matrix.cpp

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/*
 *  GiNaC Copyright (C) 1999-2026 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
 *  the Free Software Foundation; either version 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License
 *  along with this program.  If not, see <https://www.gnu.org/licenses/>.
 */
 
#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 "archive.h"
#include "utils.h"
 
#include <map>
#include <sstream>
#include <stdexcept>
#include <string>
 
namespace GiNaC {
 
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 constructor
 
matrix::matrix() : row(1), col(1), m(1, _ex0)
{
    setflag(status_flags::not_shareable);
}
 
// other constructors
 
// public
 
matrix::matrix(unsigned r, unsigned c) : row(r), col(c), m(r*c, _ex0)
{
    setflag(status_flags::not_shareable);
}
 
matrix::matrix(unsigned r, unsigned c, const lst & l)
  : row(r), col(c), m(r*c, _ex0)
{
    setflag(status_flags::not_shareable);
 
    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] = it;
        ++i;
    }
}
 
matrix::matrix(std::initializer_list<std::initializer_list<ex>> l)
  : row(l.size()), col(l.begin()->size())
{
    setflag(status_flags::not_shareable);
 
    m.reserve(row*col);
    for (const auto & r : l) {
        unsigned c = 0;
        for (const auto & e : r) {
            m.push_back(e);
            ++c;
        }
        if (c != col)
            throw std::invalid_argument("matrix::matrix{{}}: wrong dimension");
    }
}
 
// protected
 
matrix::matrix(unsigned r, unsigned c, const exvector & m2)
  : row(r), col(c), m(m2)
{
    setflag(status_flags::not_shareable);
}
matrix::matrix(unsigned r, unsigned c, exvector && m2)
  : row(r), col(c), m(std::move(m2))
{
    setflag(status_flags::not_shareable);
}
 
// archiving
 
void matrix::read_archive(const archive_node &n, lst &sym_lst)
{
    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);
    // XXX: default ctor inserts a zero element, we need to erase it here.
    m.pop_back();
    auto range = n.find_property_range("m", "m");
    for (auto i=range.begin; i != range.end; ++i) {
        ex e;
        n.find_ex_by_loc(i, e, sym_lst);
        m.emplace_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);
    for (auto & i : m) {
        n.add_ex("m", i);
    }
}
 
// functions overriding virtual functions from base classes
 
// public
 
void matrix::print_elements(const print_context & c, const char *row_start, const char *row_end, const char *row_sep, const char *col_sep) const
{
    for (unsigned ro=0; ro<row; ++ro) {
        c.s << row_start;
        for (unsigned co=0; co<col; ++co) {
            m[ro*col+co].print(c);
            if (co < col-1)
                c.s << col_sep;
            else
                c.s << row_end;
        }
        if (ro < row-1)
            c.s << row_sep;
    }
}
 
void matrix::do_print(const print_context & c, unsigned level) const
{
    c.s << "[";
    print_elements(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 << ')';
}
 
size_t matrix::nops() const
{
    return static_cast<size_t>(row) * static_cast<size_t>(col);
}
 
ex matrix::op(size_t i) const
{
    GINAC_ASSERT(i<nops());
    
    return m[i];
}
 
ex & matrix::let_op(size_t i)
{
    GINAC_ASSERT(i<nops());
    
    ensure_if_modifiable();
    return m[i];
}
 
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(mp, options);
 
    return matrix(row, col, std::move(m2)).subs_one_level(mp, options);
}
 
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));
}
 
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
 
int matrix::compare_same_type(const basic & other) const
{
    GINAC_ASSERT(is_exactly_a<matrix>(other));
    const matrix &o = static_cast<const matrix &>(other);
    
    // compare number of rows
    if (row != o.rows())
        return row < o.rows() ? -1 : 1;
    
    // compare number of columns
    if (col != o.cols())
        return col < o.cols() ? -1 : 1;
    
    // equal number of rows and columns, compare individual elements
    int cmpval;
    for (unsigned r=0; r<row; ++r) {
        for (unsigned c=0; c<col; ++c) {
            cmpval = ((*this)(r,c)).compare(o(r,c));
            if (cmpval!=0) return cmpval;
        }
    }
    // all elements are equal => matrices are equal;
    return 0;
}
 
bool matrix::match_same_type(const basic & other) const
{
    GINAC_ASSERT(is_exactly_a<matrix>(other));
    const matrix & o = static_cast<const matrix &>(other);
    
