X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fmatrix.cpp;h=66f3999a4e37574d142f57d4dc68d486c572864c;hp=f92a7c56a481ec61f935f692d0a5690cad5a78ff;hb=4bce949dac6328bc9fcfcd3f875fd422055745f4;hpb=e7cc6a764ff67b5885d6633385fac23ccc1dc9a7

diff --git a/ginac/matrix.cpp b/ginac/matrix.cpp
index f92a7c56..66f3999a 100644
--- a/ginac/matrix.cpp
+++ b/ginac/matrix.cpp
@@ -3,7 +3,7 @@
  *  Implementation of symbolic matrices */
 
 /*
- *  GiNaC Copyright (C) 1999-2002 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2010 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
@@ -17,52 +17,49 @@
  *
  *  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 <iostream>
-#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 <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)
-{
-	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
@@ -71,33 +68,36 @@ 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)
 {
-	m.resize(r*c, _ex0);
+	setflag(status_flags::not_shareable);
 }
 
 // protected
 
 /** Ctor from representation, for internal use only. */
 matrix::matrix(unsigned r, unsigned c, const exvector & m2)
-  : inherited(TINFO_matrix), row(r), col(c), m(m2) {}
+  : row(r), col(c), m(m2)
+{
+	setflag(status_flags::not_shareable);
+}
 
 /** Construct matrix from (flat) list of elements. If the list has fewer
  *  elements than the matrix, the remaining matrix elements are set to zero.
  *  If the list has more elements than the matrix, the excessive elements are
  *  thrown away. */
 matrix::matrix(unsigned r, unsigned c, const lst & l)
-  : inherited(TINFO_matrix), row(r), col(c)
+  : row(r), col(c), m(r*c, _ex0)
 {
-	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 (lst::const_iterator it = l.begin(); it != l.end(); ++it, ++i) {
+		size_t x = i % c;
+		size_t y = i / c;
 		if (y >= r)
 			break; // matrix smaller than list: throw away excessive elements
-		m[y*c+x] = l.op(i);
+		m[y*c+x] = *it;
 	}
 }
 
@@ -105,19 +105,25 @@ matrix::matrix(unsigned r, unsigned c, const lst & l)
 // 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)
 {
+	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();
+	archive_node::archive_node_cit first = n.find_first("m");
+	archive_node::archive_node_cit last = n.find_last("m");
+	++last;
+	for (archive_node::archive_node_cit 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
 {
@@ -131,67 +137,69 @@ void matrix::archive(archive_node &n) const
 	}
 }
 
-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
 {
-	if (is_a<print_tree>(c)) {
-
-		inherited::print(c, level);
-
-	} else {
-
-		if (is_a<print_python_repr>(c))
-			c.s << class_name() << '(';
-
-		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 << ",";
+	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;
 		}
-		m[row*col-1].print(c);
-		c.s << "]]";
+		if (ro < row-1)
+			c.s << row_sep;
+	}
+}
 
-		if (is_a<print_python_repr>(c))
-			c.s << ')';
+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 << ')';
 }
 
 /** 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];
 }
 
@@ -214,17 +222,63 @@ ex matrix::eval(int level) const
 			m2[r*col+c] = m[r*col+c].eval(level);
 	
 	return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
-											   status_flags::evaluated );
+	                                           status_flags::evaluated);
 }
 
-ex matrix::subs(const lst & ls, const lst & lr, bool no_pattern) const
+ex matrix::subs(const exmap & mp, unsigned options) const
 {
 	exvector m2(row * col);
 	for (unsigned r=0; r<row; ++r)
 		for (unsigned c=0; c<col; ++c)
-			m2[r*col+c] = m[r*col+c].subs(ls, lr, no_pattern);
+			m2[r*col+c] = m[r*col+c].subs(mp, options);
 
