Formal analysis of the Schulz matrix inversion algorithm: A paradigm towards computer aided verification of general matrix flow solvers

Vassilios A. Tsachouridis; Georgios Giantamidis; Stylianos Basagiannis; Kostas Kouramas

doi:10.3934/naco.2019047

Article Contents

2020, Volume 10, Issue 2: 177-206. Doi: 10.3934/naco.2019047

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Formal analysis of the Schulz matrix inversion algorithm: A paradigm towards computer aided verification of general matrix flow solvers

United Technologies Research Center Ltd., 2nd Floor Penrose Wharf Business Centre, Penrose Quay, Cork, T23 XN53, Ireland

^* Corresponding author: Vassilios A. Tsachouridis (tsachov@utrc.utc.com)
^* Corresponding author: Vassilios A. Tsachouridis (tsachov@utrc.utc.com)

^** Also with Aalto University, Finland

Received: January 2019

Revised: July 2019

Published: April 2020

This work was supported by the Irish Development Agency (IDA) under the program "Network of Excellence in Aerospace Cyber-Physical Systems", 2015-2019.

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Abstract

This paper pilots Schulz generalised matrix inverse algorithm as a paradigm in demonstrating how computer aided reachability analysis and theoretical numerical analysis can be combined effectively in developing verification methodologies and tools for matrix iterative solvers. It is illustrated how algorithmic convergence to computed solutions with required accuracy is mathematically quantified and used within computer aided reachability analysis tools to formally verify convergence over predefined sets of multiple problem data. In addition, some numerical analysis results are used to form computational reliability monitors to escort the algorithm on-line and monitor the numerical performance, accuracy and stability of the entire computational process. For making the paper self-contained, formal verification preliminaries and background on tools and approaches are reported together with the detailed numerical analysis in basic mathematical language. For demonstration purposes, a custom made reachability analysis program based on affine arithmetic is applied to numerical examples.

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Mathematics Subject Classification: Primary: 65F20, 65F30; Secondary: 68Q45.

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Figure 1. Hybrid automaton for a $ 1 \times 2 $ matrix $ A $

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Figure 2. Hybrid automaton for a $ 2 \times 2 $ matrix $ A $

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Figure 3. Hybrid automaton for a $ 2 \times 3 $ matrix $ A $

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Figure 4. Affine arithmetic approach for bound convergence expression (34) for a $ 2 \times 2 $ interval matrix

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Figure 5. Affine arithmetic approach for bound convergence expression (34) for a $ 2 \times 3 $ interval matrix

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Figure 6. Variation of condition number of $ A $ as function of $ {A_{11}} $

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Figure 7. $ 43 \times 68 $ matrix lpkb2 University of Florida sparse matrix collection

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Figure 8. $ 43 \times 68 $ matrix example in Breach

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Figure 9. Computational reliability monitor for $ 43 \times 68 $ matrix Example 6

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