    // The number of rows and columns must be the same. This is necessary to
    // prevent a 2x3 matrix from matching a 3x2 one.
    return row == o.rows() && col == o.cols();
}
 
ex matrix::eval_indexed(const basic & i) const
{
    GINAC_ASSERT(is_a<indexed>(i));
    GINAC_ASSERT(is_a<matrix>(i.op(0)));
 
    bool all_indices_unsigned = static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint);
 
    // Check indices
    if (i.nops() == 2) {
 
        // One index, must be one-dimensional vector
        if (row != 1 && col != 1)
            throw (std::runtime_error("matrix::eval_indexed(): vector must have exactly 1 index"));
 
        const idx & i1 = ex_to<idx>(i.op(1));
 
        if (col == 1) {
 
            // Column vector
            if (!i1.get_dim().is_equal(row))
                throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
 
            // Index numeric -> return vector element
            if (all_indices_unsigned) {
                unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
                if (n1 >= row)
                    throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
                return (*this)(n1, 0);
            }
 
        } else {
 
            // Row vector
            if (!i1.get_dim().is_equal(col))
                throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
 
            // Index numeric -> return vector element
            if (all_indices_unsigned) {
                unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
                if (n1 >= col)
                    throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
                return (*this)(0, n1);
            }
        }
 
    } else if (i.nops() == 3) {
 
        // Two indices
        const idx & i1 = ex_to<idx>(i.op(1));
        const idx & i2 = ex_to<idx>(i.op(2));
 
        if (!i1.get_dim().is_equal(row))
            throw (std::runtime_error("matrix::eval_indexed(): dimension of first index must match number of rows"));
        if (!i2.get_dim().is_equal(col))
            throw (std::runtime_error("matrix::eval_indexed(): dimension of second index must match number of columns"));
 
        // Pair of dummy indices -> compute trace
        if (is_dummy_pair(i1, i2))
            return trace();
 
        // Both indices numeric -> return matrix element
        if (all_indices_unsigned) {
            unsigned n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
            if (n1 >= row)
                throw (std::runtime_error("matrix::eval_indexed(): value of first index exceeds number of rows"));
            if (n2 >= col)
                throw (std::runtime_error("matrix::eval_indexed(): value of second index exceeds number of columns"));
            return (*this)(n1, n2);
        }
 
    } else
        throw (std::runtime_error("matrix::eval_indexed(): matrix must have exactly 2 indices"));
 
    return i.hold();
}
 
ex matrix::add_indexed(const ex & self, const ex & other) const
{
    GINAC_ASSERT(is_a<indexed>(self));
    GINAC_ASSERT(is_a<matrix>(self.op(0)));
    GINAC_ASSERT(is_a<indexed>(other));
    GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
 
    // Only add two matrices
    if (is_a<matrix>(other.op(0))) {
        GINAC_ASSERT(other.nops() == 2 || other.nops() == 3);
 
        const matrix &self_matrix = ex_to<matrix>(self.op(0));
        const matrix &other_matrix = ex_to<matrix>(other.op(0));
 
        if (self.nops() == 2 && other.nops() == 2) { // vector + vector
 
            if (self_matrix.row == other_matrix.row)
                return indexed(self_matrix.add(other_matrix), self.op(1));
            else if (self_matrix.row == other_matrix.col)
                return indexed(self_matrix.add(other_matrix.transpose()), self.op(1));
 
        } else if (self.nops() == 3 && other.nops() == 3) { // matrix + matrix
 
            if (self.op(1).is_equal(other.op(1)) && self.op(2).is_equal(other.op(2)))
                return indexed(self_matrix.add(other_matrix), self.op(1), self.op(2));
            else if (self.op(1).is_equal(other.op(2)) && self.op(2).is_equal(other.op(1)))
                return indexed(self_matrix.add(other_matrix.transpose()), self.op(1), self.op(2));
 
