X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fmatrix.cpp;h=fb6750602caf1e36a34bc93eace613b24fea8323;hp=5b9822510c6642c340511d502483248a0f468d5c;hb=f7884835d397de85e648d1957c058b7d4c0948ba;hpb=a74dc4ea339457bef06d47a1ebca21d614bda279

diff --git a/ginac/matrix.cpp b/ginac/matrix.cpp
index 5b982251..fb675060 100644
--- a/ginac/matrix.cpp
+++ b/ginac/matrix.cpp
@@ -3,7 +3,7 @@
  *  Implementation of symbolic matrices */
 
 /*
- *  GiNaC Copyright (C) 1999-2018 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2019 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
@@ -986,7 +986,7 @@ matrix matrix::inverse(unsigned algo) const
 /** Solve a linear system consisting of a m x n matrix and a m x p right hand
  *  side by applying an elimination scheme to the augmented matrix.
  *
- *  @param vars n x p matrix, all elements must be symbols 
+ *  @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
@@ -1002,8 +1002,8 @@ matrix matrix::solve(const matrix & vars,
 	const unsigned n = this->cols();
 	const unsigned p = rhs.cols();
 	
-	// syntax checks    
-	if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
+	// 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)
@@ -1018,41 +1018,9 @@ matrix matrix::solve(const matrix & vars,
 		for (unsigned c=0; c<p; ++c)
 			aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
 	}
-	
-	// Gather some statistical information about the augmented matrix:
-	bool numeric_flag = true;
-	for (auto & r : aug.m) {
-		if (!r.info(info_flags::numeric)) {
-			numeric_flag = false;
-			break;
-		}
-	}
-	
-	// Here is the heuristics in case this routine has to decide:
-	if (algo == solve_algo::automatic) {
-		// Bareiss (fraction-free) elimination is generally a good guess:
-		algo = solve_algo::bareiss;
-		// For m<3, Bareiss elimination is equivalent to division free
-		// elimination but has more logistic overhead
-		if (m<3)
-			algo = solve_algo::divfree;
-		// This overrides any prior decisions.
-		if (numeric_flag)
-			algo = solve_algo::gauss;
-	}
-	
+
 	// Eliminate the augmented matrix:
-	switch(algo) {
-		case solve_algo::gauss:
-			aug.gauss_elimination();
-			break;
-		case solve_algo::divfree:
-			aug.division_free_elimination();
-			break;
-		case solve_algo::bareiss:
-		default:
-			aug.fraction_free_elimination();
-	}
+	auto colid = aug.echelon_form(algo, n);
 	
 	// assemble the solution matrix:
 	matrix sol(n,p);
@@ -1071,37 +1039,40 @@ matrix matrix::solve(const matrix & vars,
 				// assign solutions for vars between fnz+1 and
 				// last_assigned_sol-1: free parameters
 				for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
-					sol(c,co) = vars.m[c*p+co];
+					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[c*p+co];
-				sol(fnz-1,co) = (e/(aug.m[r*(n+p)+(fnz-1)])).normal();
+					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(ro,co) = vars(ro,co);
+			sol(colid[ro],co) = vars(colid[ro],co);
 	}
 	
 	return sol;
 }
 
-
 /** Compute the rank of this matrix. */
 unsigned matrix::rank() const
+{
+	return rank(solve_algo::automatic);
+}
+
+/** Compute the rank of this matrix using the given algorithm,
+ *  which should be a member of enum solve_algo. */
+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());
 
-	// 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();
+	to_eliminate.echelon_form(solve_algo, col);
 
 	unsigned r = row*col;  // index of last non-zero element
 	while (r--) {
@@ -1126,17 +1097,8 @@ unsigned matrix::rank() const
  *  @see matrix::determinant() */
 ex matrix::determinant_minor() const
 {
-	// for small matrices the algorithm does not make any sense:
 	const unsigned n = this->cols();
-	if (n==1)
-		return m[0].expand();
-	if (n==2)
-		return (m[0]*m[3]-m[2]*m[1]).expand();
-	if (n==3)
-		return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
-		        m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
-		        m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
-	
+
 	// This algorithm can best be understood by looking at a naive
 	// implementation of Laplace-expansion, like this one:
 	// ex det;
@@ -1169,70 +1131,133 @@ ex matrix::determinant_minor() const
 	// calculated in step c-1.  We therefore only have to store at most 
 	// 2*binomial(n,n/2) minors.
 	
