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Formal analysis of the Schulz matrix inversion algorithm: a paradigm towards computer aided verification of general matrix flow solvers. (English) Zbl 07220004
Summary: 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.
##### MSC:
 65F20 Numerical solutions to overdetermined systems, pseudoinverses 65F30 Other matrix algorithms (MSC2010) 68Q45 Formal languages and automata
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##### References:
 [1] AAFLIB - An Affine Arithmetic C++, 2019. Available from: http://aaflib.sourceforge.net. [2] R. Alur; C. Courcoubetis; N. Halbwachs; T. Henzinger; P. H. Ho; X. Nicollin; A. Olivero; J. Sifakis; S. Yovine, The algorithmic analysis of hybrid systems, Theoretical Computer Science, 138, 3-34 (1995) · Zbl 0874.68206 [3] R. Alur; T. A. Henzinger; G. Lafferriere; G. Pappas, Discrete abstractions of hybrid systems, Proceedings of the IEEE 88, 7, 971-984 (2000) [4] R. Alur; T. Dang; F. Ivancic, Counter example-guided predicate abstraction of hybrid systems, Theoretical Computer Science, 354, 250-271 (2006) · Zbl 1088.68096 [5] Y. Annapureddy, C. Liu, G. Fainekos and S. Sankaranarayanan, S-TaLiRo: A tool for temporal logic falsification for hybrid systems, Proc. of Tools and Algorithms for the Construction and Analysis of Systems, (2011), 254-257. · Zbl 1316.68069 [6] E. Asarin, T. Dang and O. Maler, The d/dt tool for verification of hybrid systems, Int. Conf. on Computer Aided Verification, LNCS, Springer-Verlag, (2002), 365-350. · Zbl 1010.68796 [7] A. Ben-Israel; D. Cohen, On iterative computation of generalized inverses and associated projections, SIAM J. Numer. Anal., 3, 410-419 (1966) · Zbl 0143.37402 [8] A. Ben-Israel and T. N. E. Greville, Generalized Inverses Theory and Applications, Springer, 2003, ISBN 978-0-387-00293-4. · Zbl 1026.15004 [9] A. Bhatia and E. Frazzoli, Incremental search methods for reachability analysis of continuous and hybrid systems, Hybrid Systems: Computation and Control, LNCS, Springer-Verlag, 2993 (2004), 142-156. · Zbl 1135.93316 [10] O. Botchkarev and S. Tripakis, Verification of hybrid systems with linear differential inclusions using ellipsoidal approximations, Hybrid Systems: Computation and Control, LNCS, Springer-Verlag, 1790 (2000), 73-78. · Zbl 1037.93510 [11] M. S. Branicky; M. M. Curtiss; J. Levine; S. Morgan, Sampling-based planning, control, and verification of hybrid systems, Control Theory and Applications, 153, 575-590 (2006) [12] C. Bu; X. Zhang; J. Zhou; W. Wang; Y. Wei, The inverse, rank and product of tensors, Linear Algebra and Its Applications, 446, 269-280 (2014) · Zbl 1286.15030 [13] X. Chen, E. Abraham and S. Sankaranarayanan, Flow* An analyzer for non-linear hybrid systems, Proc. of CAV13, LNCS, Springer, 8044 (2013), 258-263. [14] X. Chen, Reachability Analysis of Non-Linear Hybrid Systems Using Taylor Models, PhD Thesis in RWTH Aachen University, Germany, 2015. [15] C. Chutinan; B. H. Krogh, Computational techniques for hybrid system verification, IEEE Transactions on Automatic Control, 48, 64-75 (2003) · Zbl 1364.93457 [16] E. M Clarke, W. Klieber, M. Novcek and P. Zuliani, Model checking and the state explosion problem, Tools for Practical Software Verification, Springer, (2012), 1-30. [17] E. M. Clarke, Th. A. Henzinger, H. Veith and R. Bloem, Handbook of Model Checking, Springer International Publishing, 2018. · Zbl 1390.68001 [18] J-J. Climent; N. Thome; Y. Wei, A geometrical approach on generalized inverses by Neumann-type series, Linear Algebra and Its Applications, 1, 533-540 (2001) · Zbl 0986.15005 [19] M. Daumas; D. Lester; C. Muoz, Verified real number calculations, A library for interval arithmetic, IEEE Transactions on Computers, 58, 226-237 (2009) · Zbl 1367.65213 [20] A. Donze, Breach, a toolbox for verification and parameter synthesis of hybrid systems, Proc. of Computer Aided Verification, (2010), 167-170. [21] P. Duggirala, S. Mitra, M. Viswanathan and M. Potok, C2E2 A verification tool for Stateflow models, Proc. of TACAS15, LNCS, Springer, 9035 (2015), 68-82. [22] 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. [23] 