m.resize(r*c, _ex0());
}
- // protected
+// protected
/** Ctor from representation, for internal use only. */
matrix::matrix(unsigned r, unsigned c, const exvector & m2)
exvector::const_iterator i = m.begin(), iend = m.end();
while (i != iend) {
n.add_ex("m", *i);
- i++;
+ ++i;
}
}
os << "[[ ";
for (unsigned r=0; r<row-1; ++r) {
os << "[[";
- for (unsigned c=0; c<col-1; ++c) {
+ for (unsigned c=0; c<col-1; ++c)
os << m[r*col+c] << ",";
- }
os << m[col*(r+1)-1] << "]], ";
}
os << "[[";
- for (unsigned c=0; c<col-1; ++c) {
+ for (unsigned c=0; c<col-1; ++c)
os << m[(row-1)*col+c] << ",";
- }
os << m[row*col-1] << "]] ]]";
}
os << "matrix(" << row << "," << col <<",";
for (unsigned r=0; r<row-1; ++r) {
os << "(";
- for (unsigned c=0; c<col-1; ++c) {
+ for (unsigned c=0; c<col-1; ++c)
os << m[r*col+c] << ",";
- }
os << m[col*(r-1)-1] << "),";
}
os << "(";
- for (unsigned c=0; c<col-1; ++c) {
+ for (unsigned c=0; c<col-1; ++c)
os << m[(row-1)*col+c] << ",";
- }
os << m[row*col-1] << "))";
}
/** returns matrix entry at position (i/col, i%col). */
ex & matrix::let_op(int i)
{
+ GINAC_ASSERT(i>=0);
+ GINAC_ASSERT(i<nops());
+
return m[i];
}
ex matrix::expand(unsigned options) const
{
exvector tmp(row*col);
- for (unsigned i=0; i<row*col; ++i) {
- tmp[i]=m[i].expand(options);
- }
+ for (unsigned i=0; i<row*col; ++i)
+ tmp[i] = m[i].expand(options);
+
return matrix(row, col, tmp);
}
if (is_equal(*other.bp)) return true;
// search all the elements
- for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
+ for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r)
if ((*r).has(other)) return true;
- }
+
return false;
}
// eval() entry by entry
exvector m2(row*col);
- --level;
- for (unsigned r=0; r<row; ++r) {
- for (unsigned c=0; c<col; ++c) {
+ --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, m2))->setflag(status_flags::dynallocated |
status_flags::evaluated );
// evalf() entry by entry
exvector m2(row*col);
--level;
- for (unsigned r=0; r<row; ++r) {
- for (unsigned c=0; c<col; ++c) {
+ for (unsigned r=0; r<row; ++r)
+ for (unsigned c=0; c<col; ++c)
m2[r*col+c] = m[r*col+c].evalf(level);
- }
- }
+
return matrix(row, col, m2);
}
exvector sum(this->m);
exvector::iterator i;
exvector::const_iterator ci;
- for (i=sum.begin(), ci=other.m.begin();
- i!=sum.end();
- ++i, ++ci) {
+ for (i=sum.begin(), ci=other.m.begin(); i!=sum.end(); ++i, ++ci)
(*i) += (*ci);
- }
+
return matrix(row,col,sum);
}
exvector dif(this->m);
exvector::iterator i;
exvector::const_iterator ci;
- for (i=dif.begin(), ci=other.m.begin();
- i!=dif.end();
- ++i, ++ci) {
+ for (i=dif.begin(), ci=other.m.begin(); i!=dif.end(); ++i, ++ci)
(*i) -= (*ci);
- }
+
return matrix(row,col,dif);
}
* @exception logic_error (incompatible matrices) */
matrix matrix::mul(const matrix & other) const
{
- if (col != other.row)
+ if (this->cols() != other.rows())
throw (std::logic_error("matrix::mul(): incompatible matrices"));
- exvector prod(row*other.col);
+ exvector prod(this->rows()*other.cols());
- for (unsigned r1=0; r1<rows(); ++r1) {
- for (unsigned c=0; c<cols(); ++c) {
+ for (unsigned r1=0; r1<this->rows(); ++r1) {
+ for (unsigned c=0; c<this->cols(); ++c) {
if (m[r1*col+c].is_zero())
continue;
- for (unsigned r2=0; r2<other.col; ++r2)
- prod[r1*other.col+r2] += m[r1*col+c] * other.m[c*other.col+r2];
+ for (unsigned r2=0; r2<other.cols(); ++r2)
+ prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]).expand();
}
}
return matrix(row, other.col, prod);
* represents the transposed. */
matrix matrix::transpose(void) const
{
- exvector trans(col*row);
+ exvector trans(this->cols()*this->rows());
- for (unsigned r=0; r<col; ++r)
- for (unsigned c=0; c<row; ++c)
- trans[r*row+c] = m[c*col+r];
+ for (unsigned r=0; r<this->cols(); ++r)
+ for (unsigned c=0; c<this->rows(); ++c)
+ trans[r*this->rows()+c] = m[c*this->cols()+r];
- return matrix(col,row,trans);
+ return matrix(this->cols(),this->rows(),trans);
}
/** Determinant of square matrix. This routine doesn't actually calculate the
* determinant, it only implements some heuristics about which algorithm to
- * call. If all the elements of the matrix are elements of an integral domain
+ * run. If all the elements of the matrix are elements of an integral domain
* the determinant is also in that integral domain and the result is expanded
* only. If one or more elements are from a quotient field the determinant is
* usually also in that quotient field and the result is normalized before it
* [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
* behaves like MapleV and unlike Mathematica.)
