[Paper Review] Real dqds for the nonsymmetric tridiagonal eigenvalue problem

Carla Ferreira, Beresford Ν. Parlett|arXiv (Cornell University)|Jan 24, 2012

Matrix Theory and Algorithms18 references3 citations

TL;DR

This paper introduces a real dqds algorithm for computing eigenvalues of real nonsymmetric tridiagonal matrices using only real arithmetic, enabling efficient complex shifts via a triple-step transform. It achieves performance comparable to complex dqds codes and is up to 100 times faster than the Ehrlich-Aberth method, with robust element growth control and dynamic transform rejection for stability and speed.

ABSTRACT

We present a new transform, triple dqds, to help to compute the eigenvalues of a real tridiagonal matrix C using real arithmetic. The algorithm uses the real dqds transform to shift by a real number and triple dqds to shift by a complex conjugate pair. We present what seems to be a new criteria for splitting the current pair L,U. The algorithm rejects any transform which suffers from excessive element growth and then tries a new transform. Our numerical tests show that the algorithm is about 100 times faster than the Ehrlich-Aberth method of D. A. Bini, L. Gemignani and F. Tisseur. Our code is comparable in performance to a complex dqds code and is sometimes 3 times faster.

Motivation & Objective

• To develop a fast and accurate eigenvalue solver for real nonsymmetric tridiagonal matrices using only real arithmetic.
• To enable complex shifts (via conjugate pairs) while avoiding complex arithmetic, preserving numerical stability and efficiency.
• To design a robust transform rejection mechanism that detects and discards steps with excessive element growth.
• To achieve performance competitive with complex dqds algorithms while maintaining real arithmetic advantages.
• To provide a practical, high-performance alternative to existing methods like Ehrlich-Aberth and complex QR/dqds for nonsymmetric tridiagonal eigenvalue problems.

Proposed method

• The algorithm uses a triple-step transform called tridqds, combining real dqds and a complex shift via conjugate pairs to simulate double shifts in real arithmetic.
• It applies a sequence of similarity transformations using matrices $ Z_i $, $ ilde{L}_i^{-1} $, $ ilde{L}_i $, $ Y_i^{-1} $, and $ Y_i $, with recursive updates to the state variables $ x_r, y_r, z_r, x_l, y_l $.
• Element growth is monitored via the ratio $ \hat{u}_i / \text{max}(u_i, l_i) $, and transforms are rejected if growth exceeds a threshold.
• The algorithm uses a dynamic shift strategy based on the current $ L,U $ pair, selecting shifts to maximize convergence while avoiding instability.
• The method is implemented in MATLAB and includes deflation and splitting criteria to isolate converged eigenvalues.
• The core computation uses recursive updates of the $ x, y, z $ state vectors, with normalization and scaling at each step to maintain numerical stability.

Experimental results

Research questions

• RQ1 Can a real arithmetic algorithm achieve performance comparable to complex dqds for nonsymmetric tridiagonal eigenvalue problems?
• RQ2 How can complex shifts be effectively applied in real arithmetic without introducing numerical instability?
• RQ3 What criteria can reliably detect and reject transforms with excessive element growth to ensure robustness?
• RQ4 How does the triple-step tridqds transform compare in speed and accuracy to existing methods like Ehrlich-Aberth and complex dqds?
• RQ5 Can the algorithm maintain high relative accuracy while avoiding the computational cost of complex arithmetic?

Key findings

• The proposed real tridqds algorithm is approximately 100 times faster than the Ehrlich-Aberth method of Bini, Gemignani, and Tisseur.
• The algorithm's performance is comparable to complex dqds codes, and in some cases it is up to three times faster.
• The dynamic rejection of transforms with excessive element growth significantly improves numerical stability and convergence reliability.
• The algorithm successfully computes all eigenvalues of real nonsymmetric tridiagonal matrices using only real arithmetic, even when eigenvalues are complex conjugate pairs.
• The splitting and deflation strategy effectively isolates converged eigenvalues, enabling efficient computation of the full spectrum.
• The method maintains high relative accuracy, consistent with the known benefits of dqds for singular value computation, and extends them to the nonsymmetric eigenvalue problem.