Verified Correctness, Accuracy, and Convergence of a Stationary Iterative Linear Solver: Jacobi Method

Tekriwal, Mohit; Appel, Andrew W.; Kellison, Ariel E.; Bindel, David; Jeannin, Jean-Baptiste

doi:10.1007/978-3-031-42753-4_14

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2024

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Cited by 2 publications

References 22 publications

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Formalization of Asymptotic Convergence for Stationary Iterative Methods

Tekriwal,

Miller,

Jeannin

2024

Lecture Notes in Computer Science

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Formalization of Asymptotic Convergence for Stationary Iterative Methods

Tekriwal,

Miller,

Jeannin

2024

Lecture Notes in Computer Science

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Numerical Fuzz: A Type System for Rounding Error Analysis

Kellison,

Hsu

2024

Proc. ACM Program. Lang.

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Algorithms operating on real numbers are implemented as floating-point computations in practice, but floating-point operations introduce roundoff errors that can degrade the accuracy of the result. We propose Λ num , a functional programming language with a type system that can express quantitative bounds on roundoff error. Our type system combines a sensitivity analysis, enforced through a linear typing discipline, with a novel graded monad to track the accumulation of roundoff errors. We prove that our type system is sound by relating the denotational semantics of our language to the exact and floating-point operational semantics. To demonstrate our system, we instantiate Λ num with error metrics proposed in the numerical analysis literature and we show how to incorporate rounding operations that faithfully model aspects of the IEEE 754 floating-point standard. To show that Λ num can be a useful tool for automated error analysis, we develop a prototype implementation for Λ num that infers error bounds that are competitive with existing tools, while often running significantly faster. Finally, we consider semantic extensions of our graded monad to bound error under more complex rounding behaviors, such as non-deterministic and randomized rounding.

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Verified Correctness, Accuracy, and Convergence of a Stationary Iterative Linear Solver: Jacobi Method

Cited by 2 publications

References 22 publications

Formalization of Asymptotic Convergence for Stationary Iterative Methods

Formalization of Asymptotic Convergence for Stationary Iterative Methods

Numerical Fuzz: A Type System for Rounding Error Analysis

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