*
+ * @param algo allows to chose an algorithm
* @return the determinant as a new expression
- * @exception logic_error (matrix not square) */
-ex matrix::determinant(void) const
+ * @exception logic_error (matrix not square)
+ * @see determinant_algo */
+ex matrix::determinant(unsigned algo) const
{
if (row!=col)
throw (std::logic_error("matrix::determinant(): matrix not square"));
GINAC_ASSERT(row*col==m.capacity());
- if (this->row==1) // continuation would be pointless
- return m[0];
- // Gather some information about the matrix:
+ // Gather some statistical information about this matrix:
bool numeric_flag = true;
bool normal_flag = false;
- unsigned sparse_count = 0; // count non-zero elements
+ unsigned sparse_count = 0; // counts non-zero elements
for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
- if (!(*r).is_zero())
+ lst srl; // symbol replacement list
+ ex rtest = (*r).to_rational(srl);
+ if (!rtest.is_zero())
++sparse_count;
- if (!(*r).info(info_flags::numeric))
+ if (!rtest.info(info_flags::numeric))
numeric_flag = false;
- if ((*r).info(info_flags::rational_function) &&
- !(*r).info(info_flags::crational_polynomial))
+ if (!rtest.info(info_flags::crational_polynomial) &&
+ rtest.info(info_flags::rational_function))
normal_flag = true;
}
- // Purely numeric matrix handled by Gauss elimination
- if (numeric_flag) {
- ex det = 1;
- matrix tmp(*this);
- int sign = tmp.gauss_elimination();
- for (int d=0; d<row; ++d)
- det *= tmp.m[d*col+d];
- return (sign*det);
+ // Here is the heuristics in case this routine has to decide:
+ if (algo == determinant_algo::automatic) {
+ // Minor expansion is generally a good guess:
+ algo = determinant_algo::laplace;
+ // Does anybody know when a matrix is really sparse?
+ // Maybe <~row/2.236 nonzero elements average in a row?
+ if (row>3 && 5*sparse_count<=row*col)
+ algo = determinant_algo::bareiss;
+ // Purely numeric matrix can be handled by Gauss elimination.
+ // This overrides any prior decisions.
+ if (numeric_flag)
+ algo = determinant_algo::gauss;
}
- // Does anybody know when a matrix is really sparse?
- // Maybe <~row/2.2 nonzero elements average in a row?
- if (5*sparse_count<=row*col) {
- // copy *this:
- matrix tmp(*this);
- int sign;
- sign = tmp.fraction_free_elimination(true);
+ // Trap the trivial case here, since some algorithms don't like it
+ if (this->row==1) {
+ // for consistency with non-trivial determinants...
if (normal_flag)
- return (sign*tmp.m[row*col-1]).normal();
+ return m[0].normal();
else
- return (sign*tmp.m[row*col-1]).expand();
+ return m[0].expand();
}
- // Now come the minor expansion schemes. We always develop such that the
- // smallest minors (i.e, the trivial 1x1 ones) are on the rightmost column.
- // For this to be efficient it turns out that the emptiest columns (i.e.
- // the ones with most zeros) should be the ones on the right hand side.
- // Therefore we presort the columns of the matrix:
- typedef std::pair<unsigned,unsigned> uintpair; // # of zeros, column
- std::vector<uintpair> c_zeros; // number of zeros in column
- for (unsigned c=0; c<col; ++c) {
- unsigned acc = 0;
- for (unsigned r=0; r<row; ++r)
- if (m[r*col+c].is_zero())
- ++acc;
- c_zeros.push_back(uintpair(acc,c));
- }
- sort(c_zeros.begin(),c_zeros.end());
- // unfortunately std::vector<uintpair> can't be used for permutation_sign.
