LAProof: A Library of Formal Proofs of Accuracy and Correctness for Linear Algebra Programs

Ariel E. Kellison, Andrew W. Appel, Mohit Tekriwal, David Bindel

Abstract: The LAProof library provides formal machine-checked proofs of the accuracy of basic linear algebra operations: inner product using conventional multiply and add, inner product using fused multiply-add, scaled matrix-vector and matrix-matrix multiplication, and scaled vector and matrix addition. These proofs can connect to concrete implementations of low-level basic linear algebra subprograms; as a proof of concept we present a machine-checked correctness proof of a C function implementing sparse matrix-vector multiplication using the compressed sparse row format. Our accuracy proofs are backward error bounds and mixed backward-forward error bounds that account for underflow, proved subject to no assumptions except a low-level formal model of IEEE-754 arithmetic. We treat low-order error terms concretely, not approximating as $\mathcal{O}\left( {{u^2}} \right)$.