Deriving Algorithms for Triangular Tridiagonalization a Skew-Symmetric Matrix

TL;DR

The paper addresses the problem of triangular tridiagonalization of skew-symmetric matrices by deriving a spectrum of $LTL^T$ factorization algorithms within the FLAME framework. It systematically builds unblocked and blocked variants, including right-looking, left-looking, and Wimmer-inspired approaches, and extends them with fused computations to reduce memory traffic; pivoting is then incorporated into several variants. The core contributions are the Partitioned Matrix Expressions, loop invariants, and explicit derivations of multiple algorithmic variants (including a novel blocked left-looking method) with detailed update rules and complexity insights. The work provides a path toward high-performance, BLAS-3–friendly implementations, clarifies the connections between different algorithms, and offers a foundation for automatic generation of skew-symmetric factorizations in future libraries, while highlighting open questions about pivoting applicability. Overall, it broadens the toolkit for robust, efficient factorization of skew-symmetric matrices and their Pfaffians in scientific computing.

Abstract

This paper provides technical details regarding the application of the FLAME methodology to derive algorithms hand in hand with their proofs of correctness for the computation of the $ L T L^T $ decomposition (with and without pivoting) of a skew-symmetric matrix. The approach yields known as well as new algorithms, presented using the FLAME notation, enabling comparing and contrasting. A number of BLAS-like primitives are exposed at the core of the resulting unblocked and blocked algorithms.

Figures (8)

• Figure 1: The unblocked right-looking (modified Parlett-Reid) and left-looking (modified Aasen) algorithms.
• Figure 2: The FLAME methodology workflow.
• Figure 3: Worksheet for deriving the unblocked right-looking algorithm.
• Figure 4: Two-step unblocked (Wimmer's) algorithm.
• Figure 5: Blocked right- and left-looking algorithms. In the blocked right-looking algorithm, the suffix "_0" indicates that $l_{21}$, $\lambda_{31}$, and $l_{41}$ are implicitly equal to zero during the factorization sub-problem.

Theorems & Definitions (19)

• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• proof
• Theorem 2.4
• Definition 2.5
• Lemma 2.6
• Theorem 2.7
• Corollary 2.8
• Corollary 2.9