QR Factorizations Using a Restricted Set of Rotations

Published

January 6, 2005

Author(s)

Dianne M. O'Leary, Stephen Bullock

Abstract

Any matrix of dimension m x n (m greater than or equal to n) can be reduced to upper triangular form by multiplying by a sequence of mn - n(n + 1) / 2 appropriately chosen rotation matrices. In this work, we address the question of whether such a factorization exists when the set of allowed rotation planes is restricted. We introduce the rotation graph as a tool to devise elimination orderings in QR factorizations. Properties of this graph characterize sets of rotation planes that are sufficient (or sufficient under permutation) and identify rotation planes to add to complete a deficient set. We also devise a constructive way to determine all feasible rotation sequences for performing the QR factorization using a restricted set of rotation planes. We present applications to quantum circuit design and parallel QR factorization.

Citation

Electronic Transactions on Numerical Analysis

Volume

21

Pub Type

Journals

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Keywords

Givens rotations, parallel QR decomposition, plane rotations, QR decomposition, quantum circuit design, qudits

Citation

O'Leary, D. and Bullock, S. (2005), QR Factorizations Using a Restricted Set of Rotations, Electronic Transactions on Numerical Analysis, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=150891 (Accessed October 27, 2025)