- std::vector<unsigned> pre_sort;
- for (std::vector<uintpair>::iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
- pre_sort.push_back(i->second);
- int sign = permutation_sign(pre_sort);
- exvector result(row*col); // represents sorted matrix
- unsigned c = 0;
- for (std::vector<unsigned>::iterator i=pre_sort.begin();
- i!=pre_sort.end();
- ++i,++c) {
- for (unsigned r=0; r<row; ++r)
- result[r*col+c] = m[r*col+(*i)];
+ // Compute the determinant
+ switch(algo) {
+ case determinant_algo::gauss: {
+ ex det = 1;
+ matrix tmp(*this);
+ int sign = tmp.gauss_elimination(true);
+ for (unsigned d=0; d<row; ++d)
+ det *= tmp.m[d*col+d];
+ if (normal_flag)
+ return (sign*det).normal();
+ else
+ return (sign*det).normal().expand();
+ }
+ case determinant_algo::bareiss: {
+ matrix tmp(*this);
+ int sign;
+ sign = tmp.fraction_free_elimination(true);
+ if (normal_flag)
+ return (sign*tmp.m[row*col-1]).normal();
+ else
+ return (sign*tmp.m[row*col-1]).expand();
+ }
+ case determinant_algo::divfree: {
+ matrix tmp(*this);
+ int sign;
+ sign = tmp.division_free_elimination(true);
+ if (sign==0)
+ return _ex0();
+ ex det = tmp.m[row*col-1];
+ // factor out accumulated bogus slag
+ for (unsigned d=0; d<row-2; ++d)
+ for (unsigned j=0; j<row-d-2; ++j)
+ det = (det/tmp.m[d*col+d]).normal();
+ return (sign*det);
+ }
+ case determinant_algo::laplace:
+ default: {
+ // This is the minor expansion scheme. We always develop such
+ // that the smallest minors (i.e, the trivial 1x1 ones) are on the
+ // rightmost column. For this to be efficient it turns out that
+ // the emptiest columns (i.e. the ones with most zeros) should be
+ // the ones on the right hand side. Therefore we presort the
+ // columns of the matrix:
+ typedef std::pair<unsigned,unsigned> uintpair;
+ std::vector<uintpair> c_zeros; // number of zeros in column
+ for (unsigned c=0; c<col; ++c) {
+ unsigned acc = 0;
+ for (unsigned r=0; r<row; ++r)
+ if (m[r*col+c].is_zero())
+ ++acc;
+ c_zeros.push_back(uintpair(acc,c));
+ }
+ sort(c_zeros.begin(),c_zeros.end());
+ std::vector<unsigned> pre_sort;
+ for (std::vector<uintpair>::iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
+ pre_sort.push_back(i->second);
+ int sign = permutation_sign(pre_sort);
+ exvector result(row*col); // represents sorted matrix
+ unsigned c = 0;
+ for (std::vector<unsigned>::iterator i=pre_sort.begin();
+ i!=pre_sort.end();
+ ++i,++c) {
+ for (unsigned r=0; r<row; ++r)
+ result[r*col+c] = m[r*col+(*i)];
+ }
+
+ if (normal_flag)
+ return (sign*matrix(row,col,result).determinant_minor()).normal();
+ else
+ return sign*matrix(row,col,result).determinant_minor();
+ }
}
-
- if (normal_flag)
- return sign*matrix(row,col,result).determinant_minor().normal();
- return sign*matrix(row,col,result).determinant_minor();
}
{
if (row != col)
throw (std::logic_error("matrix::trace(): matrix not square"));
- GINAC_ASSERT(row*col==m.capacity());
ex tr;
for (unsigned r=0; r<col; ++r)
if (row != col)
throw (std::logic_error("matrix::inverse(): matrix not square"));
+ // NOTE: the Gauss-Jordan elimination used here can in principle be
+ // replaced this by two clever calls to gauss_elimination() and some to
+ // transpose(). Wouldn't be more efficient (maybe less?), just more
+ // orthogonal.
matrix tmp(row,col);
// set tmp to the unit matrix
for (unsigned i=0; i<col; ++i)
tmp.m[i*col+i] = _ex1();
-
+
// create a copy of this matrix
matrix cpy(*this);
for (unsigned r1=0; r1<row; ++r1) {
- int indx = cpy.pivot(r1);
+ int indx = cpy.pivot(r1, r1);
if (indx == -1) {
throw (std::runtime_error("matrix::inverse(): singular matrix"));
}
ex a2 = cpy.m[r2*col+r1];
for (unsigned c=0; c<col; ++c) {
cpy.m[r2*col+c] -= a2 * cpy.m[r1*col+c];
+ if (!cpy.m[r2*col+c].info(info_flags::numeric))
+ cpy.m[r2*col+c] = cpy.m[r2*col+c].normal();
tmp.m[r2*col+c] -= a2 * tmp.m[r1*col+c];
+ if (!tmp.m[r2*col+c].info(info_flags::numeric))
+ tmp.m[r2*col+c] = tmp.m[r2*col+c].normal();
}
}
}
}
- return tmp;
-}
-
-// superfluous helper function, to be removed:
-void matrix::swap(unsigned r1, unsigned c1, unsigned r2 ,unsigned c2)
-{
- ensure_if_modifiable();
- ex tmp = (*this)(r1,c1);
- set(r1,c1,(*this)(r2,c2));
- set(r2,c2,tmp);
+ return tmp;
}
-/** Solve a set of equations for an m x n matrix by fraction-free Gaussian
- * elimination. Based on algorithm 9.1 from 'Algorithms for Computer Algebra'
- * by Keith O. Geddes et al.
