MA51: Auxiliary for MA48 and MA50: identify ignored rows or columns in the rectangular or rank-deficient case, compute determinant

This is for use in conjunction with the MA48 and MA50 packages for solving sparse unsymmetric sets of linear equations. If the matrix is singular or rectangular, it identifies the rows and columns that are treated specially. If the matrix is square, it computes the determinant.

It identifies which equations are ignored when solving \(\bf Ax = \bf b\) and which solution components are always set to zero. The roles are reversed for \(\mathbf{ A} ^T \mathbf {x} = \mathbf {b}\). There are such equations and/or components in the singular or rectangular case. Note that if \(\bf Ax = \bf b\) or \(\mathbf{ A}^T \bf x = \bf b\) is not consistent, there may be large residuals for the equations that are ignored.