Exact QR factorizations of rectangular matrices

Abstract

QR factorization is a key tool in mathematics, computer science, operations research, and engineering. This paper presents the roundoff-error-free (REF) QR factorization framework comprising integer-preserving versions of the standard and the thin QR factorizations and associated algorithms to compute them. Specifically, the standard REF QR factorization factors a given matrix \(A\in {\mathbb {Z}}^{m\times n}\) as \(A=QDR\), where \(Q\in {\mathbb {Z}}^{m\times m}\) has pairwise orthogonal columns, D is a diagonal matrix, and \(R\in {\mathbb {Z}}^{m\times n}\) is an upper trapezoidal matrix; notably, the entries of Q and R are integral, while the entries of D are reciprocals of integers. In the thin REF QR factorization, \(Q\in {\mathbb {Z}}^{m\times n}\) also has pairwise orthogonal columns, and \(R\in {\mathbb {Z}}^{n\times n}\) is also an upper triangular matrix. In contrast to traditional (i.e., floating-point) QR factorizations, every operation used to compute these factors is integral; thus, REF QR is guaranteed to be an exact orthogonal decomposition. Importantly, the bit-length of every entry in the REF QR factorizations (and within the algorithms to compute them) is bounded polynomially. Notable applications of our REF QR factorizations include finding exact least squares or exact basic solutions, \({\textbf{x}}\in {\mathbb {Q}}^n\), to any given full column rank or rank deficient linear system \(A {\textbf{x}}= {\textbf{b}}\), respectively. In addition, our exact factorizations can be used as a subroutine within exact and/or high-precision quadratic programming. Altogether, REF QR provides a framework to obtain exact orthogonal factorizations of any rational matrix (as any rational/decimal matrix can be easily transformed into an integral matrix).