+/** 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
+ * @param vars n x p matrix, all elements must be symbols
* @param rhs m x p matrix
+ * @return n x p solution matrix
* @exception logic_error (incompatible matrices)
- * @exception runtime_error (singular matrix) */
-matrix matrix::fraction_free_elim(const matrix & vars,
- const matrix & rhs) const
+ * @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
{
- // FIXME: use implementation of matrix::fraction_free_elimination
- if ((row != rhs.row) || (col != vars.row) || (rhs.col != vars.col))
- throw (std::logic_error("matrix::fraction_free_elim(): incompatible matrices"));
-
- matrix a(*this); // make a copy of the matrix
- matrix b(rhs); // make a copy of the rhs vector
+ const unsigned m = this->rows();
+ const unsigned n = this->cols();
+ const unsigned p = rhs.cols();
- // given an m x n matrix a, reduce it to upper echelon form
- unsigned m = a.row;
- unsigned n = a.col;
- int sign = 1;
- ex divisor = 1;
- unsigned r = 0;
+ // syntax checks
+ if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
+ throw (std::logic_error("matrix::solve(): incompatible matrices"));
+ for (unsigned ro=0; ro<n; ++ro)
+ for (unsigned co=0; co<p; ++co)
+ if (!vars(ro,co).info(info_flags::symbol))
+ throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
- // eliminate below row r, with pivot in column k
- for (unsigned k=0; (k<n)&&(r<m); ++k) {
- // find a nonzero pivot
- unsigned p;
- for (p=r; (p<m)&&(a(p,k).is_zero()); ++p) {}
- // pivot is in row p
- if (p<m) {
- if (p!=r) {
- // swap rows p and r
- for (unsigned j=k; j<n; ++j)
- a.swap(p,j,r,j);
- b.swap(p,0,r,0);
- // keep track of sign changes due to row exchange
- sign *= -1;
- }
- for (unsigned i=r+1; i<m; ++i) {
- for (unsigned j=k+1; j<n; ++j) {
- a.set(i,j,(a(r,k)*a(i,j)
- -a(r,j)*a(i,k))/divisor);
- a.set(i,j,a(i,j).normal());
- }
- b.set(i,0,(a(r,k)*b(i,0)
- -b(r,0)*a(i,k))/divisor);
- b.set(i,0,b(i,0).normal());
- a.set(i,k,0);
- }
- divisor = a(r,k);
- ++r;
- }
+ // build the augmented matrix of *this with rhs attached to the right
+ matrix aug(m,n+p);
+ for (unsigned r=0; r<m; ++r) {
+ for (unsigned c=0; c<n; ++c)
+ aug.m[r*(n+p)+c] = this->m[r*n+c];
+ for (unsigned c=0; c<p; ++c)
+ aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
}
-#ifdef DO_GINAC_ASSERT
- // test if we really have an upper echelon matrix
- int zero_in_last_row = -1;
- for (unsigned r=0; r<m; ++r) {
- int zero_in_this_row=0;
- for (unsigned c=0; c<n; ++c) {
- if (a(r,c).is_zero())
- zero_in_this_row++;
- else
- break;
- }
- GINAC_ASSERT((zero_in_this_row>zero_in_last_row)||(zero_in_this_row=n));
- zero_in_last_row = zero_in_this_row;
+ // Gather some statistical information about the augmented matrix:
+ bool numeric_flag = true;
+ for (exvector::const_iterator r=aug.m.begin(); r!=aug.m.end(); ++r) {
+ if (!(*r).info(info_flags::numeric))
+ numeric_flag = false;
}
-#endif // def DO_GINAC_ASSERT
- // assemble solution
- matrix sol(n,1);
- unsigned last_assigned_sol = n+1;
- for (int r=m-1; r>=0; --r) {
- unsigned first_non_zero = 1;
- while ((first_non_zero<=n)&&(a(r,first_non_zero-1).is_zero()))
- first_non_zero++;
- if (first_non_zero>n) {
- // row consists only of zeroes, corresponding rhs must be 0 as well
- if (!b(r,0).is_zero()) {
- throw (std::runtime_error("matrix::fraction_free_elim(): singular matrix"));
- }
- } else {
- // assign solutions for vars between first_non_zero+1 and
- // last_assigned_sol-1: free parameters
- for (unsigned c=first_non_zero; c<last_assigned_sol-1; ++c)
- sol.set(c,0,vars(c,0));
- ex e = b(r,0);
- for (unsigned c=first_non_zero; c<n; ++c)
- e -= a(r,c)*sol(c,0);
- sol.set(first_non_zero-1,0,
- (e/a(r,first_non_zero-1)).normal());
- last_assigned_sol = first_non_zero;
- }
+ // 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;
}
- // assign solutions for vars between 1 and
- // last_assigned_sol-1: free parameters
- for (unsigned c=0; c<last_assigned_sol-1; ++c)
- sol.set(c,0,vars(c,0));
-#ifdef DO_GINAC_ASSERT
- // test solution with echelon matrix
- for (unsigned r=0; r<m; ++r) {
- ex e = 0;
- for (unsigned c=0; c<n; ++c)
- e += a(r,c)*sol(c,0);
- if (!(e-b(r,0)).normal().is_zero()) {
- cout << "e=" << e;
- cout << "b(" << r <<",0)=" << b(r,0) << endl;
- cout << "diff=" << (e-b(r,0)).normal() << endl;
- }
- GINAC_ASSERT((e-b(r,0)).normal().is_zero());
+ // Eliminate the augmented matrix:
+ switch(algo) {
+ case solve_algo::gauss:
+ aug.gauss_elimination();
+ case solve_algo::divfree:
+ aug.division_free_elimination();