        }
    }
 
    // Don't know what to do, return unevaluated sum
    return self + other;
}
 
ex matrix::scalar_mul_indexed(const ex & self, const numeric & other) const
{
    GINAC_ASSERT(is_a<indexed>(self));
    GINAC_ASSERT(is_a<matrix>(self.op(0)));
    GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
 
    const matrix &self_matrix = ex_to<matrix>(self.op(0));
 
    if (self.nops() == 2)
        return indexed(self_matrix.mul(other), self.op(1));
    else // self.nops() == 3
        return indexed(self_matrix.mul(other), self.op(1), self.op(2));
}
 
bool matrix::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
{
    GINAC_ASSERT(is_a<indexed>(*self));
    GINAC_ASSERT(is_a<indexed>(*other));
    GINAC_ASSERT(self->nops() == 2 || self->nops() == 3);
    GINAC_ASSERT(is_a<matrix>(self->op(0)));
 
    // Only contract with other matrices
    if (!is_a<matrix>(other->op(0)))
        return false;
 
    GINAC_ASSERT(other->nops() == 2 || other->nops() == 3);
 
    const matrix &self_matrix = ex_to<matrix>(self->op(0));
    const matrix &other_matrix = ex_to<matrix>(other->op(0));
 
    if (self->nops() == 2) {
 
        if (other->nops() == 2) { // vector * vector (scalar product)
 
            if (self_matrix.col == 1) {
                if (other_matrix.col == 1) {
                    // Column vector * column vector, transpose first vector
                    *self = self_matrix.transpose().mul(other_matrix)(0, 0);
                } else {
                    // Column vector * row vector, swap factors
                    *self = other_matrix.mul(self_matrix)(0, 0);
                }
            } else {
                if (other_matrix.col == 1) {
                    // Row vector * column vector, perfect
                    *self = self_matrix.mul(other_matrix)(0, 0);
                } else {
                    // Row vector * row vector, transpose second vector
                    *self = self_matrix.mul(other_matrix.transpose())(0, 0);
                }
            }
            *other = _ex1;
            return true;
 
        } else { // vector * matrix
 
            // B_i * A_ij = (B*A)_j (B is row vector)
            if (is_dummy_pair(self->op(1), other->op(1))) {
                if (self_matrix.row == 1)
                    *self = indexed(self_matrix.mul(other_matrix), other->op(2));
                else
                    *self = indexed(self_matrix.transpose().mul(other_matrix), other->op(2));
                *other = _ex1;
                return true;
            }
 
            // B_j * A_ij = (A*B)_i (B is column vector)
            if (is_dummy_pair(self->op(1), other->op(2))) {
                if (self_matrix.col == 1)
                    *self = indexed(other_matrix.mul(self_matrix), other->op(1));
                else
                    *self = indexed(other_matrix.mul(self_matrix.transpose()), other->op(1));
                *other = _ex1;
                return true;
            }
        }
 
    } else if (other->nops() == 3) { // matrix * matrix
 
        // A_ij * B_jk = (A*B)_ik
        if (is_dummy_pair(self->op(2), other->op(1))) {
            *self = indexed(self_matrix.mul(other_matrix), self->op(1), other->op(2));
            *other = _ex1;
            return true;
        }
 
        // A_ij * B_kj = (A*Btrans)_ik
        if (is_dummy_pair(self->op(2), other->op(2))) {
            *self = indexed(self_matrix.mul(other_matrix.transpose()), self->op(1), other->op(1));
            *other = _ex1;
            return true;
        }
 
        // A_ji * B_jk = (Atrans*B)_ik
        if (is_dummy_pair(self->op(1), other->op(1))) {
            *self = indexed(self_matrix.transpose().mul(other_matrix), self->op(2), other->op(2));
            *other = _ex1;
            return true;
        }
 
        // A_ji * B_kj = (B*A)_ki
        if (is_dummy_pair(self->op(1), other->op(2))) {
            *self = indexed(other_matrix.mul(self_matrix), other->op(1), self->op(2));
            *other = _ex1;
            return true;
        }
    }
 
    return false;
}
 
 
// non-virtual functions in this class
 
// public
 
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);
    auto ci = other.m.begin();
    for (auto & i : sum)
        i += *ci++;
    
    return matrix(row, col, std::move(sum));
}
 
 
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);
    auto ci = other.m.begin();
    for (auto & i : dif)
        i -= *ci++;
    
    return matrix(row, col, std::move(dif));
}
 
 
matrix matrix::mul(const matrix & other) const
{
    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) {
            // 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]);
        }
    }
    return matrix(row, other.col, std::move(prod));
}
 
