Shifted CholeskyQR for computing the QR factorization of ill-conditioned matrices

TL;DR

This paper introduces shiftedCholeskyQR, an improved algorithm for QR factorization of ill-conditioned matrices that extends applicability, enhances stability, and achieves significant speedups over existing methods.

Contribution

It extends the applicability of CholeskyQR to higher condition numbers by introducing a shift, ensuring numerical stability and enabling efficient parallel computation.

Findings

Achieves stable QR factorization for matrices with condition number up to O(u^{-1})

Provides theoretical analysis confirming numerical stability and orthogonality

Demonstrates up to 40x speedup over alternative methods in experiments

Abstract

The Cholesky QR algorithm is an efficient communication-minimizing algorithm for computing the QR factorization of a tall-skinny matrix. Unfortunately it has the inherent numerical instability and breakdown when the matrix is ill-conditioned. A recent work establishes that the instability can be cured by repeating the algorithm twice (called CholeskyQR2). However, the applicability of CholeskyQR2 is still limited by the requirement that the Cholesky factorization of the Gram matrix runs to completion, which means it does not always work for matrices $X$ with $\kappa_2(X)\gtrsim {{\bf u}}^{-\frac{1}{2}}$ where ${{\bf u}}$ is the unit roundoff. In this work we extend the applicability to $\kappa_2(X)=\mathcal{O}({\bf u}^{-1})$ by introducing a shift to the computed Gram matrix so as to guarantee the Cholesky factorization $R^TR= A^TA+sI$ succeeds numerically. We show that the computed $AR^{-1}$ has reduced condition number $\leq {{\bf u}}^{-\frac{1}{2}}$, for which CholeskyQR2 safely computes the QR factorization, yielding a computed $Q$ of orthogonality $\|Q^TQ-I\|_2$ and residual $\|A-QR\|_F/\|A\|_F$ both $\mathcal{O}({{\bf u}})$. Thus we obtain the required QR factorization by essentially running Cholesky QR thrice. We extensively analyze the resulting algorithm shiftedCholeskyQR to reveal its excellent numerical stability. shiftedCholeskyQR is also highly parallelizable, and applicable and effective also when working in an oblique inner product space. We illustrate our findings through experiments, in which we achieve significant (up to x40) speedup over alternative methods.