PFI (Product Form of the Inverse)

The Product Form of the Inverse (PFI) is based on the fact that, if we already have the inverse B of an n×n matrix C, we can obtain the inverse B ^* of a new matrix C ^* , formed by replacing column r of C by a vector ξ, by evaluating the matrix product υ = Bξ and then performing a pivot operation expressed by the formulae

b _ij ^* = b _ij - ( υ _i / υ _r ) b _rj, ∀ i∈{0,...,n-1}, i≠r ;
b _rj ^* = b _rj / υ _r } ∀ j∈{0,...,n-1}

An M_OPS function block (FB) performs the PIVOT operation by taking the ξ vector from the IorM-th column of its A MATRIX input, and pivoting it into the JorN-th column of its B output. In this case, the IorM and JorN inputs of the FB do not correspond to the indices i, j or n in the algorithm above:

The IorM input specifies the column of the A input matrix to be used as the ξ vector.
The JorN input specifies the pivot column index r in the B matrix output.

An M_OPS FB performs the matrix inversion (INV) function by initializing the B matrix output to an identity matrix of the same order as the A matrix input, and then pivoting the columns of A into B one at a time.

A maximum pivot divisor strategy is used to provide the highest accuracy of the inverse. That is, at each iteration, the index r of the pivot column is chosen to maximize the absolute value of the divisor υ_r from the set of as-yet unpivoted columns.
If the absolute value of the maximum divisor υ_r is less than the tolerance specified by the FB's X input, the A matrix is declared to be singular and the inversion operation terminates gracefully with an appropriate STATUS output code.