Inversion with Partial Evaluation

Abstract

The complete inversion A^–1 is required if all components of $x = {A^{ - 1}}b$ are needed. If only a smaller part of the solution x or a number of functionals ${\varphi _i}\left( x \right)$ is of interest, the question arises as to whether the computation of the complete inversion can be avoided and whether a partial evaluation of A^–1 is cheaper.

The conceptuality in linear algebra and analysis is quite contrastive. The usual understanding in linear algebra is that the respective data (vectors, matrices) must be defined completely. In the case of a linear system Ax = b, only the full solution vector $x \in {^I}$ is the correct answer, while the inversion of A requires all entries of the matrix ${A^{ - 1}} \in {^{I \times I}}$.