## Roundoff error analysis of the CholeskyQR2 algorithm

Yusaku Yamamoto, Yuji Nakatsukasa, Yuka Yanagisawa, and Takeshi Fukaya

### Abstract

We consider the QR decomposition of an $m\times n$ matrix $X$ with full column rank, where $m\ge n$. Among the many algorithms available, the Cholesky QR algorithm is ideal from the viewpoint of high performance computing since it consists entirely of standard level 3 BLAS operations with large matrix sizes, and requires only one reduce and broadcast in parallel environments. Unfortunately, it is well-known that the algorithm is not numerically stable and the deviation from orthogonality of the computed $Q$ factor is of order $O((\kappa_2(X))^2{\bf u})$, where $\kappa_2(X)$ is the 2-norm condition number of $X$ and ${\bf u}$ is the unit roundoff. In this paper, we show that if the condition number of $X$ is not too large, we can greatly improve the stability by iterating the Cholesky QR algorithm twice. More specifically, if $\kappa_2(X)$ is at most $O({\bf u}^{-\frac{1}{2}})$, both the residual and deviation from orthogonality are shown to be of order $O({\bf u})$. Numerical results support our theoretical analysis.

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### Key words

QR decomposition, Cholesky QR, communication-avoiding algorithms, roundoff error analysis.

### AMS subject classifications

15A23, 65F25, 65G50.

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