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An efficient method for computing the outer inverse \(A_{T,S}^{(2)}\) through Gauss-Jordan elimination

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Abstract

In this paper, we derive a novel expression for the computation of the outer inverse \(A_{T,S}^{(2)}\). Based on this expression, we present a new Gauss-Jordan elimination method for computing \(A_{R(G),N(G)}^{(2)}\). The analysis of computational complexity indicates that our algorithm is more efficient than the existing Gauss-Jordan elimination algorithms for \(A_{R(G),N(G)}^{(2)}\) in the literature for a large class of problems. Especially for the case when G is a Hermitian matrix, our algorithm has the lowest computational complexity among the existing algorithms. Finally, numerical experiments show that our method for the outer inverse \(A_{R(G),N(G)}^{(2)}\) generally is more efficient than that of the other existing methods in the cases of matrices A with m < n or square matrices G with high rank or Hermitian matrices G in practice.

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Acknowledgments

The first author is supported by the Fundamental Research Funds for the Central Universities (JBK1901028).

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Authors and Affiliations

  1. School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, China

    Jie Ma

  2. Faculty of Geosciences and Environmental Engineering, Southwest Jiaotong University, Chengdu, China

    Jie Ma & Yongshu Li

  3. School of Statistics, Southwestern University of Finance and Economics, Chengdu, China

    Feng Gao

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  1. Jie Ma
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  2. Feng Gao
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  3. Yongshu Li
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Correspondence to Jie Ma.

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Ma, J., Gao, F. & Li, Y. An efficient method for computing the outer inverse \(A_{T,S}^{(2)}\) through Gauss-Jordan elimination. Numer Algor 85, 77–106 (2020). https://doi.org/10.1007/s11075-019-00803-w

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