-	// Unique flipper counter for partitioning into minors
-	std::vector<unsigned> Pkey;
-	Pkey.reserve(n);
-	// key for minor determinant (a subpartition of Pkey)
-	std::vector<unsigned> Mkey;
+	// 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);
-	// we store our subminors in maps, keys being the rows they arise from
-	typedef std::map<std::vector<unsigned>,class ex> Rmap;
-	typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
-	Rmap A;
-	Rmap B;
+	Nkey.reserve(n);
+
 	ex det;
-	// initialize A with last column:
-	for (unsigned r=0; r<n; ++r) {
-		Pkey.erase(Pkey.begin(),Pkey.end());
-		Pkey.push_back(r);
-		A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
-	}
 	// proceed from right to left through matrix
-	for (int c=n-2; c>=0; --c) {
-		Pkey.erase(Pkey.begin(),Pkey.end());  // don't change capacity
-		Mkey.erase(Mkey.begin(),Mkey.end());
+	for (int c=n-1; c>=0; --c) {
+		Nkey.clear();
+		Mkey.clear();
 		for (unsigned i=0; i<n-c; ++i)
-			Pkey.push_back(i);
-		unsigned fc = 0;  // controls logic for our strange flipper counter
+			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[Pkey[r]*n+c].is_zero())
+				if (m[Nkey[r]*n+c].is_zero())
 					continue;
-				// create the sorted key for all possible minors
-				Mkey.erase(Mkey.begin(),Mkey.end());
-				for (unsigned i=0; i<n-c; ++i)
-					if (i!=r)
-						Mkey.push_back(Pkey[i]);
-				// Fetch the minors and compute the new determinant
+				// 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[Pkey[r]*n+c]*A[Mkey];
+					det -= m[Nkey[r]*n+c]*M[Mkey];
 				else
-					det += m[Pkey[r]*n+c]*A[Mkey];
+					det += m[Nkey[r]*n+c]*M[Mkey];
 			}
-			// prevent build-up of deep nesting of expressions saves time:
+			// prevent nested expressions to save time
 			det = det.expand();
-			// store the new determinant at its place in B:
+			// 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())
-				B.insert(Rmap_value(Pkey,det));
-			// increment our strange flipper counter
+				N[Nkey] = det;
+			// compute next minor key
 			for (fc=n-c; fc>0; --fc) {
-				++Pkey[fc-1];
-				if (Pkey[fc-1]<fc+c)
+				++Nkey[fc-1];
+				if (Nkey[fc-1]<fc+c)
 					break;
 			}
 			if (fc<n-c && fc>0)
 				for (unsigned j=fc; j<n-c; ++j)
-					Pkey[j] = Pkey[j-1]+1;
+					Nkey[j] = Nkey[j-1]+1;
 		} while(fc);
-		// next column, clear B and change the role of A and B:
-		A = std::move(B);
+		// 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;
+}
 
 /** Perform the steps of an ordinary Gaussian elimination to bring the m x n
  *  matrix into an upper echelon form.  The algorithm is ok for matrices
@@ -1293,6 +1318,119 @@ int matrix::gauss_elimination(const bool det)
 	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;
+}
 
 /** Perform the steps of division free elimination to bring the m x n matrix
  *  into an upper echelon form.
@@ -1323,7 +1461,7 @@ int matrix::division_free_elimination(const bool det)
 				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]).expand();
+					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;