* Implementation of symbolic matrices */
/*
- * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2015 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
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
- * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
-#include <algorithm>
-#include <map>
-#include <stdexcept>
-
#include "matrix.h"
#include "numeric.h"
#include "lst.h"
#include "idx.h"
#include "indexed.h"
+#include "add.h"
#include "power.h"
#include "symbol.h"
+#include "operators.h"
#include "normal.h"
-#include "print.h"
#include "archive.h"
#include "utils.h"
-#include "debugmsg.h"
+
+#include <algorithm>
+#include <iostream>
+#include <map>
+#include <sstream>
+#include <stdexcept>
+#include <string>
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>(&matrix::do_print_tree).
+ print_func<print_python_repr>(&matrix::do_print_python_repr))
//////////
-// default ctor, dtor, copy ctor, assignment operator and helpers:
+// default constructor
//////////
/** Default ctor. Initializes to 1 x 1-dimensional zero-matrix. */
-matrix::matrix() : inherited(TINFO_matrix), row(1), col(1)
-{
- debugmsg("matrix default ctor",LOGLEVEL_CONSTRUCT);
- m.push_back(_ex0());
-}
-
-void matrix::copy(const matrix & other)
+matrix::matrix() : row(1), col(1), m(1, _ex0)
{
- inherited::copy(other);
- row = other.row;
- col = other.col;
- m = other.m; // STL's vector copying invoked here
+ setflag(status_flags::not_shareable);
}
-DEFAULT_DESTROY(matrix)
-
//////////
-// other ctors
+// other constructors
//////////
// public
*
* @param r number of rows
* @param c number of cols */
-matrix::matrix(unsigned r, unsigned c)
- : inherited(TINFO_matrix), row(r), col(c)
+matrix::matrix(unsigned r, unsigned c) : row(r), col(c), m(r*c, _ex0)
{
- debugmsg("matrix ctor from unsigned,unsigned",LOGLEVEL_CONSTRUCT);
- m.resize(r*c, _ex0());
+ setflag(status_flags::not_shareable);
}
// 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)
+ : row(r), col(c), m(m2)
{
- debugmsg("matrix ctor from unsigned,unsigned,exvector",LOGLEVEL_CONSTRUCT);
+ setflag(status_flags::not_shareable);
}
/** Construct matrix from (flat) list of elements. If the list has fewer
* 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)
+ : row(r), col(c), m(r*c, _ex0)
{
- debugmsg("matrix ctor from unsigned,unsigned,lst",LOGLEVEL_CONSTRUCT);
- m.resize(r*c, _ex0());
+ setflag(status_flags::not_shareable);
- for (unsigned i=0; i<l.nops(); i++) {
- unsigned x = i % c;
- unsigned y = i / c;
+ size_t i = 0;
+ for (auto & it : l) {
+ 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] = l.op(i);
+ m[y*c+x] = it;
+ ++i;
}
}
// archiving
//////////
-matrix::matrix(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
+void matrix::read_archive(const archive_node &n, lst &sym_lst)
{
- debugmsg("matrix ctor from archive_node", LOGLEVEL_CONSTRUCT);
+ inherited::read_archive(n, sym_lst);
+
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++) {
+ // XXX: default ctor inserts a zero element, we need to erase it here.