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. [24] L. H. de Figueiredo; J. Stolfi, Affine arithmetic: concepts and applications, Numerical Algorithms, 37, 147-158 (2004) · Zbl 1074.65050 [25] 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. [26] G. Frehse; R. Kateja; C. Le Guernic, Flowpipe approximation and clustering in space-time, Proc. of HSCC13, 9035, 203-212 (2013) · Zbl 1362.93015 [27] M. Gameiro and P. Manolios, Formally verifying an algorithm based on interval arithmetic for checking transversality, Workshop on ACL2 Prover and Applications, 2004. [28] 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. [29] C. Le Guernic, Reachability Analysis of Hybrid Systems with Linear Continuous Dynamics, PhD Thesis in Universit Joseph-Fourier-Grenoble I, France, 2009. · Zbl 1242.93059 [30] T. Henzinger, P. Kopke, A. Puri and P. Varaiya, What’s decidable about hybrid automata?, ACM Symposium on Theory of Computing, (1995), 273-282. · Zbl 0978.68534 [31] T. Henzinger, The theory of hybrid automata, Symposium on Logic in Computer Science, (1996), 278-292. · Zbl 0959.68073 [32] 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. [33] 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. · Zbl 1221.93076 [34] S. Kong, S. Gao and W. Chen, Reachability analysis for hybrid systems, Proc. of TACAS15, LNCS, Springer, 9035 (2015), 200-205. [35] M. Konstantinov, D. Gu, V. Mehrmann and P. Petkov, Perturbation Theory for Matrix Equations, $$2^{nd}$$ edition, Elsevier, Amsterdam, 2003, ISBN-9780444513151. · Zbl 1025.15017 [36] G. A. Kumar; T. V. Subbareddy; B. M. Redd; N. Raju; V. Elamaran, An approach to design a matrix inversion hardware module using FPGA, Int. Conf. on Control, Instrumentation, Comm. and Comput. Technologies, 230, 87-90 (2014) [37] G. Lafferriere; G. Pappas; S. Yovine, A new class of decidable hybrid systems, Hybrid Systems: Computation and Control, LNCS, 1569, 137-151 (1999) · Zbl 0926.93036 [38] LAPACK-Linear Algebra PACKage, 2019. Available from: http://www.netlib.org/lapack/. [39] W. Levine, The Control Handbook, IEEE Press, 1996. · Zbl 0859.93001 [40] C. Livadas; N. Lynch, A new class of decidable hybrid systems, Hybrid Systems: Computation and Control, LNCS, 1386, 253-272 (1998) [41] Matlab-Mathworks, 2019. Available from: https://www.mathworks.com [42] F. Messine; A. Touhami, A general reliable quadratic form: an extension of affine arithmetic, Reliable Computing, 12, 171-192 (2006) · Zbl 1106.65042 [43] D. Monniaux, Toward verifiably correct control implementations, The impact of control technology Part 2: Challenges for control research, Report of IEEE Control Systems Society, 2nd Ed, 2011. Available from: http://ieeecss.org/general/IoCT2-report. [44] D. Monniaux and A. Mine, Verification of control system software, the impact of control technology, Part 1: Success stories for control, Report of IEEE Control Systems Society, 2nd Ed, 2011. Available from: http://ieeecss.org/general/IoCT2-report. [45] R. E. Moore, Methods and Applications of Interval Analysis, SIAM Studies in Applied and Numerical Mathematics, Philadelphia, 1987, ISBN-10: 0898711614. [46] R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, SIAM, 2009. · Zbl 1168.65002 [47] C. Munoz and D. Lester, Real number calculations and theorem proving, 18th Int. Conf. on Theorem Proving in Higher Order Logics, England, (2005), 239-254. [48] T. Nahhal; T. Dang, Test coverage for continuous and hybrid systems, Int. Conf. on Computer Aided Verification, LNCS, 4590, 449-462 (2007) · Zbl 1135.68346 [49] I. Pasca, Formal Verifcation for Numerical Methods, PhD Thesis in Universit Nice Sophia Antipolis, France, 2010. [50] P. Petkov, M. Konstantinov and N. Christov, Computational Methods for Linear Control Systems, $$2^{nd}$$ edition, Prentice- Hall, Hemel Hempstead, 1991, ISBN-9780444513151. · Zbl 0790.93001 [51] M. D. Petkovic; P. S. Stanimirovic, Generalized matrix inversion is not harder than matrix multiplication, J. Comput. and Applied Math., 230, 270-282 (2009) · Zbl 1170.65020 [52] M. D. Petkovic; P. S. Stanimirovic, Iterative method for computing Moore-Penrose inverse based on Penrose