Abstract
Solving a sparse linear system of the form \(Ax=b\) is a common engineering task, e.g., as a step in approximating solutions of differential equations. Inverting a large matrix A is often too expensive, and instead engineers rely on iterative methods, which progressively approximate the solution x of the linear system in several iterations, where each iteration is a much less expensive (sparse) matrix-vector multiplication.
We present a formal proof in the Coq proof assistant of the correctness, accuracy and convergence of one prominent iterative method, the Jacobi iteration. The accuracy and convergence properties of Jacobi iteration are well-studied, but most past analyses were performed in real arithmetic; instead, we study those properties, and prove our results, in floating-point arithmetic. We then show that our results are properly reflected in a concrete implementation in the C language. Finally, we show that the iteration will not overflow, under assumptions that we make explicit. Notably, our proofs are faithful to the details of the implementation, including C program semantics and floating-point arithmetic.
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Notes
- 1.
A an \(n\times n\) matrix; b and x dimension n; \(0<n<2^{32}\); A, b, x all finite; A, b, x stored in memory in the right places—but nothing else about the values of A, b, x.
References
Ahlberg, J.H., Nilson, E.N.: Convergence properties of the spline fit. J. Soc. Indust. Appl. Math. 11, 95–104 (1963)
Appel, A.W., Bertot, Y.: C-language floating-point proofs layered with VST and Flocq. J. Formalized Reason. 13(1), 1–16 (2020)
Appel, A.W., Kellison, A.E.: VCFloat2: floating-point error analysis in Coq (2022). https://github.com/VeriNum/vcfloat/blob/master/doc/vcfloat2.pdf
Barrett, R., et al.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM (1994)
Boldo, S., Clément, F., Filliâtre, J.C., Mayero, M., Melquiond, G., Weis, P.: Wave equation numerical resolution: a comprehensive mechanized proof of a C program. J. Autom. Reason. 50(4), 423–456 (2013)
Boldo, S., Clément, F., Filliâtre, J.C., Mayero, M., Melquiond, G., Weis, P.: Trusting computations: a mechanized proof from partial differential equations to actual program. Comput. Math. Appl. 68(3), 325–352 (2014)
Boldo, S., Lelay, C., Melquiond, G.: Coquelicot: a user-friendly library of real analysis for Coq. Math. Comput. Sci. 9(1), 41–62 (2015)
Boldo, S., Melquiond, G.: Flocq: a unified library for proving floating-point algorithms in Coq. In: 2011 IEEE 20th Symposium on Computer Arithmetic, pp. 243–252. IEEE (2011)
Cano, G., Dénès, M.: Matrices à blocs et en forme canonique. In: Pous, D., Tasson, C. (eds.) JFLA - Journées francophones des langages applicatifs. Aussois, France (2013). https://hal.inria.fr/hal-00779376
Cao, Q., Beringer, L., Gruetter, S., Dodds, J., Appel, A.W.: VST-Floyd: a separation logic tool to verify correctness of C programs. J. Autom. Reason. 61(1–4), 367–422 (2018)
Garillot, F., Gonthier, G., Mahboubi, A., Rideau, L.: Packaging mathematical structures. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 327–342. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03359-9_23
Gleich, D.F.: Pagerank beyond the web. SIAM Rev. 57(3), 321–363 (2015)
Higham, N.J.: Accuracy and Stability of Numerical Algorithms. SIAM (2002)
Higham, N.J., Knight, P.A.: Componentwise error analysis for stationary iterative methods. In: Meyer, C.D., Plemmons, R.J. (eds.) Linear Algebra, Markov Chains, and Queueing Models, pp. 29–46. Springer, New York (1993). https://doi.org/10.1007/978-1-4613-8351-2_3
Immler, F.: A Verified ODE Solver and Smale’s 14th Problem. Dissertation, Technische Universität München, München (2018)
Immler, F., Hölzl, J.: Numerical analysis of ordinary differential equations in Isabelle/HOL. In: Beringer, L., Felty, A. (eds.) ITP 2012. LNCS, vol. 7406, pp. 377–392. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32347-8_26
Immler, F., Traut, C.: The flow of ODEs: formalization of variational equation and Poincaré map. J. Autom. Reason. 62(2), 215–236 (2019)
Katz, L.: A new status index derived from sociometric analysis. Psychometrika 18(1), 39–43 (1953)
Kellison, A., Tekriwal, M., Jeannin, J.B., Hulette, G.: Towards verified rounding error analysis for stationary iterative methods. In: 2022 IEEE/ACM Sixth International Workshop on Software Correctness for HPC Applications (Correctness), pp. 10–17 (2022). https://doi.org/10.1109/Correctness56720.2022.00007
Kellison, A.E., Appel, A.W.: Verified numerical methods for ordinary differential equations. In: 15th International Workshop on Numerical Software Verification (2022)
Kellison, A.E., Appel, A.W., Tekriwal, M., Bindel, D.: LAProof: a library of formal accuracy and correctness proofs for sparse linear algebra programs (2023). https://www.cs.princeton.edu/~appel/papers/LAProof.pdf
Mahboubi, A., Tassi, E.: Mathematical components. Online book (2021)
Martin-Dorel, É., Rideau, L., Théry, L., Mayero, M., Pasca, I.: Certified, efficient and sharp univariate Taylor models in Coq. In: 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, pp. 193–200. IEEE (2013)
McKenzie, L.: Matrices with dominant diagonals and economic theory. In: Arroa, K., Karlin, S., Puppes, S. (eds.) Mathematical Methods in the Social Sciences, pp. 47–60. Stanford University Press (1960)
O’Connor, R.: Certified exact transcendental real number computation in Coq. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 246–261. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-71067-7_21
Pasca, I.: Formal verification for numerical methods. Ph.D. thesis, Université Nice Sophia Antipolis (2010)
Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM (2003)
Tekriwal, M., Duraisamy, K., Jeannin, J.-B.: A formal proof of the lax equivalence theorem for finite difference schemes. In: Dutle, A., Moscato, M.M., Titolo, L., Muñoz, C.A., Perez, I. (eds.) NFM 2021. LNCS, vol. 12673, pp. 322–339. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-76384-8_20
Tekriwal, M., Miller, J., Jeannin, J.B.: Formal verification of iterative convergence of numerical algorithms (2022). https://doi.org/10.48550/arXiv.2202.05587
Thiemann, R.: A Perron-Frobenius theorem for deciding matrix growth. J. Log. Algebraic Methods Program. 100699 (2021)
Acknowledgement
We thank Yves Bertot for feedback on earlier drafts of this paper. This research was supported in part by NSF Grants CCF-2219997 and CCF-2219757, by a US Department of Energy Computational Science Fellowship DE-SC0021110, and by the Chateaubriand fellowship program.
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Tekriwal, M., Appel, A.W., Kellison, A.E., Bindel, D., Jeannin, JB. (2023). Verified Correctness, Accuracy, and Convergence of a Stationary Iterative Linear Solver: Jacobi Method. In: Dubois, C., Kerber, M. (eds) Intelligent Computer Mathematics. CICM 2023. Lecture Notes in Computer Science(), vol 14101. Springer, Cham. https://doi.org/10.1007/978-3-031-42753-4_14
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