+ m.pop_back();
+ auto first = n.find_first("m");
+ auto last = n.find_last("m");
+ ++last;
+ for (auto i=first; i != last; ++i) {
ex e;
- if (n.find_ex("m", e, sym_lst, i))
- m.push_back(e);
- else
- break;
+ n.find_ex_by_loc(i, e, sym_lst);
+ m.push_back(e);
}
}
+GINAC_BIND_UNARCHIVER(matrix);
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;
+ for (auto & i : m) {
+ n.add_ex("m", i);
}
}
-DEFAULT_UNARCHIVE(matrix)
-
//////////
// functions overriding virtual functions from base classes
//////////
// public
-void matrix::print(const print_context & c, unsigned level) 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
{
- debugmsg("matrix print", LOGLEVEL_PRINT);
-
- if (is_a<print_tree>(c)) {
-
- inherited::print(c, level);
+ 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;
+ }
+}
- } else {
+void matrix::do_print(const print_context & c, unsigned level) const
+{
+ c.s << "[";
+ print_elements(c, "[", "]", ",", ",");
+ c.s << "]";
+}
- 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 << "]]";
+void matrix::do_print_latex(const print_latex & c, unsigned level) const
+{
+ c.s << "\\left(\\begin{array}{" << std::string(col,'c') << "}";
+ print_elements(c, "", "", "\\\\", "&");
+ c.s << "\\end{array}\\right)";
+}
- }
+void matrix::do_print_python_repr(const print_python_repr & c, unsigned level) const
+{
+ 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
{
+ 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)
{
- GINAC_ASSERT(i>=0);
GINAC_ASSERT(i<nops());
+ ensure_if_modifiable();
return m[i];
}
/** 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;
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 );
+ return (new matrix(row, col, std::move(m2)))->setflag(status_flags::dynallocated |
+ status_flags::evaluated);
}
-ex matrix::subs(const lst & ls, const lst & lr, bool no_pattern) const
+ex matrix::subs(const exmap & mp, unsigned options) const
{
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);
+ m2[r*col+c] = m[r*col+c].subs(mp, options);
+
+ return matrix(row, col, std::move(m2)).subs_one_level(mp, options);
+}
+
+/** Complex conjugate every matrix entry. */
+ex matrix::conjugate() const
+{
+ std::unique_ptr<exvector> ev(nullptr);
+ for (auto i=m.begin(); i!=m.end(); ++i) {
+ ex x = i->conjugate();
+ if (ev) {
+ ev->push_back(x);
+ continue;
+ }
+ if (are_ex_trivially_equal(x, *i)) {
+ continue;
+ }
+ ev.reset(new exvector);
+ ev->reserve(m.size());
+ for (auto j=m.begin(); j!=i; ++j) {
+ ev->push_back(*j);
+ }
+ ev->push_back(x);
+ }
+ if (ev) {
+ return matrix(row, col, std::move(*ev));
+ }
+ return *this;
+}
+
+ex matrix::real_part() const
+{
+ exvector v;
+ v.reserve(m.size());
+ for (auto & i : m)
+ v.push_back(i.real_part());
+ return matrix(row, col, std::move(v));
+}
- return matrix(row, col, m2).basic::subs(ls, lr, no_pattern);
+ex matrix::imag_part() const
+{
+ exvector v;
+ v.reserve(m.size());
+ for (auto & i : m)
+ v.push_back(i.imag_part());
+ return matrix(row, col, std::move(v));
}
// protected
GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
// Only add two matrices
- if (is_ex_of_type(other.op(0), matrix)) {
+ 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));
GINAC_ASSERT(is_a<matrix>(self->op(0)));
// Only contract with other matrices
- if (!is_ex_of_type(other->op(0), matrix))
+ if (!is_a<matrix>(other->op(0)))
return false;
GINAC_ASSERT(other->nops() == 2 || other->nops() == 3);
*self = self_matrix.mul(other_matrix.transpose())(0, 0);
}
}
- *other = _ex1();
+ *other = _ex1;
return true;
} else { // vector * matrix
*self = indexed(self_matrix.mul(other_matrix), other->op(2));
else
*self = indexed(self_matrix.transpose().mul(other_matrix), other->op(2));
- *other = _ex1();
+ *other = _ex1;
return true;
}
*self = indexed(other_matrix.mul(self_matrix), other->op(1));
else
*self = indexed(other_matrix.mul(self_matrix.transpose()), other->op(1));
- *other = _ex1();
+ *other = _ex1;
return true;
}
}
// 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();
+ *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();
+ *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();
+ *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();
+ *other = _ex1;
return true;
}
}
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++;
+ auto ci = other.m.begin();
+ for (auto & i : sum)
+ i += *ci++;
- return matrix(row,col,sum);
+ return matrix(row, col, std::move(sum));
}
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++;
+ auto ci = other.m.begin();
+ for (auto & i : dif)
+ i -= *ci++;
- return matrix(row,col,dif);
+ return matrix(row, col, std::move(dif));
}
for (unsigned r1=0; r1<this->rows(); ++r1) {
for (unsigned c=0; c<this->cols(); ++c) {
+ // Quick test: can we shortcut?
