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On Two-Stage Householder Orthogonalization

Zhuang-Ao He, Meiyue Shao

Abstract

Two-stage orthogonalization is essential in numerical algorithms such as Krylov subspace methods. For this task we need to orthogonalize a matrix $A$ against another matrix $V$ with orthonormal columns. A common approach is to employ the block Gram--Schmidt algorithm. However, its stability largely depends on the condition number of $[V,A]$. While performing a Householder orthogonalization on $[V,A]$ is unconditionally stable, it does not utilize the knowledge that $V$ has orthonormal columns. To address these issues, we propose a two-stage Householder orthogonalization algorithm based on the generalized Householder transformation. Instead of explicitly orthogonalizing the entire $V$, our algorithm only needs to orthogonalizes a square submatrix of $V$. Theoretical analysis and numerical experiments demonstrate that our method is also unconditionally stable.

On Two-Stage Householder Orthogonalization

Abstract

Two-stage orthogonalization is essential in numerical algorithms such as Krylov subspace methods. For this task we need to orthogonalize a matrix against another matrix with orthonormal columns. A common approach is to employ the block Gram--Schmidt algorithm. However, its stability largely depends on the condition number of . While performing a Householder orthogonalization on is unconditionally stable, it does not utilize the knowledge that has orthonormal columns. To address these issues, we propose a two-stage Householder orthogonalization algorithm based on the generalized Householder transformation. Instead of explicitly orthogonalizing the entire , our algorithm only needs to orthogonalizes a square submatrix of . Theoretical analysis and numerical experiments demonstrate that our method is also unconditionally stable.
Paper Structure (20 sections, 8 theorems, 81 equations, 2 figures, 4 tables, 4 algorithms)

This paper contains 20 sections, 8 theorems, 81 equations, 2 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Existing approaches
  3. BCGS
  4. BMGS
  5. Cholesky-QR
  6. Two-stage Householder-QR
  7. Generalized Householder transformation
  8. Choices of $P$
  9. First choice
  10. Second choice
  11. Third choice
  12. Rounding error analysis
  13. Extension to non-standard inner products
  14. Numerical experiments
  15. Stability tests
  16. ...and 5 more sections

Key Result

Theorem 1

Let $X$, $Y\in\mathbb{C}^{n\times k}$ and $W=X-Y$. Suppose that $T=W^* X$ is nonsingular and $\lVert X^* X-Y^* Y\rVert_2\le\delta$. Then the matrix $H=I_n-WT^{-1}W^*$ satisfies and $HX=Y$. In particular, if $X^* X=Y^* Y$, then $H$ is unitary.

Figures (2)

  • Figure 1: Losses of orthogonality and relative residuals with different $\kappa_2(T)$'s.
  • Figure 2: Execution time relative to the Householder-QR performed on $[V,A]$. Cases 1--3 correspond to real arithmetic, and Cases 4--6 to complex arithmetic. In both settings $k_0=100$ and $k=50$, $100$, $200$.

Theorems & Definitions (23)

  • Theorem 1
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Remark 2
  • Remark 3
  • ...and 13 more