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.
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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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