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();
+ prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]);
}
}
- return matrix(row, other.col, prod);
+ return matrix(row, other.col, std::move(prod));
}
for (unsigned c=0; c<col; ++c)
prod[r*col+c] = m[r*col+c] * other;
- return matrix(row, col, prod);
+ return matrix(row, col, std::move(prod));
}
for (unsigned c=0; c<col; ++c)
prod[r*col+c] = m[r*col+c] * other;
- return matrix(row, col, prod);
+ return matrix(row, col, std::move(prod));
}
if (col!=row)
throw (std::logic_error("matrix::pow(): matrix not square"));
- if (is_ex_exactly_of_type(expn, numeric)) {
+ 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)) {
}
matrix C(row,col);
for (unsigned r=0; r<row; ++r)
- C(r,r) = _ex1();
+ 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!=1) {
+ while (b!=*_num1_p) {
if (b.is_odd()) {
C = C.mul(A);
- b -= 1;
+ --b;
}
- b *= _num1_2(); // b /= 2, still integer.
+ b /= *_num2_p; // still integer.
A = A.mul(A);
}
return A.mul(C);
/** 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(this->cols()*this->rows());
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);
+ return matrix(this->cols(), this->rows(), std::move(trans));
}
/** Determinant of square matrix. This routine doesn't actually calculate the
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);
+ for (auto r : m) {
+ if (!r.info(info_flags::numeric))
+ numeric_flag = false;
+ 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))
+ rtest.info(info_flags::rational_function))
normal_flag = true;
- ++r;
}
// Here is the heuristics in case this routine has to decide:
else
return m[0].expand();
}
-
+
// Compute the determinant
switch(algo) {
case determinant_algo::gauss: {
int sign;
sign = tmp.division_free_elimination(true);
if (sign==0)
- return _ex0();
+ return _ex0;
ex det = tmp.m[row*col-1];
// factor out accumulated bogus slag
for (unsigned d=0; d<row-2; ++d)
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:
+ // 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) {
++acc;
c_zeros.push_back(uintpair(acc,c));
}
- sort(c_zeros.begin(),c_zeros.end());
+ 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);
+ for (auto & i : c_zeros)
+ 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 (auto & it : pre_sort) {
for (unsigned r=0; r<row; ++r)
- result[r*col+c] = m[r*col+(*i)];
+ result[r*col+c] = m[r*col+it];
+ ++c;
}
if (normal_flag)
- return (sign*matrix(row,col,result).determinant_minor()).normal();
+ return (sign*matrix(row, col, std::move(result)).determinant_minor()).normal();
else
- return sign*matrix(row,col,result).determinant_minor();
+ return sign*matrix(row, col, std::move(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"));
tr += m[r*col+r];
if (tr.info(info_flags::rational_function) &&
- !tr.info(info_flags::crational_polynomial))
+ !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
+ * characteristic polynomial of a matrix M is defined as the determinant 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
* @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;
- exvector::const_iterator r = m.begin(), rend = m.end();
- while (r!=rend && numeric_flag==true) {
- if (!r->info(info_flags::numeric))
+ for (auto & r : m) {
+ if (!r.info(info_flags::numeric)) {
numeric_flag = false;
- ++r;
+ break;
+ }
}
// 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);
+ 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);
+ 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;
+ matrix M(*this);
+ for (unsigned r=0; r<col; ++r)
+ M.m[r*col+r] -= lambda;
- return M.determinant().collect(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"));
// 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();
+ 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
*
* @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 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 matrix & rhs,
+ unsigned algo) const
{
const unsigned m = this->rows();
const unsigned n = this->cols();
// 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))
+ for (auto & r : aug.m) {
+ if (!r.info(info_flags::numeric)) {
numeric_flag = false;
- ++r;
+ break;
+ }
}
// Here is the heuristics in case this routine has to decide:
}
+/** Compute the rank of this matrix. */
+unsigned matrix::rank() const
+{
+ // Method:
+ // Transform this matrix into upper echelon form and then count the
+ // number of non-zero rows.
+
+ GINAC_ASSERT(row*col==m.capacity());
+
+ // Actually, any elimination scheme will do since we are only
+ // interested in the echelon matrix' zeros.
+ matrix to_eliminate = *this;
+ to_eliminate.fraction_free_elimination();
+
+ unsigned r = row*col; // index of last non-zero element
+ while (r--) {
+ if (!to_eliminate.m[r].is_zero())
+ return 1+r/col;
+ }
+ return 0;
+}
+
+
// protected
/** Recursive determinant for small matrices having at least one symbolic
* 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.
