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Accurate solutions of structured generalized Kronecker product linear systems

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Abstract

In this paper, we consider the generalized Kronecker product (GKP) linear system associated with a class of consecutive-rank-descending (CRD) matrices arising from bivariate interpolation problems. Relying on the sign sequences of CRD matrices, we show that the associated GKP linear system is accurately solved with an “ideal” componentwise forward error. In particular, a pleasantly small componentwise relative forward error is provided to illustrate that each component of the solution is computed to high relative accuracy. We then present the sign sequences of generalized Vandermonde matrices to show that the associated GKP linear system is accurately solved with the desired componentwise forward errors. Numerical experiments are performed to confirm the high relative accuracy.

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Acknowledgments

The authors would like to thank the Editor and the anonymous referees for their valuable comments and suggestions which have helped to improve the overall presentation of the paper.

Funding

Research supported by National Natural Science Foundations of China (Grants No. 11871020 and 11471279), Natural Science Foundation for Distinguished Young Scholars of Hunan Province (Grant No. 2017JJ1025), Research Foundation of Education Bureau of Hunan Province (Grant No. 18A198), Hunan Provincial Innovation Foundation for Postgraduate (Grant No. XDCX2019B062).

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, China

    Zhao Yang & Wei Zhu

  2. School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan, 411201, Hunan, China

    Rong Huang

  3. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, 411105, Hunan, China

    Jianzhou Liu

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  1. Zhao Yang
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  2. Rong Huang
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  3. Wei Zhu
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  4. Jianzhou Liu
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Correspondence to Rong Huang.

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Yang, Z., Huang, R., Zhu, W. et al. Accurate solutions of structured generalized Kronecker product linear systems. Numer Algor 87, 797–818 (2021). https://doi.org/10.1007/s11075-020-00988-5

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  • DOI: https://doi.org/10.1007/s11075-020-00988-5

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