A heuristics is applied for automatic determination of a suitable algorithm.
gauss
Gauss elimination.
If are the entries of the original matrix, then the matrix is transformed into triangular form by applying the rules
This algorithm is well-suited for numerical matrices but generally suffers from the expensive division (and computation of GCDs) at each step.
divfree
Division-free elimination.
This is a modification of Gauss elimination where the division by the pivot element is not carried out. If are the entries of the original matrix, then the matrix is transformed into triangular form by applying the rules
This algorithm is only there for the purpose of cross-checks. It suffers from exponential intermediate expression swell. Use it only for small systems.
bareiss
Bareiss fraction-free elimination.
This is a modification of Gauss elimination where the division by the pivot element is delayed until it can be carried out without computing GCDs. If are the entries of the original matrix, then the matrix is transformed into triangular form by applying the rules
(We have set in order to avoid a case distinction in above formula.) It can be shown that nothing more than polynomial long division is needed for carrying out the division. This is generally the fastest algorithm for solving linear systems. In contrast to division-free elimination it only has a linear expression swell. For two-dimensional systems, the two algorithms are equivalent, however.
markowitz
Markowitz-ordered Gaussian elimination.
Same as the usual Gaussian elimination, but with additional effort spent on selecting pivots that minimize fill-in. Faster than the methods above for large sparse matrices (particularly with symbolic coefficients), otherwise slightly slower than Gaussian elimination.
![\[
m_{i,j}^{(k+1)} = m_{i,j}^{(k)} - m_{i,k}^{(k)} m_{k,j}^{(k)} / m_{k,k}^{(k)}
\]](form_1_dark.png)
![\[
m_{i,j}^{(k+1)} = m_{i,j}^{(k)} m_{k,k}^{(k)} - m_{i,k}^{(k)} m_{k,j}^{(k)}
\]](form_2_dark.png)
![\[
m_{i,j}^{(k+1)} = (m_{i,j}^{(k)} m_{k,k}^{(k)} - m_{i,k}^{(k)} m_{k,j}^{(k)}) / m_{k-1,k-1}^{(k-1)}
\]](form_3_dark.png)