 
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, std::move(prod));
}
 
 
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, std::move(prod));
}
 
 
matrix matrix::pow(const ex & expn) const
{
    if (col!=row)
        throw (std::logic_error("matrix::pow(): matrix not square"));
    
    if (is_exactly_a<numeric>(expn)) {
        // Integer cases are computed by successive multiplication, using the
        // obvious shortcut of storing temporaries, like A^4 == (A*A)*(A*A).
        if (expn.info(info_flags::integer)) {
            numeric b = ex_to<numeric>(expn);
            matrix A(row,col);
            if (expn.info(info_flags::negative)) {
                b *= -1;
                A = this->inverse();
            } else {
                A = *this;
            }
            matrix C(row,col);
            for (unsigned r=0; r<row; ++r)
                C(r,r) = _ex1;
            if (b.is_zero())
                return C;
            // This loop computes the representation of b in base 2 from right
            // to left and multiplies the factors whenever needed.  Note
            // that this is not entirely optimal but close to optimal and
            // "better" algorithms are much harder to implement.  (See Knuth,
            // TAoCP2, section "Evaluation of Powers" for a good discussion.)
            while (b!=*_num1_p) {
                if (b.is_odd()) {
                    C = C.mul(A);
                    --b;
                }
                b /= *_num2_p;  // still integer.
                A = A.mul(A);
            }
            return A.mul(C);
        }
    }
    throw (std::runtime_error("matrix::pow(): don't know how to handle exponent"));
}
 
 
const ex & matrix::operator() (unsigned ro, unsigned co) const
{
    if (ro>=row || co>=col)
        throw (std::range_error("matrix::operator(): index out of range"));
 
    return m[ro*col+co];
}
 
 
ex & matrix::operator() (unsigned ro, unsigned co)
{
    if (ro>=row || co>=col)
        throw (std::range_error("matrix::operator(): index out of range"));
 
    ensure_if_modifiable();
    return m[ro*col+co];
}
 
 
matrix matrix::transpose() const
{
    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(), std::move(trans));
}
 
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
    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::crational_polynomial) &&
             rtest.info(info_flags::rational_function))
            normal_flag = true;
    }
    
    // Here is the heuristics in case this routine has to decide:
    if (algo == determinant_algo::automatic) {
        // Minor expansion is generally a good guess:
        algo = determinant_algo::laplace;
        // Does anybody know when a matrix is really sparse?
        // Maybe <~row/2.236 nonzero elements average in a row?
        if (row>3 && 5*sparse_count<=row*col)
            algo = determinant_algo::bareiss;
        // Purely numeric matrix can be handled by Gauss elimination.
        // This overrides any prior decisions.
        if (numeric_flag)
            algo = determinant_algo::gauss;
    }
    
    // Trap the trivial case here, since some algorithms don't like it
    if (this->row==1) {
        // for consistency with non-trivial determinants...
        if (normal_flag)
            return m[0].normal();
        else
            return m[0].expand();
    }
 
    // Compute the determinant
    switch(algo) {
        case determinant_algo::gauss: {
            ex det = 1;
            matrix tmp(*this);
            int sign = tmp.gauss_elimination(true);
            for (unsigned d=0; d<row; ++d)
                det *= tmp.m[d*col+d];
            if (normal_flag)
                return (sign*det).normal();
            else
                return (sign*det).normal().expand();
        }
        case determinant_algo::bareiss: {
            matrix tmp(*this);
            int sign;
            sign = tmp.fraction_free_elimination(true);
            if (normal_flag)
                return (sign*tmp.m[row*col-1]).normal();
            else
                return (sign*tmp.m[row*col-1]).expand();
        }
        case determinant_algo::divfree: {
            matrix tmp(*this);
            int sign;
            sign = tmp.division_free_elimination(true);
            if (sign==0)
                return _ex0;
            ex det = tmp.m[row*col-1];
            // factor out accumulated bogus slag
            for (unsigned d=0; d<row-2; ++d)
                for (unsigned j=0; j<row-d-2; ++j)
                    det = (det/tmp.m[d*col+d]).normal();
            return (sign*det);
        }
        case determinant_algo::laplace:
        default: {
            // This is the minor expansion scheme.  We always develop such
            // that the smallest minors (i.e, the trivial 1x1 ones) are on the
            // rightmost column.  For this to be efficient, empirical tests
            // have shown that the emptiest columns (i.e. the ones with most
            // zeros) should be the ones on the right hand side -- although
            // this might seem counter-intuitive (and in contradiction to some
            // literature like the FORM manual).  Please go ahead and test it
            // if you don't believe me!  Therefore we presort the columns of
            // the matrix:
            typedef std::pair<unsigned,unsigned> uintpair;
            std::vector<uintpair> c_zeros;  // number of zeros in column
            for (unsigned c=0; c<col; ++c) {
                unsigned acc = 0;
                for (unsigned r=0; r<row; ++r)
                    if (m[r*col+c].is_zero())
                        ++acc;
                c_zeros.push_back(uintpair(acc,c));
            }
            std::sort(c_zeros.begin(),c_zeros.end());
            std::vector<unsigned> pre_sort;
            for (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 (auto & it : pre_sort) {
                for (unsigned r=0; r<row; ++r)
                    result[r*col+c] = m[r*col+it];
                ++c;
            }
            