+ * matrix' dimension is larger than 7.
*
* @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:
const unsigned n = this->cols();
Pkey.push_back(i);
unsigned fc = 0; // controls logic for our strange flipper counter
do {
- det = _ex0();
+ det = _ex0;
for (unsigned r=0; r<n-c; ++r) {
// maybe there is nothing to do?
if (m[Pkey[r]*n+c].is_zero())
Pkey[j] = Pkey[j-1]+1;
} while(fc);
// next column, so change the role of A and B:
- A = B;
+ A.swap(B);
B.clear();
}
int sign = 1;
unsigned r0 = 0;
- for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
- int indx = pivot(r0, r1, true);
+ for (unsigned c0=0; c0<n && r0<m-1; ++c0) {
+ int indx = pivot(r0, c0, true);
if (indx == -1) {
sign = 0;
if (det)
if (indx > 0)
sign = -sign;
for (unsigned r2=r0+1; r2<m; ++r2) {
- if (!this->m[r2*n+r1].is_zero()) {
+ if (!this->m[r2*n+c0].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) {
+ ex piv = this->m[r2*n+c0] / this->m[r0*n+c0];
+ for (unsigned c=c0+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();
+ for (unsigned c=r0; c<=c0; ++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();
+ this->m[r0*n+c] = _ex0;
}
++r0;
}
}
-
+ // clear remaining rows
+ for (unsigned r=r0+1; r<m; ++r) {
+ for (unsigned c=0; c<n; ++c)
+ this->m[r*n+c] = _ex0;
+ }
+
return sign;
}
int sign = 1;
unsigned r0 = 0;
- for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
- int indx = pivot(r0, r1, true);
+ for (unsigned c0=0; c0<n && r0<m-1; ++c0) {
+ int indx = pivot(r0, c0, true);
if (indx==-1) {
sign = 0;
if (det)
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();
+ for (unsigned c=c0+1; c<n; ++c)
+ this->m[r2*n+c] = (this->m[r0*n+c0]*this->m[r2*n+c] - this->m[r2*n+c0]*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();
+ for (unsigned c=r0; c<=c0; ++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();
+ this->m[r0*n+c] = _ex0;
}
++r0;
}
}
-
+ // clear remaining rows
+ for (unsigned r=r0+1; r<m; ++r) {
+ for (unsigned c=0; c<n; ++c)
+ this->m[r*n+c] = _ex0;
+ }
+
return sign;
}
//
// 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).
+ // 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
// 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;
+ exmap srl; // symbol replacement list
+ auto tmp_n_it = tmp_n.m.begin(), tmp_d_it = tmp_d.m.begin();
+ for (auto & it : this->m) {
+ ex nd = it.normal().to_rational(srl).numer_denom();
*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) {
+ for (unsigned c0=0; c0<n && r0<m-1; ++c0) {
+ // When trying to find a pivot, we should try a bit harder than expand().
+ // Searching the first non-zero element in-place here instead of calling
+ // pivot() allows us to do no more substitutions and back-substitutions
+ // than are actually necessary.
+ unsigned indx = r0;
+ while ((indx<m) &&
+ (tmp_n[indx*n+c0].subs(srl, subs_options::no_pattern).expand().is_zero()))
+ ++indx;
+ if (indx==m) {
+ // all elements in column c0 below row r0 vanish
sign = 0;
if (det)
return 0;
- }
- if (indx>=0) {
- if (indx>0) {
+ } else {
+ if (indx>r0) {
+ // Matrix needs pivoting, swap rows r0 and indx of tmp_n and tmp_d.
