Topic
QR decomposition
About: QR decomposition is a research topic. Over the lifetime, 3504 publications have been published within this topic receiving 100599 citations. The topic is also known as: QR factorization.
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TL;DR: This article reviews low rank approximation techniques briefly and gives extensive references of many techniques which give the low-rank approximation with linear complexity in n.
Abstract: Low rank approximation of matrices has been well studied in literature. Singular value decomposition, QR decomposition with column pivoting, rank revealing QR factorization, Interpolative decomposi...
111 citations
TL;DR: By repeatedly applying the Wedderburn rank-one reduction formula to reduce ranks, a biconjugation process analogous to the Gram–Schmidt process with oblique projections can be developed.
Abstract: Let $A \in R^{m \times n} $ denote an arbitrary matrix. If $x \in R^n $ and $y \in R^m $ are vectors such that $\omega = y^T Ax
e 0$, then the matrix $B: = A - \omega ^{ - 1} Axy^T A$ A has rank exactly one less than the rank of A. This Wedderburn rank-one reduction formula is easy to prove, yet the idea is so powerful that perhaps all matrix factorizations can be derived from it. The formula also appears in places such as the positive definite secant updates BFGS and DFP as well as the ABS methods. By repeatedly applying the formula to reduce ranks, a biconjugation process analogous to the Gram–Schmidt process with oblique projections can be developed. This process provides a mechanism for constructing factorizations such as ${\text{LDM}}^T $, QR, and SVD under a common framework of a general biconjugate decomposition $V^T AU = \Omega $ that is diagonal and nonsingular. Two characterizations of biconjugation provide new insight into the Lanczos method and its breakdown. One characterization shows that ...
110 citations
TL;DR: In this article, a matrix pencil factorization algorithm based on QR decomposition of a rectangular Hankel matrix is proposed for noiseless sampled data, which can be applied to sparse Fourier approximation and nonlinear approximation.
110 citations
TL;DR: In this article, the authors studied the use of unitary unitary similarity transformations in the QR decomposition, where S is either unitary or unitary symplectic, respectively.
110 citations
TL;DR: The Hamiltonian-Schur decomposition (HSC) decomposition as discussed by the authors is a variant of Schur decompositions for Hamiltonian matrices that arise from single input control systems.
Abstract: This paper presents a variant $QR$ algorithm for calculating a Hamiltonian–Schur decomposition [10]. It is defined for Hamiltonian matrices that arise from single input control systems. Numerical stability and Hamiltonian structure are preserved by using unitary symplectic similarity transformations. Following a finite step reduction to a Hessenberg-like condensed form, a sequence of similarity transformations yields a permuted triangular matrix. As the iteration converges, it deflates into problems of lower dimension. Convergence is accelerated by varying a scalar shift. When the Hamiltonian matrix is real, complex arithmetic can be avoided by using an implicit double shift technique. The Hamiltonian-Schur decomposition yields the same invariant subspace information as a Schur decomposition but requires significantly less work and storage for problems of size greater than about 20.
109 citations