Make .eval() evaluate top-level only.
[ginac.git] / ginac / matrix.cpp
index df26223..c842885 100644 (file)
@@ -3,7 +3,7 @@
  *  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
@@ -72,20 +68,9 @@ DEFAULT_DESTROY(matrix)
  *
  *  @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);
-}
-
-// 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)
-{
-       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
@@ -93,143 +78,207 @@ matrix::matrix(unsigned r, unsigned c, const exvector & m2)
  *  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;
+       }
+}
+
+/** Construct a matrix from an 2 dimensional initializer list.
+ *  Throws an exception if some row has a different length than all the others.
+ */
+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
+
+/** Ctor from representation, for internal use only. */
+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
 //////////
 
-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
+ex matrix::subs(const exmap & mp, unsigned options) 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;
+       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].eval(level);
-       
-       return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
-                                                                                          status_flags::evaluated );
+                       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::subs(const lst & ls, const lst & lr, bool no_pattern) const
+/** Complex conjugate every matrix entry. */
+ex matrix::conjugate() 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);
+       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
@@ -355,7 +404,7 @@ ex matrix::add_indexed(const ex & self, const ex & other) const
        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));
@@ -406,7 +455,7 @@ bool matrix::contract_with(exvector::iterator self, exvector::iterator other, ex
        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);
@@ -511,12 +560,11 @@ matrix matrix::add(const matrix & other) const
                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));
 }
 
 
@@ -529,12 +577,11 @@ matrix matrix::sub(const matrix & other) const
                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));
 }
 
 
@@ -550,13 +597,14 @@ matrix matrix::mul(const matrix & other) const
        
        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));
 }
 
 
@@ -569,7 +617,7 @@ matrix matrix::mul(const numeric & other) const
                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));
 }
 
 
@@ -585,7 +633,7 @@ matrix matrix::mul_scalar(const ex & other) const
                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));
 }
 
 
@@ -595,7 +643,7 @@ 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)) {
+       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)) {
@@ -610,17 +658,19 @@ matrix matrix::pow(const ex & expn) const
                        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!=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);
@@ -661,7 +711,7 @@ ex & matrix::operator() (unsigned ro, unsigned co)
 
 /** Transposed of an m x n matrix, producing a new n x m matrix object that
  *  represents the transposed. */
-matrix matrix::transpose(void) const
+matrix matrix::transpose() const
 {
        exvector trans(this->cols()*this->rows());
        
@@ -669,7 +719,7 @@ matrix matrix::transpose(void) const
                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
@@ -696,18 +746,16 @@ ex matrix::determinant(unsigned algo) const
        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:
@@ -732,7 +780,7 @@ ex matrix::determinant(unsigned algo) const
                else
                        return m[0].expand();
        }
-       
+
        // Compute the determinant
        switch(algo) {
                case determinant_algo::gauss: {
@@ -788,25 +836,24 @@ ex matrix::determinant(unsigned algo) const
                                                ++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();
                }
        }
 }
@@ -818,7 +865,7 @@ ex matrix::determinant(unsigned algo) const
  *
  *  @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"));
@@ -828,7 +875,7 @@ ex matrix::trace(void) const
                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();
@@ -836,7 +883,7 @@ ex matrix::trace(void) const
 
 
 /** 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
@@ -846,44 +893,47 @@ ex matrix::trace(void) const
  *  @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);
+       }
 }
 
 
@@ -892,7 +942,7 @@ ex matrix::charpoly(const symbol & lambda) const
  *  @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"));
@@ -931,14 +981,15 @@ matrix matrix::inverse(void) const
  *
  *  @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();
@@ -963,11 +1014,11 @@ matrix matrix::solve(const matrix & vars,
        
        // 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:
@@ -1031,6 +1082,29 @@ matrix matrix::solve(const matrix & vars,
 }
 
 
+/** 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
@@ -1039,11 +1113,11 @@ matrix matrix::solve(const matrix & vars,
  *  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();
@@ -1145,9 +1219,8 @@ ex matrix::determinant_minor(void) const
                                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();
+               // next column, clear B and change the role of A and B:
+               A = std::move(B);
        }
        
        return det;
@@ -1172,8 +1245,8 @@ int matrix::gauss_elimination(const bool det)
        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)
@@ -1183,17 +1256,17 @@ int matrix::gauss_elimination(const bool 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)
+                               for (unsigned c=r0; c<=c0; ++c)
                                        this->m[r2*n+c] = _ex0;
                        }
                        if (det) {
@@ -1204,7 +1277,12 @@ int matrix::gauss_elimination(const bool det)
                        ++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;
 }
 
@@ -1226,8 +1304,8 @@ int matrix::division_free_elimination(const bool det)
        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)
@@ -1237,10 +1315,10 @@ int matrix::division_free_elimination(const bool 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)
+                               for (unsigned c=r0; c<=c0; ++c)
                                        this->m[r2*n+c] = _ex0;
                        }
                        if (det) {
@@ -1251,7 +1329,12 @@ int matrix::division_free_elimination(const bool det)
                        ++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;
 }
 
@@ -1276,7 +1359,7 @@ int matrix::fraction_free_elimination(const bool det)
        //
        // 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
@@ -1313,39 +1396,46 @@ int matrix::fraction_free_elimination(const bool det)
        // 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,
@@ -1353,13 +1443,13 @@ int matrix::fraction_free_elimination(const bool det)
                                        GINAC_ASSERT(check);
                                }
                                // fill up left hand side with zeros
-                               for (unsigned c=0; c<=r1; ++c)
+                               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) {
@@ -1371,12 +1461,17 @@ int matrix::fraction_free_elimination(const bool det)
                        ++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;
 }
@@ -1392,7 +1487,7 @@ int matrix::fraction_free_elimination(const bool det)
  *  @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)
@@ -1404,11 +1499,11 @@ 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;
@@ -1433,36 +1528,168 @@ int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
        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 = 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)
 {
-       unsigned dim = l.nops();
+       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;
+}
 
-       matrix &m = *new matrix(dim, dim);
-       m.setflag(status_flags::dynallocated);
-       for (unsigned i=0; i<dim; i++)
-               m(i, i) = l.op(i);
+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;
+       return M;
 }
 
 } // namespace GiNaC