equations, J. Comput. and Applied Math., 235, 1604-1613 (2011) · Zbl 1206.65139 [53] M. S. Petkovic, Iterative methods for bounding the inverse of a matrix (a survey), FILOMAT Nis, Algebra Logic & Discrete Mathematics, 9, 543-577 (1995) · Zbl 0845.65014 [54] E. Plaku; L. Kavraki; M. Vardi, Hybrid systems: from verification to falsification by combining motion planning and discrete search, Formal Methods in System Design, 34, 157-182 (2009) · Zbl 1192.68692 [55] A. Platzer and J-D Quesel, KeYmaera a hybrid theorem prover for hybrid systems (system description), Automated Reasoning, IJCAR 2008, Lecture Notes in Computer Science (eds. M. Frehse and M. AlthoffArmando A., Baumgartner P., Dowek G.), Springer Berlin Heidelberg, 5195 (2008), 163-183. [56] A. Puri, Theory of Hybrid Systems and Discrete Event Systems, PhD Thesis in University of California, Berkeley, 1995. [57] PyInterval - Interval arithmetic in Python, 2019. Available from: https://github.com/taschini/pyinterval [58] S. Qiao; X. Wang; Y. Wei, Two finite-time convergent Zhang neural network models for time-varying complex matrix Drazin inverse, Linear Algebra and Its Applications, 542, 101-117 (2018) · Zbl 1415.15007 [59] Y. Rahman, A. Xie and S. Bernstein, Retrospective cost adaptive control: pole placement, frequency response, and connections with LQG control, IEEE Control System Magazine, (2017), 28-69. [60] S. Ratschan and Z. She, Safety verification of hybrid systems by constraint propagation based abstraction refinement, ACM Transactions on Embedded Computing Systems, 6 (2007), Article No. 8. · Zbl 1078.93508 [61] RTCA Formal Methods Supplement to DO-178C and DO-278A, RTCA DO-333, December 13, 2011. [62] RTCA, Software Considerations in Airborne Systems and Equipment Certification, RTCA DO-178C, December 13, 2011. [63] S. Schupp, E. Abraham, X. Chen, I. B. Makhlouf, G. Frehse, S. Sankaranarayanan and S. Kowalewski, Current challenges in the verification of hybrid systems, Workshop on Design, Modeling, and Evaluation of Cyber Physical Systems, (2015), 8-24. [64] S. Schupp and E. Abraham, Efficient dynamic error reduction for hybrid systems reachability analysis, Proc. of the 24th International Conference on Tools and Algorithms for the Construction and Analysis of Systems TACAS18, LNCS, Springer, 2018. · Zbl 1423.68290 [65] B. I. Silva and B. H. Krogh, Formal verification of hybrid systems using CheckMate: A case study, American Control Conference, (2000), 1679-1683. [66] G. Stewart and J. Sun, Matrix Perturbation Theory, Academic Press, New York, 1990. · Zbl 0706.65013 [67] O. Stursberg; B. H. Krogh, Efficient representation and computation of reachable sets for hybrid systems, Hybrid Systems: Computation and Control, LNCS, 2623, 482-497 (2003) · Zbl 1032.93037 [68] C. J. Tomlin, I. Mitchell, A. Bayen and M. Oishi, Computational techniques for the verification and control of hybrid systems, Proceedings of the IEEE 91, Springer-Verlag, 7 (2003), 986-1001. [69] V. A. Tsachouridis, Numerical analysis of $$H_{\infty}$$ filter for system parameter identification, Int. J. Modelling, Identification and Control, 30, 163-183 (2018) [70] University of Florida Sparse Matrix Collection, 2019. Available from: https://www.cise.ufl.edu/research/sparse/matrices/list_by_id.html [71] G. Wang, Y. Wei and S. Qiao, Generalized Inverses: Theory and Computations, Springer, Singapore: Science Press Beijing, Beijing, 2018. · Zbl 1395.15002 [72] Y. Wei, P. Stanimirovic and M. Petkovic, Numerical and Symbolic Computations of Generalized Inverses, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018. · Zbl 1404.65002 [73] J. Wilkinson, Rounding Errors in Algebraic Processes, Prentice Hall, Englewood Cliff, NJ, 1963. · Zbl 1041.65502 [74] H. Wolkovicz; G. P. H. Stayan, Bounds for eigenvalues using traces, Linear Algebra and Its Applications, 29, 471-506 (1980) · Zbl 0435.15015 [75] P. Xie, H. Xiang and Y. Wei, Randomized algorithms for total least squares problems, Numerical Linear Algebra with Applications, 26 (2019), e2219, Available from: https://doi.org/10.1002/nla.2219 · Zbl 07031719
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