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Orthonormal integrators based on Householder and Givens transformations

Luca Dieci, Erik S. Van Vleck

TL;DR

The article develops orthonormal integrators for QR-flow on the Stiefel manifold by expressing the evolving orthonormal factor $Q$ through Householder reflectors or Givens rotations, and by using a reimbedding strategy to ensure stable local coordinates. It introduces new formulations, notably $w$-variables for Householder and $\theta$-variables for Givens, and provides a complete FORTRAN code suite with fixed and adaptive Runge-Kutta schemes to test the methods on dense coefficient matrices. Numerical experiments indicate that variable-step Dormand–Prince schemes with these representations achieve high accuracy and robustness, often outperforming projection-based RKF45, while $u$-variable Householder representations may be unstable. The work demonstrates practical effectiveness for solving $\dot X = A(t)X$ with $X=QR$, delivering orthonormal solutions efficiently and suggesting directions for optimization and structured-matrix extensions.

Abstract

We carry further our work [DV2] on orthonormal integrators based on Householder and Givens transformations. We propose new algorithms and pay particular attention to appropriate implementation of these techniques. We also present a suite of Fortran codes and provide numerical testing to show the efficiency and accuracy of our techniques.

Orthonormal integrators based on Householder and Givens transformations

TL;DR

The article develops orthonormal integrators for QR-flow on the Stiefel manifold by expressing the evolving orthonormal factor through Householder reflectors or Givens rotations, and by using a reimbedding strategy to ensure stable local coordinates. It introduces new formulations, notably -variables for Householder and -variables for Givens, and provides a complete FORTRAN code suite with fixed and adaptive Runge-Kutta schemes to test the methods on dense coefficient matrices. Numerical experiments indicate that variable-step Dormand–Prince schemes with these representations achieve high accuracy and robustness, often outperforming projection-based RKF45, while -variable Householder representations may be unstable. The work demonstrates practical effectiveness for solving with , delivering orthonormal solutions efficiently and suggesting directions for optimization and structured-matrix extensions.

Abstract

We carry further our work [DV2] on orthonormal integrators based on Householder and Givens transformations. We propose new algorithms and pay particular attention to appropriate implementation of these techniques. We also present a suite of Fortran codes and provide numerical testing to show the efficiency and accuracy of our techniques.

Paper Structure

This paper contains 7 sections, 3 theorems, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and Householder and Givens representations
  3. Householder transformations
  4. Givens transformations
  5. Implementation
  6. Codes & Examples
  7. Conclusions

Key Result

Lemma 2.1

The choice (sigmas) is the same as (idealsigma).

Figures (1)

  • Figure 1: Plot of $t$ vs. $\log_{10}(||D(t)-\diag(\tilde{A}(t))||_\infty)$ for vtdp5 applied to Example \ref{['Ex7']} for different tolerances using a "random" $Q(0)$.

Theorems & Definitions (4)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • proof