+ case solve_algo::bareiss:
+ default:
+ aug.fraction_free_elimination();
}
- // test solution with original matrix
- for (unsigned r=0; r<m; ++r) {
- ex e = 0;
- for (unsigned c=0; c<n; ++c)
- e += this->m[r*cols()+c]*sol(c,0);
- try {
- if (!(e-rhs(r,0)).normal().is_zero()) {
- cout << "e==" << e << endl;
- e.printtree(cout);
- ex en = e.normal();
- cout << "e.normal()=" << en << endl;
- en.printtree(cout);
- cout << "rhs(" << r <<",0)=" << rhs(r,0) << endl;
- cout << "diff=" << (e-rhs(r,0)).normal() << endl;
+ // assemble the solution matrix:
+ matrix sol(n,p);
+ for (unsigned co=0; co<p; ++co) {
+ unsigned last_assigned_sol = n+1;
+ for (int r=m-1; r>=0; --r) {
+ unsigned fnz = 1; // first non-zero in row
+ while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].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()) {
+ 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.set(c,co,vars.m[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.set(fnz-1,co,
+ (e/(aug.m[r*(n+p)+(fnz-1)])).normal());
+ last_assigned_sol = fnz;
}
- } catch (...) {
- ex xxx = e - rhs(r,0);
- cerr << "xxx=" << xxx << endl << endl;
}
- GINAC_ASSERT((e-rhs(r,0)).normal().is_zero());
+ // assign solutions for vars between 1 and
+ // last_assigned_sol-1: free parameters
+ for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
+ sol.set(ro,co,vars(ro,co));
}
-#endif // def DO_GINAC_ASSERT
return sol;
}
-/** Solve a set of equations for an m x n matrix.
- *
- * @param vars n x p matrix
- * @param rhs m x p matrix
- * @exception logic_error (incompatible matrices)
- * @exception runtime_error (singular matrix) */
-matrix matrix::solve(const matrix & vars,
- const matrix & rhs) const
-{
- if ((row != rhs.row) || (col != vars.row) || (rhs.col != vars.col))
- throw (std::logic_error("matrix::solve(): incompatible matrices"));
-
- throw (std::runtime_error("FIXME: need implementation."));
-}
-
-/** Old and obsolete interface: */
-matrix matrix::old_solve(const matrix & v) const
-{
- if ((v.row != col) || (col != v.row))
- throw (std::logic_error("matrix::solve(): incompatible matrices"));
-
- // build the augmented matrix of *this with v attached to the right
- matrix tmp(row,col+v.col);
- for (unsigned r=0; r<row; ++r) {
- for (unsigned c=0; c<col; ++c)
- tmp.m[r*tmp.col+c] = this->m[r*col+c];
- for (unsigned c=0; c<v.col; ++c)
- tmp.m[r*tmp.col+c+col] = v.m[r*v.col+c];
- }
- // cout << "augmented: " << tmp << endl;
- tmp.gauss_elimination();
- // cout << "degaussed: " << tmp << endl;
- // assemble the solution matrix
- exvector sol(v.row*v.col);
- for (unsigned c=0; c<v.col; ++c) {
- for (unsigned r=row; r>0; --r) {
- for (unsigned i=r; i<col; ++i)
- sol[(r-1)*v.col+c] -= tmp.m[(r-1)*tmp.col+i]*sol[i*v.col+c];
- sol[(r-1)*v.col+c] += tmp.m[(r-1)*tmp.col+col+c];
- sol[(r-1)*v.col+c] = (sol[(r-1)*v.col+c]/tmp.m[(r-1)*tmp.col+(r-1)]).normal();
- }
- }
- return matrix(v.row, v.col, sol);
-}
-
// protected
ex matrix::determinant_minor(void) const
{
// for small matrices the algorithm does not make any sense:
- if (this->row==1)
- return m[0];
- if (this->row==2)
+ 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 (this->row==3)
+ 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;
- // matrix minorM(this->row-1,this->col-1);
- // for (unsigned r1=0; r1<this->row; ++r1) {
+ // matrix minorM(this->rows()-1,this->cols()-1);
+ // for (unsigned r1=0; r1<this->rows(); ++r1) {
// // shortcut if element(r1,0) vanishes
// if (m[r1*col].is_zero())
// continue;
// Unique flipper counter for partitioning into minors
std::vector<unsigned> Pkey;
- Pkey.reserve(this->col);
+ Pkey.reserve(n);
// key for minor determinant (a subpartition of Pkey)
std::vector<unsigned> Mkey;
- Mkey.reserve(this->col-1);
+ 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 B;
ex det;
// initialize A with last column:
- for (unsigned r=0; r<this->col; ++r) {
+ for (unsigned r=0; r<n; ++r) {
Pkey.erase(Pkey.begin(),Pkey.end());
Pkey.push_back(r);
- A.insert(Rmap_value(Pkey,m[this->col*r+this->col-1]));
+ A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
}
// proceed from right to left through matrix
- for (int c=this->col-2; c>=0; --c) {
+ 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 (unsigned i=0; i<this->col-c; ++i)
+ for (unsigned i=0; i<n-c; ++i)
Pkey.push_back(i);
unsigned fc = 0; // controls logic for our strange flipper counter
do {
det = _ex0();
- for (unsigned r=0; r<this->col-c; ++r) {
+ for (unsigned r=0; r<n-c; ++r) {
// maybe there is nothing to do?
