### 5.1.3 bareiss

qcindex Gauss

`Syntax:`
  `bareiss (` module_expression `)`
  `bareiss (` matrix_expression `)`
  `bareiss (` module_expression`,` int_expression`,` int_expression `)`
  `bareiss (` matrix_expression`,` int_expression`,` int_expression `)`
`Type:`
  list of module and intvec
`Purpose:`
  applies the sparse Gauss-Bareiss algorithm (see References, Lee and Saunders) to a module (or with type conversion to a matrix) with an 'optimal' pivot strategy. The vectors of the module are the columns of the matrix, hence elimination takes place w.r.t. rows.
  With only one parameter a complete elimination is done. Result is a list: the first entry is a module with a minimal independent set of vectors (as a matrix lower triangular), the second entry an intvec with the permutation of the rows w.r.t. the original matrix, that is, a k at position l indicates that row k was carried over to the row l.
  The further parameters control the algorithm. `bareiss(M,i,j)` does not attempt to diagonalize the last i rows in the elimination procedure and stops computing when the remaining number of vectors (columns) to reduce is at most j.
`Example:`

        ring r=0,(x,y,z),(c,dp);
        module mm;
        // ** generation of the module mm **
        int d=7;
        int b=2;
        int db=d-b;
        int i;
        for(i=d;i>0;i--){ mm[i]=3*x*gen(i); }
        for(i=db;i;i--){ mm[i]=mm[i]+7*y*gen(i+b); }
        for(i=d;i>db;i--){ mm[i]=mm[i]+7*y*gen(i-db); }
        for(i=d;i>b;i--){ mm[i]=mm[i]+11*z*gen(i-b); }
        for(i=b;i;i--){ mm[i]=mm[i]+11*z*gen(i+db); }
        // ** the generating matrix of mm **
        print(mm);
      ==> 3x, 0,  11z,0,  0,  7y, 0,
      ==> 0,  3x, 0,  11z,0,  0,  7y,
      ==> 7y, 0,  3x, 0,  11z,0,  0,
      ==> 0,  7y, 0,  3x, 0,  11z,0,
      ==> 0,  0,  7y, 0,  3x, 0,  11z,
      ==> 11z,0,  0,  7y, 0,  3x, 0,
      ==> 0,  11z,0,  0,  7y, 0,  3x
        // complete elimination
        list ss=bareiss(mm);
        print(ss[1]);
      ==> 7y, 0,     0,     0,      0,        0,     0,
      ==> 3x, -33xz, 0,     0,      0,        0,     0,
      ==> 11z,-121z2,1331z3,0,      0,        0,     0,
      ==> 0,  0,     0,     9317yz3,0,        0,     0,
      ==> 0,  21xy,  _[5,3],14641z4,-43923xz4,0,     0,
      ==> 0,  0,     0,     0,      65219y2z3,_[6,6],0,
      ==> 0,  49y2,  _[7,3],3993xz3,_[7,5],   _[7,6],_[7,7]
        ss[2];
      ==> 2,7,5,1,4,3,6
        // elimination up to 3 vectors
        ss=bareiss(mm,0,3);
        print(ss[1]);
      ==> 7y, 0,     0,     0,      0,        0,        0,
      ==> 3x, -33xz, 0,     0,      0,        0,        0,
      ==> 11z,-121z2,1331z3,0,      0,        0,        0,
      ==> 0,  0,     0,     9317yz3,0,        0,        0,
      ==> 0,  0,     0,     0,      27951xyz3,102487yz4,65219y2z3,
      ==> 0,  21xy,  _[6,3],14641z4,_[6,5],   _[6,6],   -43923xz4,
      ==> 0,  49y2,  _[7,3],3993xz3,_[7,5],   _[7,6],   _[7,7]
        ss[2];
      ==> 2,7,5,1,3,4,6
        // elimination without the last 3 rows
        ss=bareiss(mm,3,0);
        print(ss[1]);
      ==> 7y, 0,   0,      0,       0,     0,     0,
      ==> 0,  77yz,0,      0,       0,     0,     0,
      ==> 0,  0,   231xyz, 0,       0,     0,     0,
      ==> 0,  0,   0,      1617xy2z,0,     0,     0,
      ==> 11z,21xy,-1331z3,14641z4, _[5,5],_[5,6],_[5,7],
      ==> 0,  0,   539y2z, _[6,4],  _[6,5],_[6,6],-3773y3z,
      ==> 3x, 49y2,-363xz2,3993xz3, _[7,5],_[7,6],_[7,7]
        ss[2];
      ==> 2,3,4,1

See det; matrix.

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