallocated);
+ for (i=0; i<rows; i++)
+ for (j=0; j<cols; j++)
+ if (l.op(i).nops() > j)
+ m(i, j) = l.op(i).op(j);
+ else
+ m(i, j) = _ex0();
+ return m;
+}
+
+ex diag_matrix(const lst & l)
+{
+ unsigned dim = l.nops();
+
+ matrix &m = *new matrix(dim, dim);
+ m.setflag(status_flags::dynallocated);
+ for (unsigned i=0; i<dim; i++)
+ m(i, i) = l.op(i);
+
+ return m;
+}
-#ifndef NO_NAMESPACE_GINAC
} // namespace GiNaC
-#endif // ndef NO_NAMESPACE_GINAC
git://www.ginac.de / ginac.git / blobdiff