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doi: 10.3934/naco.2019047

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 and  Kostas Kouramas

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)

**Also with Aalto University, Finland

Received  January 2019 Revised  July 2019 Published  September 2019

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

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.

Keywords: Matrix iterative algorithms, Schulz algorithm, Reachability analysis, Formal verification.
Mathematics Subject Classification: Primary: 65F20, 65F30; Secondary: 68Q45.
Citation: Vassilios A. Tsachouridis, Georgios Giantamidis, Stylianos Basagiannis, Kostas Kouramas. Formal analysis of the Schulz matrix inversion algorithm: A paradigm towards computer aided verification of general matrix flow solvers. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2019047
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show all references

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J. M. Esposito, J. Kim and V. Kumar, Adaptive RRTs for validating hybrid robotic control systems, Workshop on Algorithmic Foundations of Robotics, Zeist, Netherlands, (2004), 107–132. Google Scholar

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J. Kim, J. M. Esposito and V. Kumar, An RRT-based algorithm for testing and validating multi-robot controllers, Robotics: Science and Systems, Boston, MA, (2005), 249–256. Google Scholar

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L. H. de Figueiredo and J. Stolfi, Affine arithmetic: concepts and applications, Numerical Algorithms, 37 (2004), 147-158.  doi: 10.1023/B:NUMA.0000049462.70970.b6.  Google Scholar

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G. Frehse, C. L. Guernic, A. Donz, R. Ray, O. Lebeltel, R. Ripado, A. Girard, T. Dang and O. Maler, SpaceEx Scalable verification of hybrid systems, Proc. of CAV11, LNCS, Springer, 6806 (2011), 379–395. doi: 10.1007/978-3-642-22110-1_30.  Google Scholar

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G. Frehse, R. Kateja and C. Le Guernic, Flowpipe approximation and clustering in space-time, Proc. of HSCC13, 9035 (2013), 203-212.  doi: 10.1145/2461328.2461361.  Google Scholar

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N. Giorgetti, G.J. Pappas and A. Bemporad, Bounded model checking for hybrid dynamical systems, IEEE Conference on Decision and Control, Seville, Spain, (2005), 672–677. Google Scholar

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C. Le Guernic, Reachability Analysis of Hybrid Systems with Linear Continuous Dynamics, PhD Thesis in Universit Joseph-Fourier-Grenoble I, France, 2009. Google Scholar

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T. Henzinger, P. Kopke, A. Puri and P. Varaiya, What's decidable about hybrid automata?, ACM Symposium on Theory of Computing, (1995), 273–282. doi: 10.1016/0895-7177(96)00072-6.  Google Scholar

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T. Henzinger, The theory of hybrid automata, Symposium on Logic in Computer Science, (1996), 278–292. doi: 10.1109/LICS.1996.561342.  Google Scholar

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F. Immler, Tool presentation Isabelle/hol for reachability analysis of continuous systems, in ARCH14-15, 1st and 2nd International Workshop on Applied veRification for Continuous and Hybrid Systems (eds. M. Frehse and M. Althoff), Academic Press, (1971), 33–75. EPiC Series in Computer Science, 34 (2015), 180–187.  Google Scholar

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A. A. A. Julius, G. E. Fainekos, M. Anand, I. Lee and G. J. Pappas, IRobust test generation and coverage for hybrid systems, Hybrid Systems: Computation and Control, LNCS, Springer-Verlag, 4416 (2007), 329–342. Google Scholar

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G. Lafferriere, G. Pappas and S. Yovine, A new class of decidable hybrid systems, Hybrid Systems: Computation and Control, LNCS, 1569 (1999), 137-151.   Google Scholar

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C. Livadas and N. Lynch, A new class of decidable hybrid systems, Hybrid Systems: Computation and Control, LNCS, 1386 (1998), 253-272.   Google Scholar

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F. Messine and A. Touhami, A general reliable quadratic form: an extension of affine arithmetic, Reliable Computing, 12 (2006), 171-192.  doi: 10.1007/s11155-006-7217-4.  Google Scholar

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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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