- if (m[Pkey[r]*this->col+c].is_zero())
+ if (m[Pkey[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<this->col-c; ++i)
+ for (unsigned i=0; i<n-c; ++i)
if (i!=r)
Mkey.push_back(Pkey[i]);
// Fetch the minors and compute the new determinant
if (r%2)
- det -= m[Pkey[r]*this->col+c]*A[Mkey];
+ det -= m[Pkey[r]*n+c]*A[Mkey];
else
- det += m[Pkey[r]*this->col+c]*A[Mkey];
+ det += m[Pkey[r]*n+c]*A[Mkey];
}
// prevent build-up of deep nesting of expressions saves time:
det = det.expand();
if (!det.is_zero())
B.insert(Rmap_value(Pkey,det));
// increment our strange flipper counter
- for (fc=this->col-c; fc>0; --fc) {
+ for (fc=n-c; fc>0; --fc) {
++Pkey[fc-1];
if (Pkey[fc-1]<fc+c)
break;
}
- if (fc<this->col-c)
- for (unsigned j=fc; j<this->col-c; ++j)
+ if (fc<n-c)
+ 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:
}
-/** Perform the steps of an ordinary Gaussian elimination to bring the matrix
- * into an upper echelon form.
+/** 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
+ * with numeric coefficients but quite unsuited for symbolic matrices.
*
+ * @param det may be set to true to save a lot of space if one is only
+ * interested in the diagonal elements (i.e. for calculating determinants).
+ * The others are set to zero in this case.
* @return sign is 1 if an even number of rows was swapped, -1 if an odd
* number of rows was swapped and 0 if the matrix is singular. */
-int matrix::gauss_elimination(void)
+int matrix::gauss_elimination(const bool det)
{
ensure_if_modifiable();
+ const unsigned m = this->rows();
+ const unsigned n = this->cols();
+ GINAC_ASSERT(!det || n==m);
int sign = 1;
- ex piv;
- for (unsigned r1=0; r1<row-1; ++r1) {
- int indx = pivot(r1);
- if (indx == -1)
- return 0; // Note: leaves *this in a messy state.
- if (indx > 0)
- sign = -sign;
- for (unsigned r2=r1+1; r2<row; ++r2) {
- piv = this->m[r2*col+r1] / this->m[r1*col+r1];
- for (unsigned c=r1+1; c<col; ++c)
- this->m[r2*col+c] -= piv * this->m[r1*col+c];
- for (unsigned c=0; c<=r1; ++c)
- this->m[r2*col+c] = _ex0();
+
+ unsigned r0 = 0;
+ for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
+ int indx = pivot(r0, r1, true);
+ if (indx == -1) {
+ sign = 0;
+ if (det)
+ return 0; // leaves *this in a messy state
+ }
+ if (indx>=0) {
+ if (indx > 0)
+ sign = -sign;
+ for (unsigned r2=r0+1; r2<m; ++r2) {
+ ex piv = this->m[r2*n+r1] / this->m[r0*n+r1];
+ for (unsigned c=r1+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)
+ this->m[r2*n+c] = _ex0();
+ }
+ if (det) {
+ // save space by deleting no longer needed elements
+ for (unsigned c=r0+1; c<n; ++c)
+ this->m[r0*n+c] = _ex0();
+ }
+ ++r0;
}
}
}
-/** Perform the steps of division free elimination to bring the matrix
+/** Perform the steps of division free elimination to bring the m x n matrix
* into an upper echelon form.
*
+ * @param det may be set to true to save a lot of space if one is only
+ * interested in the diagonal elements (i.e. for calculating determinants).
+ * The others are set to zero in this case.
* @return sign is 1 if an even number of rows was swapped, -1 if an odd
* number of rows was swapped and 0 if the matrix is singular. */
-int matrix::division_free_elimination(void)
+int matrix::division_free_elimination(const bool det)
{
- int sign = 1;
ensure_if_modifiable();
- for (unsigned r1=0; r1<row-1; ++r1) {
- int indx = pivot(r1);
- if (indx==-1)
- return 0; // Note: leaves *this in a messy state.