sign = -sign;
- // tmp_n's rows r0 and indx were swapped, do the same in tmp_d:
- for (unsigned c=r1; c<n; ++c)
+ for (unsigned c=c0; c<n; ++c) {
+ tmp_n.m[n*indx+c].swap(tmp_n.m[n*r0+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();
+ for (unsigned c=c0+1; c<n; ++c) {
+ dividend_n = (tmp_n.m[r0*n+c0]*tmp_n.m[r2*n+c]*
+ tmp_d.m[r2*n+c0]*tmp_d.m[r0*n+c]
+ -tmp_n.m[r2*n+c0]*tmp_n.m[r0*n+c]*
+ tmp_d.m[r0*n+c0]*tmp_d.m[r2*n+c]).expand();
+ dividend_d = (tmp_d.m[r2*n+c0]*tmp_d.m[r0*n+c]*
+ tmp_d.m[r0*n+c0]*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,
GINAC_ASSERT(check);
}
// fill up left hand side with zeros
- for (unsigned c=0; c<=r1; ++c)
- tmp_n.m[r2*n+c] = _ex0();
+ for (unsigned c=r0; c<=c0; ++c)
+ tmp_n.m[r2*n+c] = _ex0;
}
- if ((r1<n-1)&&(r0<m-1)) {
+ if (c0<n && 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();
+ divisor_n = tmp_n.m[r0*n+c0].expand();
+ divisor_d = tmp_d.m[r0*n+c0].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();
+ tmp_n.m[r0*n+c] = _ex0;
+ tmp_d.m[r0*n+c] = _ex1;
}
}
}
++r0;
}
}
+ // clear remaining rows
+ for (unsigned r=r0+1; r<m; ++r) {
+ for (unsigned c=0; c<n; ++c)
+ tmp_n.m[r*n+c] = _ex0;
+ }
+
// 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);
+ for (auto & it : this->m)
+ it = ((*tmp_n_it++)/(*tmp_d_it++)).subs(srl, subs_options::no_pattern);
return sign;
}
* @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
+ * @return 0 if no interchange occurred, -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)
++k;
} else {
// search largest element in column co beginning at row ro
- GINAC_ASSERT(is_a<numeric>(this->m[k*col+co]));
+ 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_a<numeric>(this->m[kmax*col+co]));
+ 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;
return k;
}
+/** Function to check that all elements of the matrix are zero.
+ */
+bool matrix::is_zero_matrix() const
+{
+ for (auto & i : m)
+ if (!i.is_zero())
+ return false;
+ return true;
+}
+
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();
+ size_t rows = l.nops(), cols = 0;
+ for (auto & itr : l) {
+ 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);
- 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;
+ matrix &M = *new matrix(rows, cols);
+ M.setflag(status_flags::dynallocated);
+
+ unsigned i = 0;
+ for (auto & itr : l) {
+ unsigned j = 0;
+ for (auto & itc : ex_to<lst>(itr)) {
+ M(i, j) = itc;
+ ++j;
+ }
+ ++i;
+ }
+
+ return M;
}
ex diag_matrix(const lst & l)
{
- unsigned dim = l.nops();
+ size_t 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);
+ // Allocate and fill matrix
+ matrix &M = *new matrix(dim, dim);
+ M.setflag(status_flags::dynallocated);
+
+ unsigned i = 0;
+ for (auto & it : l) {
+ M(i, i) = it;
+ ++i;
+ }
+
+ 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;
+}
+
+ex reduced_matrix(const matrix& m, unsigned r, unsigned c)
+{
+ if (r+1>m.rows() || c+1>m.cols() || m.cols()<2 || m.rows()<2)
+ throw std::runtime_error("minor_matrix(): index out of bounds");
+
+ const unsigned rows = m.rows()-1;
+ const unsigned cols = m.cols()-1;
+ matrix &M = *new matrix(rows, cols);
+ M.setflag(status_flags::dynallocated | status_flags::evaluated);
+
+ unsigned ro = 0;
+ unsigned ro2 = 0;
+ while (ro2<rows) {
+ if (ro==r)
+ ++ro;
+ unsigned co = 0;
+ unsigned co2 = 0;
+ while (co2<cols) {
+ if (co==c)
+ ++co;
+ M(ro2,co2) = m(ro, co);
+ ++co;
+ ++co2;
+ }
+ ++ro;
+ ++ro2;
+ }
+
+ return M;
+}
+
+ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc)
+{
+ if (r+nr>m.rows() || c+nc>m.cols())
+ throw std::runtime_error("sub_matrix(): index out of bounds");
+
+ matrix &M = *new matrix(nr, nc);
+ M.setflag(status_flags::dynallocated | status_flags::evaluated);
+
+ for (unsigned ro=0; ro<nr; ++ro) {
+ for (unsigned co=0; co<nc; ++co) {
+ M(ro,co) = m(ro+r,co+c);
+ }
+ }
- return m;
+ return M;
}
} // namespace GiNaC