            if (normal_flag)
                return (sign*matrix(row, col, std::move(result)).determinant_minor()).normal();
            else
                return sign*matrix(row, col, std::move(result)).determinant_minor();
        }
    }
}
 
 
ex matrix::trace() 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];
    
    if (tr.info(info_flags::rational_function) &&
       !tr.info(info_flags::crational_polynomial))
        return tr.normal();
    else
        return tr.expand();
}
 
 
ex matrix::charpoly(const ex & lambda) const
{
    if (row != col)
        throw (std::logic_error("matrix::charpoly(): matrix not square"));
    
    bool numeric_flag = true;
    for (auto & r : m) {
        if (!r.info(info_flags::numeric)) {
            numeric_flag = false;
            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);
        for (unsigned i=1; i<row; ++i) {
            for (unsigned j=0; j<row; ++j)
                B.m[j*col+j] -= c;
            B = this->mul(B);
            c = B.trace() / ex(i+1);
            poly -= c*power(lambda, row-i-1);
        }
        if (row%2)
            return -poly;
        else
            return poly;
 
    } else {
    
        matrix M(*this);
        for (unsigned r=0; r<col; ++r)
            M.m[r*col+r] -= lambda;
    
        return M.determinant().collect(lambda);
    }
}
 
 
matrix matrix::inverse() const
{
    return inverse(solve_algo::automatic);
}
 
matrix matrix::inverse(unsigned algo) const
{
    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, algo);
    } 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;
}
 
 
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.cols() != 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];
    }
 
    // Eliminate the augmented matrix:
    auto colid = aug.echelon_form(algo, n);
    
    // 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)].normal().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].normal().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(colid[c],co) = vars.m[colid[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[colid[c]*p+co];
                sol(colid[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(colid[ro],co) = vars(colid[ro],co);
    }
    
    return sol;
}
 
unsigned matrix::rank() const
{
    return rank(solve_algo::automatic);
}
 
unsigned matrix::rank(unsigned solve_algo) const
{
    // Method:
    // Transform this matrix into upper echelon form and then count the
    // number of non-zero rows.
    GINAC_ASSERT(row*col==m.capacity());
 
    matrix to_eliminate = *this;
    to_eliminate.echelon_form(solve_algo, col);
 
    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
 
ex matrix::determinant_minor() const
{
    const unsigned n = this->cols();
 
    // 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.
    
    // we store the minors in maps, keyed by the rows they arise from
    typedef std::vector<unsigned> keyseq;
    typedef std::map<keyseq, ex> Rmap;
 
    Rmap M, N;  // minors used in current and next column, respectively
    // populate M with dummy unit, to be used as factor in rightmost column
    M[keyseq{}] = _ex1;
 
