Numerical Performance of the Implicitly Restarted Arnoldi Method in OFP8, Bfloat16, Posit, and Takum Arithmetics

TL;DR

This work assesses the numerical performance of emerging number formats for eigenvalue computations using the implicitly restarted Arnoldi method. It employs the MuFoLAB framework with ARPACK to compare OFP8, bfloat16, posits, and takums against IEEE 754 across general matrices and graph Laplacians from real-world datasets, using a float128 reference and a permutation-based eigenvector alignment strategy. Key findings show that takum arithmetic consistently yields superior accuracy, while OFP8 is generally unsuitable and posits underperform at higher precisions; bfloat16 remains worse than float16. The results provide practical guidance on format choices for large-scale eigenvalue problems and highlight takums as a promising alternative for robust numerical linear algebra in graph analytics.

Abstract

The computation of select eigenvalues and eigenvectors of large, sparse matrices is fundamental to a wide range of applications. Accordingly, evaluating the numerical performance of emerging alternatives to the IEEE 754 floating-point standard -- such as OFP8 (E4M3 and E5M2), bfloat16, and the tapered-precision posit and takum formats -- is of significant interest. Among the most widely used methods for this task is the implicitly restarted Arnoldi method, as implemented in ARPACK. This paper presents a comprehensive and untailored evaluation based on two real-world datasets: the SuiteSparse Matrix Collection, which includes matrices of varying sizes and condition numbers, and the Network Repository, a large collection of graphs from practical applications. The results demonstrate that the tapered-precision posit and takum formats provide improved numerical performance, with takum arithmetic avoiding several weaknesses observed in posits. While bfloat16 performs consistently better than float16, the OFP8 types are generally unsuitable for general-purpose computations.

Figures (5)

• Figure 1: Cumulative error distribution of the relative errors of the $10$ largest eigenvalues (left) and their corresponding eigenvectors (right) of the general matrices computed using a range of machine number types. The symbol $\infty_\omega$ denotes where the Arnoldi method did not converge, $\infty_\sigma$ denotes where the dynamic range of the matrix entries exceeded the target number type.
• Figure 2: Cumulative error distribution of the relative errors of the $10$ largest eigenvalues (left) and their corresponding eigenvectors (right) of the biological graph symmetrized, normalized Laplacian matrices computed using a range of machine number types. The symbol $\infty_\omega$ denotes where the Arnoldi method did not converge.
• Figure 3: Cumulative error distribution of the relative errors of the $10$ largest eigenvalues (left) and their corresponding eigenvectors (right) of the infrastructure graph symmetrized, normalized Laplacian matrices computed using a range of machine number types. The symbol $\infty_\omega$ denotes where the Arnoldi method did not converge.
• Figure 4: Cumulative error distribution of the relative errors of the $10$ largest eigenvalues (left) and their corresponding eigenvectors (right) of the social graph symmetrized, normalized Laplacian matrices computed using a range of machine number types. The symbol $\infty_\omega$ denotes where the Arnoldi method did not converge.
• Figure 5: Cumulative error distribution of the relative errors of the $10$ largest eigenvalues (left) and their corresponding eigenvectors (right) of the miscellaneous graph symmetrized, normalized Laplacian matrices computed using a range of machine number types. The symbol $\infty_\omega$ denotes where the Arnoldi method did not converge, $\infty_\sigma$ denotes where the dynamic range of the matrix entries exceeded the target number type.