Computation of the Adjoint Matrix

TL;DR

This paper introduces a new method for computing the adjoint matrix in a commutative domain that, while requiring more operations, offers improved parallelization suitable for parallel computing systems.

Contribution

A novel algorithm for adjoint matrix computation that enhances parallelization capabilities despite increased operation count.

Findings

Operates with 1.5 times more operations than the best existing method.

Allows better parallelization of the computation process.

Suitable for implementation in parallel computational systems.

Abstract

The best method for computing the adjoint matrix of an order $n$ matrix in an arbitrary commutative ring requires $O(n^{\beta+1/3}\log n \log \log n)$ operations, provided the complexity of the algorithm for multiplying two matrices is $\gamma n^\beta+o(n^\beta)$. For a commutative domain -- and under the same assumptions -- the complexity of the best method is ${6\gamma n^\beta}/{(2^{\beta}-2)}+o(n^\beta)$. In the present work a new method is presented for the computation of the adjoint matrix in a commutative domain. Despite the fact that the number of operations required is now 1.5 times more, than that of the best method, this new method permits a better parallelization of the computational process and may be successfully employed for computations in parallel computational systems.