- if (indx>0)
- sign = -sign;
- for (unsigned r2=r1+1; r2<row; ++r2) {
- for (unsigned c=r1+1; c<col; ++c)
- this->m[r2*col+c] = this->m[r1*col+r1]*this->m[r2*col+c] - this->m[r2*col+r1]*this->m[r1*col+c];
- for (unsigned c=0; c<=r1; ++c)
- this->m[r2*col+c] = _ex0();
+ const unsigned m = this->rows();
+ const unsigned n = this->cols();
+ GINAC_ASSERT(!det || n==m);
+ int sign = 1;
+
+ unsigned r0 = 0;
+ for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
+ int indx = pivot(r0, r1, true);
+ if (indx==-1) {
+ sign = 0;
+ if (det)
+ return 0; // leaves *this in a messy state
+ }
+ if (indx>=0) {
+ if (indx>0)
+ sign = -sign;
+ for (unsigned r2=r0+1; r2<m; ++r2) {
+ for (unsigned c=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();
+ // fill up left hand side with zeros
+ for (unsigned c=0; c<=r1; ++c)
+ this->m[r2*n+c] = _ex0();
+ }
+ if (det) {
+ // save space by deleting no longer needed elements
+ for (unsigned c=r0+1; c<n; ++c)
+ this->m[r0*n+c] = _ex0();
+ }
+ ++r0;
}
}
* is possible, since we know the divisor at each step.
*
* @param det may be set to true to save a lot of space if one is only
- * interested in the last element (i.e. for calculating determinants), the
+ * interested in the last element (i.e. for calculating determinants). The
* others are set to zero in this case.
* @return sign is 1 if an even number of rows was swapped, -1 if an odd
* number of rows was swapped and 0 if the matrix is singular. */
-int matrix::fraction_free_elimination(bool det)
+int matrix::fraction_free_elimination(const bool det)
{
// Method:
// (single-step fraction free elimination scheme, already known to Jordan)
// and D{m[k+1](r,c)} by
// D{m[k-1](k-1,k-1)}.
- GINAC_ASSERT(det || row==col);
ensure_if_modifiable();
- if (rows()==1)
- return 1;
-
+ const unsigned m = this->rows();
+ const unsigned n = this->cols();
+ GINAC_ASSERT(!det || n==m);
int sign = 1;
+ if (m==1)
+ return 1;
ex divisor_n = 1;
ex divisor_d = 1;
ex dividend_n;
// need GCDs) since the elements of *this might be unnormalized, which
// makes things more complicated than they need to be.
matrix tmp_n(*this);
- matrix tmp_d(row,col); // for denominators, if needed
+ matrix tmp_d(m,n); // for denominators, if needed
lst srl; // symbol replacement list
- exvector::iterator it = m.begin();
+ exvector::iterator it = this->m.begin();
exvector::iterator tmp_n_it = tmp_n.m.begin();
exvector::iterator tmp_d_it = tmp_d.m.begin();
- for (; it!= m.end(); ++it, ++tmp_n_it, ++tmp_d_it) {
+ for (; it!= this->m.end(); ++it, ++tmp_n_it, ++tmp_d_it) {
(*tmp_n_it) = (*it).normal().to_rational(srl);
(*tmp_d_it) = (*tmp_n_it).denom();
(*tmp_n_it) = (*tmp_n_it).numer();
}
- for (unsigned r1=0; r1<row-1; ++r1) {
- int indx = tmp_n.pivot(r1);
- if (det && indx==-1)
- return 0; // FIXME: what to do if det is false, some day?
- if (indx>0) {
- sign = -sign;
- // rows r1 and indx were swapped, so pivot matrix tmp_d:
- for (unsigned c=0; c<col; ++c)
- tmp_d.m[row*indx+c].swap(tmp_d.m[row*r1+c]);
+ 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) {
+ sign = 0;
+ if (det)
+ return 0;
}
- if (r1>0) {
- divisor_n = tmp_n.m[(r1-1)*col+(r1-1)].expand();
- divisor_d = tmp_d.m[(r1-1)*col+(r1-1)].expand();
- // save space by deleting no longer needed elements:
- if (det) {
- for (unsigned c=0; c<col; ++c) {
- tmp_n.m[(r1-1)*col+c] = 0;
- tmp_d.m[(r1-1)*col+c] = 1;
+ if (indx>=0) {
+ if (indx>0) {
+ sign = -sign;
+ // tmp_n's rows r0 and indx were swapped, do the same in tmp_d:
+ for (unsigned c=r1; c<n; ++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();
+ bool check = divide(dividend_n, divisor_n,
+ tmp_n.m[r2*n+c], true);
+ check &= divide(dividend_d, divisor_d,
+ tmp_d.m[r2*n+c], true);
+ GINAC_ASSERT(check);
}
+ // fill up left hand side with zeros
+ for (unsigned c=0; c<=r1; ++c)
+ tmp_n.m[r2*n+c] = _ex0();
}
- }
- for (unsigned r2=r1+1; r2<row; ++r2) {
- for (unsigned c=r1+1; c<col; ++c) {
- dividend_n = (tmp_n.m[r1*col+r1]*tmp_n.m[r2*col+c]*
- tmp_d.m[r2*col+r1]*tmp_d.m[r1*col+c]
- -tmp_n.m[r2*col+r1]*tmp_n.m[r1*col+c]*
- tmp_d.m[r1*col+r1]*tmp_d.m[r2*col+c]).expand();
- dividend_d = (tmp_d.m[r2*col+r1]*tmp_d.m[r1*col+c]*
- tmp_d.m[r1*col+r1]*tmp_d.m[r2*col+c]).expand();
- bool check = divide(dividend_n, divisor_n,
- tmp_n.m[r2*col+c],true);
- check &= divide(dividend_d, divisor_d,
- tmp_d.m[r2*col+c],true);
- GINAC_ASSERT(check);
+ if ((r1<n-1)&&(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();
+ if (det) {
+ // save space by deleting no longer needed elements
+ for (unsigned c=0; c<n; ++c) {
+ tmp_n.m[r0*n+c] = _ex0();
+ tmp_d.m[r0*n+c] = _ex1();
+ }
+ }
}
- // fill up left hand side.