    // keys to identify minor of M and N (Mkey is a subsequence of Nkey)
    keyseq Mkey, Nkey;
    Mkey.reserve(n-1);
    Nkey.reserve(n);
 
    ex det;
    // proceed from right to left through matrix
    for (int c=n-1; c>=0; --c) {
        Nkey.clear();
        Mkey.clear();
        for (unsigned i=0; i<n-c; ++i)
            Nkey.push_back(i);
        unsigned fc = 0;  // controls logic for minor key generator
        do {
            det = _ex0;
            for (unsigned r=0; r<n-c; ++r) {
                // maybe there is nothing to do?
                if (m[Nkey[r]*n+c].is_zero())
                    continue;
                // Mkey is same as Nkey, but with element r removed
                Mkey.clear();
                Mkey.insert(Mkey.begin(), Nkey.begin(), Nkey.begin() + r);
                Mkey.insert(Mkey.end(), Nkey.begin() + r + 1, Nkey.end());
                // add product of matrix element and minor M to determinant
                if (r%2)
                    det -= m[Nkey[r]*n+c]*M[Mkey];
                else
                    det += m[Nkey[r]*n+c]*M[Mkey];
            }
            // prevent nested expressions to save time
            det = det.expand();
            // if the next computed minor is zero, don't store it in N:
            // (if key is not found, operator[] will just return a zero ex)
            if (!det.is_zero())
                N[Nkey] = det;
            // compute next minor key
            for (fc=n-c; fc>0; --fc) {
                ++Nkey[fc-1];
                if (Nkey[fc-1]<fc+c)
                    break;
            }
            if (fc<n-c && fc>0)
                for (unsigned j=fc; j<n-c; ++j)
                    Nkey[j] = Nkey[j-1]+1;
        } while(fc);
        // if N contains no minors, then they all vanished
        if (N.empty())
            return _ex0;
 
        // proceed to next column: switch roles of M and N, clear N
        M = std::move(N);
    }
    
    return det;
}
 
std::vector<unsigned>
matrix::echelon_form(unsigned algo, int n)
{
    // Here is the heuristics in case this routine has to decide:
    if (algo == solve_algo::automatic) {
        // Gather some statistical information about the augmented matrix:
        bool numeric_flag = true;
        for (const auto & r : m) {
            if (!r.info(info_flags::numeric)) {
                numeric_flag = false;
                break;
            }
        }
        unsigned density = 0;
        for (const auto & r : m) {
            density += !r.is_zero();
        }
        unsigned ncells = col*row;
        if (numeric_flag) {
            // For numerical matrices Gauss is good, but Markowitz becomes
            // better for large sparse matrices.
            if ((ncells > 200) && (density < ncells/2)) {
                algo = solve_algo::markowitz;
            } else {
                algo = solve_algo::gauss;
            }
        } else {
            // For symbolic matrices Markowitz is good, but Bareiss/Divfree
            // is better for small and dense matrices.
            if ((ncells < 120) && (density*5 > ncells*3)) {
                if (ncells <= 12) {
                    algo = solve_algo::divfree;
                } else {
                    algo = solve_algo::bareiss;
                }
            } else {
                algo = solve_algo::markowitz;
            }
        }
    }
    // Eliminate the augmented matrix:
    std::vector<unsigned> colid(col);
    for (unsigned c = 0; c < col; c++) {
        colid[c] = c;
    }
    switch(algo) {
        case solve_algo::gauss:
            gauss_elimination();
            break;
        case solve_algo::divfree:
            division_free_elimination();
            break;
        case solve_algo::bareiss:
            fraction_free_elimination();
            break;
        case solve_algo::markowitz:
            colid = markowitz_elimination(n);
            break;
        default:
            throw std::invalid_argument("matrix::echelon_form(): 'algo' is not one of the solve_algo enum");
    }
    return colid;
}
 
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 c0=0; c0<n && r0<m-1; ++c0) {
        int indx = pivot(r0, c0, 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+c0].is_zero()) {
                    // yes, there is something to do in this row
                    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] = (this->m[r2*n+c] - piv * this->m[r0*n+c]).normal();
                    }
                }
                // fill up left hand side with zeros
                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;
            }
            ++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;
}
 
