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arXiv · 2610.30227

DBDSVDMR3: An MRRR-Based Bidiagonal SVD Solver Built from DSTEGR

James Demmel·Inderjit Dhillon·Sai Surya Duvvuri·Julien Langou·Osni Marques·Ryan Schneider
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Abstract

The optimal algorithm for the symmetric tridiagonal eigenvalue problem is Multiple Relatively Robust Representations (MRRR or MR$^3$), which is capable of computing $k$ eigenvalue/eigenvector pairs in only $O(kn)$ floating-point operations. While MRRR is nearly 30 years old, it has never successfully been adapted to the bidiagonal SVD. This is not for lack of trying; two previous attempts produced implementations that were considered for LAPACK but eventually abandoned, a result of the fact that neither was reliable enough when run on test problems. The question therefore remained: is there a problem with the theory of MRRR for the bidiagonal SVD or were there simply bugs in the previous implementations that went undetected? This paper resolves this question by presenting DBDSVDMR3 -- a new implementation of the MRRR-based algorithm for the bidiagonal SVD originally proposed by Willems and Lang [ETNA 2012], which, for the first time, performs well on a variety of test problems while achieving significant speedups over both LAPACK's bisection-based solver and, in certain cases, divide-and-conquer for the bidiagonal SVD (thereby indicating that the theory is sound). DBDSVDMR3 will be available in the next release of LAPACK. In this paper, we discuss the design of our implementation, which was obtained with the help of Claude Code, and exhibit its performance on a diverse set of test problems.

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James Demmel, Inderjit Dhillon, Sai Surya Duvvuri, Julien Langou, Osni Marques, Ryan Schneider. 2026-10-21. DBDSVDMR3: An MRRR-Based Bidiagonal SVD Solver Built from DSTEGR. https://arxiv.org/abs/2610.30227

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