- for (unsigned c=0; c<=r1; ++c)
- tmp_n.m[r2*col+c] = _ex0();
+ ++r0;
}
}
// repopulate *this matrix:
- it = m.begin();
+ it = this->m.begin();
tmp_n_it = tmp_n.m.begin();
tmp_d_it = tmp_d.m.begin();
- for (; it!= m.end(); ++it, ++tmp_n_it, ++tmp_d_it)
+ for (; it!= this->m.end(); ++it, ++tmp_n_it, ++tmp_d_it)
(*it) = ((*tmp_n_it)/(*tmp_d_it)).subs(srl);
return sign;
* where the element was found. With (symbolic==true) it does the same thing
* with the first non-zero element.
*
- * @param ro is the row to be inspected
+ * @param ro is the row from where to begin
+ * @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
* a degeneracy) and positive integer k means that rows ro and k were swapped.
*/
-int matrix::pivot(unsigned ro, bool symbolic)
+int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
{
unsigned k = ro;
-
- if (symbolic) { // search first non-zero
- for (unsigned r=ro; r<row; ++r) {
- if (!m[r*col+ro].is_zero()) {
- k = r;
- break;
- }
- }
- } else { // search largest
- numeric tmp(0);
- numeric maxn(-1);
- for (unsigned r=ro; r<row; ++r) {
- GINAC_ASSERT(is_ex_of_type(m[r*col+ro],numeric));
- if ((tmp = abs(ex_to_numeric(m[r*col+ro]))) > maxn &&
- !tmp.is_zero()) {
- maxn = tmp;
- k = r;
+ if (symbolic) {
+ // search first non-zero element in column co beginning at row ro
+ while ((k<row) && (this->m[k*col+co].expand().is_zero()))
+ ++k;
+ } else {
+ // search largest element in column co beginning at row ro
+ GINAC_ASSERT(is_ex_of_type(this->m[k*col+co],numeric));
+ unsigned kmax = k+1;
+ numeric mmax = abs(ex_to_numeric(m[kmax*col+co]));
+ while (kmax<row) {
+ GINAC_ASSERT(is_ex_of_type(this->m[kmax*col+co],numeric));
+ numeric tmp = ex_to_numeric(this->m[kmax*col+co]);
+ if (abs(tmp) > mmax) {
+ mmax = tmp;
+ k = kmax;
}
+ ++kmax;
}
+ if (!mmax.is_zero())
+ k = kmax;
}
- if (m[k*col+ro].is_zero())
+ if (k==row)
+ // all elements in column co below row ro vanish
return -1;
- if (k!=ro) { // swap rows
- ensure_if_modifiable();
- for (unsigned c=0; c<col; ++c) {
- m[k*col+c].swap(m[ro*col+c]);
- }
- return k;
- }
- return 0;
+ if (k==ro)
+ // matrix needs no pivoting
+ return 0;
+ // matrix needs pivoting, so swap rows k and ro
+ ensure_if_modifiable();
+ for (unsigned c=0; c<col; ++c)
+ m[k*col+c].swap(m[ro*col+c]);
+
+ return k;
}
/** Convert list of lists to matrix. */
ex lst_to_matrix(const ex &l)
{
- if (!is_ex_of_type(l, lst))
- throw(std::invalid_argument("argument to lst_to_matrix() must be a lst"));
-
- // 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();
-
- // Allocate and fill matrix
- matrix &m = *new matrix(rows, cols);
- for (i=0; i<rows; i++)
- for (j=0; j<cols; j++)
- if (l.op(i).nops() > j)
- m.set(i, j, l.op(i).op(j));
- else
- m.set(i, j, ex(0));
- return m;
+ if (!is_ex_of_type(l, lst))
+ throw(std::invalid_argument("argument to lst_to_matrix() must be a lst"));
+
+ // 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();
+
+ // Allocate and fill matrix
+ matrix &m = *new matrix(rows, cols);
+ for (i=0; i<rows; i++)
+ for (j=0; j<cols; j++)
+ if (l.op(i).nops() > j)
+ m.set(i, j, l.op(i).op(j));
+ else
+ m.set(i, j, ex(0));
+ return m;
}
//////////
git://www.ginac.de / ginac.git / blobdiff