/* Perform Markowitz-ordered Gaussian elimination (with full
 * pivoting) on a matrix, constraining the choice of pivots to
 * the first n columns (this simplifies handling of augmented
 * matrices). Return the column id vector v, such that v[column]
 * is the original number of the column before shuffling (v[i]==i
 * for i >= n). */
std::vector<unsigned>
matrix::markowitz_elimination(unsigned n)
{
    GINAC_ASSERT(n <= col);
    std::vector<int> rowcnt(row, 0);
    std::vector<int> colcnt(col, 0);
    // Normalize everything before start. We'll keep all the
    // cells normalized throughout the algorithm to properly
    // handle unnormal zeros.
    for (unsigned r = 0; r < row; r++) {
        for (unsigned c = 0; c < col; c++) {
            if (!m[r*col + c].is_zero()) {
                m[r*col + c] = m[r*col + c].normal();
                rowcnt[r]++;
                colcnt[c]++;
            }
        }
    }
    std::vector<unsigned> colid(col);
    for (unsigned c = 0; c < col; c++) {
        colid[c] = c;
    }
    exvector ab(row);
    for (unsigned k = 0; (k < col) && (k < row - 1); k++) {
        // Find the pivot that minimizes (rowcnt[r]-1)*(colcnt[c]-1).
        unsigned pivot_r = row + 1;
        unsigned pivot_c = col + 1;
        int pivot_m = row*col;
        for (unsigned r = k; r < row; r++) {
            for (unsigned c = k; c < n; c++) {
                const ex &mrc = m[r*col + c];
                if (mrc.is_zero())
                    continue;
                GINAC_ASSERT(rowcnt[r] > 0);
                GINAC_ASSERT(colcnt[c] > 0);
                int measure = (rowcnt[r] - 1)*(colcnt[c] - 1);
                if (measure < pivot_m) {
                    pivot_m = measure;
                    pivot_r = r;
                    pivot_c = c;
                }
            }
        }
        if (pivot_m == row*col) {
            // The rest of the matrix is zero.
            break;
        }
        GINAC_ASSERT(k <= pivot_r && pivot_r < row);
        GINAC_ASSERT(k <= pivot_c && pivot_c < col);
        // Swap the pivot into (k, k).
        if (pivot_c != k) {
            for (unsigned r = 0; r < row; r++) {
                m[r*col + pivot_c].swap(m[r*col + k]);
            }
            std::swap(colid[pivot_c], colid[k]);
            std::swap(colcnt[pivot_c], colcnt[k]);
        }
        if (pivot_r != k) {
            for (unsigned c = k; c < col; c++) {
                m[pivot_r*col + c].swap(m[k*col + c]);
            }
            std::swap(rowcnt[pivot_r], rowcnt[k]);
        }
        // No normalization before is_zero() here, because
        // we maintain the matrix normalized throughout the
        // algorithm.
        ex a = m[k*col + k];
        GINAC_ASSERT(!a.is_zero());
        // Subtract the pivot row KJI-style (so: loop by pivot, then
        // column, then row) to maximally exploit pivot row zeros (at
        // the expense of the pivot column zeros). The speedup compared
        // to the usual KIJ order is not really significant though...
        for (unsigned r = k + 1; r < row; r++) {
            const ex &b = m[r*col + k];
            if (!b.is_zero()) {
                ab[r] = b/a;
                rowcnt[r]--;
            }
        }
        colcnt[k] = rowcnt[k] = 0;
        for (unsigned c = k + 1; c < col; c++) {
            const ex &mr0c = m[k*col + c];
            if (mr0c.is_zero())
                continue;
            colcnt[c]--;
            for (unsigned r = k + 1; r < row; r++) {
                if (ab[r].is_zero())
                    continue;
                bool waszero = m[r*col + c].is_zero();
                m[r*col + c] = (m[r*col + c] - ab[r]*mr0c).normal();
                bool iszero = m[r*col + c].is_zero();
                if (waszero && !iszero) {
                    rowcnt[r]++;
                    colcnt[c]++;
                }
                if (!waszero && iszero) {
                    rowcnt[r]--;
                    colcnt[c]--;
                }
            }
        }
        for (unsigned r = k + 1; r < row; r++) {
            ab[r] = m[r*col + k] = _ex0;
        }
    }
    return colid;
}
 
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 c0=0; c0<n && r0<m-1; ++c0) {
        int indx = pivot(r0, c0, 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=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]).normal();
                // fill up left hand side with zeros
                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;
            }
            ++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 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 identity that this really divides m[k+1](r,c).
    //
    // We also allow rational functions where the original prove still holds.
    // However, we must care for numerator and denominator separately and
    // "manually" work in the integral domains because of subtle cancellations
    // (see below).  This blows up the bookkeeping a bit and the formula has
    // to be modified to expand like this (N{x} stands for numerator of x,
    // D{x} for denominator of x):
    //     N{m[k+1](r,c)} = N{m[k](k,k)}*N{m[k](r,c)}*D{m[k](r,k)}*D{m[k](k,c)}
    //                     -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
    //     D{m[k+1](r,c)} = D{m[k](k,k)}*D{m[k](r,c)}*D{m[k](r,k)}*D{m[k](k,c)}
    // where for k>1 we now divide N{m[k+1](r,c)} by
    //     N{m[k-1](k-1,k-1)}
    // and D{m[k+1](r,c)} by
    //     D{m[k-1](k-1,k-1)}.
    
