Product Form of the Inverse (PFI)

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:

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.