* Implementation of symbolic matrices */
/*
- * GiNaC Copyright (C) 1999-2015 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2018 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
setflag(status_flags::not_shareable);
}
-// 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);
-}
-
/** 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
}
}
+/** 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
//////////
return m[i];
}
-/** Evaluate matrix entry by entry. */
-ex matrix::eval(int level) const
-{
- // 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;
- 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, std::move(m2)))->setflag(status_flags::dynallocated |
- status_flags::evaluated);
-}
-
ex matrix::subs(const exmap & mp, unsigned options) const
{
exvector m2(row * col);
}
+/** Inverse of this matrix, with automatic algorithm selection. */
+matrix matrix::inverse() const
+{
+ return inverse(solve_algo::automatic);
+}
+
/** Inverse of this matrix.
*
+ * @param algo selects the algorithm (one of solve_algo)
* @return the inverted matrix
* @exception logic_error (matrix not square)
* @exception runtime_error (singular matrix) */
-matrix matrix::inverse() const
+matrix matrix::inverse(unsigned algo) const
{
if (row != col)
throw (std::logic_error("matrix::inverse(): matrix not square"));
matrix sol(row,col);
try {
- sol = this->solve(vars,identity);
+ 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"));
/** 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
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)
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);
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)].is_zero()))
+ 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].is_zero()) {
+ 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(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--) {
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.swap(B);
- B.clear();
+ // next column, clear B and change the role of A and B:
+ A = std::move(B);
}
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 (auto & r : m) {
+ if (!r.info(info_flags::numeric)) {
+ numeric_flag = false;
+ break;
+ }
+ }
+ // Bareiss (fraction-free) elimination is generally a good guess:
+ algo = solve_algo::bareiss;
+ // For row<3, Bareiss elimination is equivalent to division free
+ // elimination but has more logistic overhead
+ if (row<3)
+ algo = solve_algo::divfree;
+ // This overrides any prior decisions.
+ if (numeric_flag)
+ algo = solve_algo::gauss;
+ }
+ // 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
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.
}
// Allocate and fill matrix
- matrix &M = *new matrix(rows, cols);
- M.setflag(status_flags::dynallocated);
+ matrix & M = dynallocate<matrix>(rows, cols);
unsigned i = 0;
for (auto & itr : l) {
size_t dim = l.nops();
// Allocate and fill matrix
- matrix &M = *new matrix(dim, dim);
- M.setflag(status_flags::dynallocated);
+ 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) {
ex unit_matrix(unsigned r, unsigned c)
{
- matrix &Id = *new matrix(r, c);
- Id.setflag(status_flags::dynallocated);
+ matrix & Id = dynallocate<matrix>(r, c);
+ Id.setflag(status_flags::evaluated);
for (unsigned i=0; i<r && i<c; i++)
Id(i,i) = _ex1;
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);
+ 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);
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);
+ matrix & M = dynallocate<matrix>(rows, cols);
+ M.setflag(status_flags::evaluated);
unsigned ro = 0;
unsigned ro2 = 0;
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);
+ 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) {
git://www.ginac.de / ginac.git / blobdiff