June  2020, 10(2): 177-206. doi: 10.3934/naco.2019047

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)

** 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.

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 and Optimization, 2020, 10 (2) : 177-206. doi: 10.3934/naco.2019047
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show all references

References:
[1]

AAFLIB - An Affine Arithmetic C++, 2019. Available from: http://aaflib.sourceforge.net.

[2]

R. AlurC. CourcoubetisN. HalbwachsT. HenzingerP. H. HoX. NicollinA. OliveroJ. Sifakis and S. Yovine, The algorithmic analysis of hybrid systems, Theoretical Computer Science, 138 (1995), 3-34.  doi: 10.1016/0304-3975(94)00202-T.

[3]

R. AlurT. A. HenzingerG. Lafferriere and G. Pappas, Discrete abstractions of hybrid systems, Proceedings of the IEEE 88, 7 (2000), 971-984. 

[4]

R. AlurT. Dang and F. Ivancic, Counter example-guided predicate abstraction of hybrid systems, Theoretical Computer Science, 354 (2006), 250-271.  doi: 10.1016/j.tcs.2005.11.026.

[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.

[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.

[7]

A. Ben-Israel and D. Cohen, On iterative computation of generalized inverses and associated projections, SIAM J. Numer. Anal., 3 (1966), 410-419.  doi: 10.1137/0703035.

[8]

A. Ben-Israel and T. N. E. Greville, Generalized Inverses Theory and Applications, Springer, 2003, ISBN 978-0-387-00293-4.

[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.

[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.

[11]

M. S. BranickyM. M. CurtissJ. Levine and S. Morgan, Sampling-based planning, control, and verification of hybrid systems, Control Theory and Applications, 153 (2006), 575-590. 

[12]

C. BuX. ZhangJ. ZhouW. Wang and Y. Wei, The inverse, rank and product of tensors, Linear Algebra and Its Applications, 446 (2014), 269-280.  doi: 10.1016/j.laa.2013.12.015.

[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. doi: 10.1007/978-3-642-39799-8_18.

[14]

X. Chen, Reachability Analysis of Non-Linear Hybrid Systems Using Taylor Models, PhD Thesis in RWTH Aachen University, Germany, 2015.

[15]

C. Chutinan and B. H. Krogh, Computational techniques for hybrid system verification, IEEE Transactions on Automatic Control, 48 (2003), 64-75.  doi: 10.1109/TAC.2002.806655.

[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. doi: 10.1007/978-3-319-10575-8.

[18]

J-J. ClimentN. Thome and Y. Wei, A geometrical approach on generalized inverses by Neumann-type series, Linear Algebra and Its Applications, 1 (2001), 533-540.  doi: 10.1016/S0024-3795(01)00309-3.

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M. DaumasD. Lester and C. Muoz, Verified real number calculations, A library for interval arithmetic, IEEE Transactions on Computers, 58 (2009), 226-237.  doi: 10.1109/TC.2008.213.

[21]

A. Donze, Breach, a toolbox for verification and parameter synthesis of hybrid systems, Proc. of Computer Aided Verification, (2010), 167–170.

[22]

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.

[23]

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.

[24]

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.

[25]

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.

[26]

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.

[27]

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

[28]

M. Gameiro and P. Manolios, Formally verifying an algorithm based on interval arithmetic for checking transversality, Workshop on ACL2 Prover and Applications, 2004.

[29]

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.

[30]

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

[31]

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.

[32]

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

[33]

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.

[34]

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.

[35]

S. Kong, S. Gao and W. Chen, Reachability analysis for hybrid systems, Proc. of TACAS15, LNCS, Springer, 9035 (2015), 200–205.

[36]

M. Konstantinov, D. Gu, V. Mehrmann and P. Petkov, Perturbation Theory for Matrix Equations, 2$^{nd}$ edition, Elsevier, Amsterdam, 2003, ISBN-9780444513151.

[37]

G. A. KumarT. V. SubbareddyB. M. ReddN. Raju and V. Elamaran, An approach to design a matrix inversion hardware module using FPGA, Int. Conf. on Control, Instrumentation, Comm. and Comput. Technologies, 230 (2014), 87-90. 

[38]

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

[39]

LAPACK-Linear Algebra PACKage, 2019. Available from: http://www.netlib.org/lapack/.

[40]

W. Levine, The Control Handbook, IEEE Press, 1996.

[41]

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

[42]

Matlab-Mathworks, 2019. Available from: https://www.mathworks.com

[43]

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.

[44]

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.

[45]

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.

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R. E. Moore, Methods and Applications of Interval Analysis, SIAM Studies in Applied and Numerical Mathematics, Philadelphia, 1987, ISBN-10: 0898711614.

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R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, SIAM, 2009. doi: 10.1137/1.9780898717716.

[48]

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. doi: 10.1007/11541868_13.

[49]

T. Nahhal and T. Dang, Test coverage for continuous and hybrid systems, Int. Conf. on Computer Aided Verification, LNCS, 4590 (2007), 449-462. 

[50]

I. Pasca, Formal Verifcation for Numerical Methods, PhD Thesis in Universit Nice Sophia Antipolis, France, 2010.

[51]

P. Petkov, M. Konstantinov and N. Christov, Computational Methods for Linear Control Systems, 2$^{nd}$ edition, Prentice- Hall, Hemel Hempstead, 1991, ISBN-9780444513151.

[52]

M. D. Petkovic and P. S. Stanimirovic, Generalized matrix inversion is not harder than matrix multiplication, J. Comput. and Applied Math., 230 (2009), 270-282.  doi: 10.1016/j.cam.2008.11.012.

[53]

M. D. Petkovic and P. S. Stanimirovic, Iterative method for computing Moore-Penrose inverse based on Penrose equations, J. Comput. and Applied Math., 235 (2011), 1604-1613.  doi: 10.1016/j.cam.2010.08.042.

[54]

M. S. Petkovic, Iterative methods for bounding the inverse of a matrix (a survey), FILOMAT Nis, Algebra Logic & Discrete Mathematics, 9 (1995), 543-577. 

[55]

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Figure 1.  Hybrid automaton for a $ 1 \times 2 $ matrix $ A $
Figure 2.  Hybrid automaton for a $ 2 \times 2 $ matrix $ A $
Figure 3.  Hybrid automaton for a $ 2 \times 3 $ matrix $ A $
Figure 4.  Affine arithmetic approach for bound convergence expression (34) for a $ 2 \times 2 $ interval matrix
Figure 5.  Affine arithmetic approach for bound convergence expression (34) for a $ 2 \times 3 $ interval matrix
Figure 6.  Variation of condition number of $ A $ as function of $ {A_{11}} $
Figure 7.  $ 43 \times 68 $ matrix lpkb2 University of Florida sparse matrix collection
Figure 8.  $ 43 \times 68 $ matrix example in Breach
Figure 9.  Computational reliability monitor for $ 43 \times 68 $ matrix Example 6
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