Concise row-pruning algorithm to invert a matrix

Lakshmikantham, V and Sen, SK and Sivasundaram, S (1994) Concise row-pruning algorithm to invert a matrix. In: Applied mathematics and computation, 61 (1). pp. 17-24.

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Abstract

Presented here, in a vector formulation, is an O(mn2) direct concise algorithm that prunes/identifies the linearly dependent (ld) rows of an arbitrary m X n matrix A and computes its reflexive type minimum norm inverse A(mr)-, which will be the true inverse A-1 if A is nonsingular and the Moore-Penrose inverse A+ if A is full row-rank. The algorithm, without any additional computation, produces the projection operator P = (I - A(mr)- A) that provides a means to compute any of the solutions of the consistent linear equation Ax = b since the general solution may be expressed as x = A(mr)+b + Pz, where z is an arbitrary vector. The rank r of A will also be produced in the process. Some of the salient features of this algorithm are that (i) the algorithm is concise, (ii) the minimum norm least squares solution for consistent/inconsistent equations is readily computable when A is full row-rank (else, a minimum norm solution for consistent equations is obtainable), (iii) the algorithm identifies ld rows, if any, and reduces concerned computation and improves accuracy of the result, (iv) error-bounds for the inverse as well as the solution x for Ax = b are readily computable, (v) error-free computation of the inverse, solution vector, rank, and projection operator and its inherent parallel implementation are straightforward, (vi) it is suitable for vector (pipeline) machines, and (vii) the inverse produced by the algorithm can be used to solve under-/overdetermined linear systems.

Item Type:	Journal Article
Publication:	Applied mathematics and computation
Publisher:	Elsevier science
Additional Information:	Copyright of this article belongs to Elsevier science.
Department/Centre:	Division of Interdisciplinary Sciences > Supercomputer Education & Research Centre
Date Deposited:	25 Mar 2011 09:43
Last Modified:	22 Nov 2018 15:04
URI:	http://eprints.iisc.ac.in/id/eprint/36299

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