-	return matrix(row, col, m2).basic::subs(ls, lr, no_pattern);
+	return matrix(row, col, m2).subs_one_level(mp, options);
+}
+
+/** Complex conjugate every matrix entry. */
+ex matrix::conjugate() const
+{
+	exvector * ev = 0;
+	for (exvector::const_iterator 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 = new exvector;
+		ev->reserve(m.size());
+		for (exvector::const_iterator j=m.begin(); j!=i; ++j) {
+			ev->push_back(*j);
+		}
+		ev->push_back(x);
+	}
+	if (ev) {
+		ex result = matrix(row, col, *ev);
+		delete ev;
+		return result;
+	}
+	return *this;
+}
+
+ex matrix::real_part() const
+{
+	exvector v;
+	v.reserve(m.size());
+	for (exvector::const_iterator i=m.begin(); i!=m.end(); ++i)
+		v.push_back(i->real_part());
+	return matrix(row, col, v);
+}
+
+ex matrix::imag_part() const
+{
+	exvector v;
+	v.reserve(m.size());
+	for (exvector::const_iterator i=m.begin(); i!=m.end(); ++i)
+		v.push_back(i->imag_part());
+	return matrix(row, col, v);
 }
 
 // protected
@@ -350,7 +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));
@@ -401,7 +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);
@@ -545,10 +599,11 @@ 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);
@@ -590,7 +645,7 @@ 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)) {
@@ -605,17 +660,19 @@ 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);
@@ -656,7 +713,7 @@ 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());
 	
@@ -693,12 +750,12 @@ ex matrix::determinant(unsigned algo) const
 	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
+		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))
 			normal_flag = true;
@@ -727,7 +784,7 @@ ex matrix::determinant(unsigned algo) const
 		else
 			return m[0].expand();
 	}
-	
+
 	// Compute the determinant
 	switch(algo) {
 		case determinant_algo::gauss: {
@@ -813,7 +870,7 @@ 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"));
@@ -823,7 +880,7 @@ 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();
@@ -841,7 +898,7 @@ 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"));
@@ -858,27 +915,30 @@ ex matrix::charpoly(const symbol & lambda) const
 	// 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);
+	}
 }
 
 
@@ -887,7 +947,7 @@ 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"));
@@ -926,14 +986,15 @@ 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();
@@ -1026,6 +1087,29 @@ 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
@@ -1038,7 +1122,7 @@ matrix matrix::solve(const matrix & vars,
  *
  *  @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();
@@ -1141,7 +1225,7 @@ ex matrix::determinant_minor(void) const
 					Pkey[j] = Pkey[j-1]+1;
 		} while(fc);
 		// next column, so change the role of A and B:
-		A = B;
+		A.swap(B);
 		B.clear();
 	}
 	
@@ -1167,8 +1251,8 @@ 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)
@@ -1178,17 +1262,17 @@ 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) {
@@ -1199,7 +1283,12 @@ 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;
 }
 
@@ -1221,8 +1310,8 @@ 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)
@@ -1232,10 +1321,10 @@ 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) {
@@ -1246,7 +1335,12 @@ 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;
 }
 
@@ -1271,7 +1365,7 @@ 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
@@ -1308,7 +1402,7 @@ 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
+	exmap srl;  // symbol replacement list
 	exvector::const_iterator cit = this->m.begin(), citend = this->m.end();
 	exvector::iterator tmp_n_it = tmp_n.m.begin(), tmp_d_it = tmp_d.m.begin();
 	while (cit != citend) {
@@ -1319,28 +1413,37 @@ int matrix::fraction_free_elimination(const bool det)
 	}
 	
 	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,
@@ -1348,13 +1451,13 @@ 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) {
@@ -1366,12 +1469,18 @@ 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);
+		*it++ = ((*tmp_n_it++)/(*tmp_d_it++)).subs(srl, subs_options::no_pattern);
 	
 	return sign;
 }
@@ -1399,11 +1508,11 @@ 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;
@@ -1428,36 +1537,152 @@ 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 (exvector::const_iterator i=m.begin(); i!=m.end(); ++i) 
+		if(!(i->is_zero()))
+			return false;
+	return true;
+}
+
 ex lst_to_matrix(const lst & l)
 {
+	lst::const_iterator itr, itc;
+
 	// Find number of rows and columns
-	unsigned rows = l.nops(), cols = 0, i, j;
-	for (i=0; i<rows; i++)
-		if (l.op(i).nops() > cols)
-			cols = l.op(i).nops();
+	size_t rows = l.nops(), cols = 0;
+	for (itr = l.begin(); itr != l.end(); ++itr) {
+		if (!is_a<lst>(*itr))
+			throw (std::invalid_argument("lst_to_matrix: argument must be a list of lists"));
+		if (itr->nops() > cols)
+			cols = itr->nops();
+	}
 