    ensure_if_modifiable();
    const unsigned m = this->rows();
    const unsigned n = this->cols();
    GINAC_ASSERT(!det || n==m);
    int sign = 1;
    if (m==1)
        return 1;
    ex divisor_n = 1;
    ex divisor_d = 1;
    ex dividend_n;
    ex dividend_d;
    
    // We populate temporary matrices to subsequently operate on.  There is
    // one holding numerators and another holding denominators of entries.
    // This is a must since the evaluator (or even earlier mul's constructor)
    // might cancel some trivial element which causes divide() to fail.  The
    // elements are normalized first (yes, even though this algorithm doesn't
    // need GCDs) since the elements of *this might be unnormalized, which
    // makes things more complicated than they need to be.
    matrix tmp_n(*this);
    matrix tmp_d(m,n);  // for denominators, if needed
    exmap srl;  // symbol replacement list
    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 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;
        } else {
            if (indx>r0) {
                // Matrix needs pivoting, swap rows r0 and indx of tmp_n and tmp_d.
                sign = -sign;
                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=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,
                                    tmp_d.m[r2*n+c], true);
                    GINAC_ASSERT(check);
                }
                // fill up left hand side with zeros
                for (unsigned c=r0; c<=c0; ++c)
                    tmp_n.m[r2*n+c] = _ex0;
            }
            if (c0<n && r0<m-1) {
                // compute next iteration's divisor
                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;
                    }
                }
            }
            ++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:
    tmp_n_it = tmp_n.m.begin();
    tmp_d_it = tmp_d.m.begin();
    for (auto & it : this->m)
        it = ((*tmp_n_it++)/(*tmp_d_it++)).subs(srl, subs_options::no_pattern);
    
    return sign;
}
 
 
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_exactly_a<numeric>(this->m[k*col+co]));
        unsigned kmax = k+1;
        numeric mmax = abs(ex_to<numeric>(m[kmax*col+co]));
        while (kmax<row) {
            GINAC_ASSERT(is_exactly_a<numeric>(this->m[kmax*col+co]));
            numeric tmp = ex_to<numeric>(this->m[kmax*col+co]);
            if (abs(tmp) > mmax) {
                mmax = tmp;
                k = kmax;
            }
            ++kmax;
        }
        if (!mmax.is_zero())
            k = kmax;
    }
    if (k==row)
        // all elements in column co below row ro vanish
        return -1;
    if (k==ro)
        // matrix needs no pivoting
        return 0;
    // matrix needs pivoting, so swap rows k and ro
    ensure_if_modifiable();
    for (unsigned c=0; c<col; ++c)
        this->m[k*col+c].swap(this->m[ro*col+c]);
    
    return k;
}
 
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
    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 = dynallocate<matrix>(rows, cols);
 
    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)
{
    size_t dim = l.nops();
 
    // Allocate and fill matrix
    matrix & M = dynallocate<matrix>(dim, dim);
 
    unsigned i = 0;
    for (auto & it : l) {
        M(i, i) = it;
        ++i;
    }
 
    return M;
}
 
ex diag_matrix(std::initializer_list<ex> l)
{
    size_t dim = l.size();
 
    // Allocate and fill matrix
    matrix & M = dynallocate<matrix>(dim, dim);
 
    unsigned i = 0;
    for (auto & it : l) {
        M(i, i) = it;
        ++i;
    }
 
    return M;
}
 
ex unit_matrix(unsigned r, unsigned c)
{
    matrix & Id = dynallocate<matrix>(r, c);
    Id.setflag(status_flags::evaluated);
    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 = dynallocate<matrix>(r, c);
    M.setflag(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 = dynallocate<matrix>(rows, cols);
    M.setflag(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 = dynallocate<matrix>(nr, nc);
    M.setflag(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;
}
 
} // namespace GiNaC