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 - ( υ_{ij} /
υ_{i} ) b_{r}, ∀ _{rj}i∈{0,...,n-1},
i≠r ;b _{rj}^{*} = b
/ υ_{rj}_{r} |
}
∀ j∈{0,...,n-1} |

An **ξ**
vector from the *i*, *j*
or *n* in the algorithm above:

`M_OPS`

function block (FB)
performs the `PIVOT`

operation by taking 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 - 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 υfrom the set of as-yet unpivoted columns._{r} - If the absolute value of the maximum divisor υ
is less than the tolerance specified by the FB's_{r}`X`

input, the`A`

matrix is declared to be singular and the inversion operation terminates gracefully with an appropriate`STATUS`

output code.

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