 	// Allocate and fill matrix
-	matrix &m = *new matrix(rows, cols);
-	m.setflag(status_flags::dynallocated);
-	for (i=0; i<rows; i++)
-		for (j=0; j<cols; j++)
-			if (l.op(i).nops() > j)
-				m(i, j) = l.op(i).op(j);
-			else
-				m(i, j) = _ex0;
-	return m;
+	matrix &M = *new matrix(rows, cols);
+	M.setflag(status_flags::dynallocated);
+
+	unsigned i;
+	for (itr = l.begin(), i = 0; itr != l.end(); ++itr, ++i) {
+		unsigned j;
+		for (itc = ex_to<lst>(*itr).begin(), j = 0; itc != ex_to<lst>(*itr).end(); ++itc, ++j)
+			M(i, j) = *itc;
+	}
+
+	return M;
 }
 
 ex diag_matrix(const lst & l)
 {
-	unsigned dim = l.nops();
+	lst::const_iterator it;
+	size_t dim = l.nops();
+
+	// Allocate and fill matrix
+	matrix &M = *new matrix(dim, dim);
+	M.setflag(status_flags::dynallocated);
 
-	matrix &m = *new matrix(dim, dim);
-	m.setflag(status_flags::dynallocated);
-	for (unsigned i=0; i<dim; i++)
-		m(i, i) = l.op(i);
+	unsigned i;
+	for (it = l.begin(), i = 0; it != l.end(); ++it, ++i)
+		M(i, i) = *it;
+
+	return M;
+}
+
+ex unit_matrix(unsigned r, unsigned c)
+{
+	matrix &Id = *new matrix(r, c);
+	Id.setflag(status_flags::dynallocated);
+	for (unsigned i=0; i<r && i<c; i++)
+		Id(i,i) = _ex1;
+
+	return Id;
+}
+
+ex symbolic_matrix(unsigned r, unsigned c, const std::string & base_name, const std::string & tex_base_name)
+{
+	matrix &M = *new matrix(r, c);
+	M.setflag(status_flags::dynallocated | status_flags::evaluated);
+
+	bool long_format = (r > 10 || c > 10);
+	bool single_row = (r == 1 || c == 1);
+
+	for (unsigned i=0; i<r; i++) {
+		for (unsigned j=0; j<c; j++) {
+			std::ostringstream s1, s2;
+			s1 << base_name;
+			s2 << tex_base_name << "_{";
+			if (single_row) {
+				if (c == 1) {
+					s1 << i;
+					s2 << i << '}';
+				} else {
+					s1 << j;
+					s2 << j << '}';
+				}
+			} else {
+				if (long_format) {
+					s1 << '_' << i << '_' << j;
+					s2 << i << ';' << j << "}";
+				} else {
+					s1 << i << j;
+					s2 << i << j << '}';
+				}
+			}
+			M(i, j) = symbol(s1.str(), s2.str());
+		}
+	}
+
+	return M;
+}
+
+ex reduced_matrix(const matrix& m, unsigned r, unsigned c)
+{
+	if (r+1>m.rows() || c+1>m.cols() || m.cols()<2 || m.rows()<2)
+		throw std::runtime_error("minor_matrix(): index out of bounds");
+
+	const unsigned rows = m.rows()-1;
+	const unsigned cols = m.cols()-1;
+	matrix &M = *new matrix(rows, cols);
+	M.setflag(status_flags::dynallocated | status_flags::evaluated);
+
+	unsigned ro = 0;
+	unsigned ro2 = 0;
+	while (ro2<rows) {
+		if (ro==r)
+			++ro;
+		unsigned co = 0;
+		unsigned co2 = 0;
+		while (co2<cols) {
+			if (co==c)
+				++co;
+			M(ro2,co2) = m(ro, co);
+			++co;
+			++co2;
+		}
+		++ro;
+		++ro2;
+	}
+
+	return M;
+}
+
+ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc)
+{
+	if (r+nr>m.rows() || c+nc>m.cols())
+		throw std::runtime_error("sub_matrix(): index out of bounds");
+
+	matrix &M = *new matrix(nr, nc);
+	M.setflag(status_flags::dynallocated | status_flags::evaluated);
+
+	for (unsigned ro=0; ro<nr; ++ro) {
+		for (unsigned co=0; co<nc; ++co) {
+			M(ro,co) = m(ro+r,co+c);
+		}
+	}
 
-	return m;
+	return M;
 }
 
 } // namespace GiNaC