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.

A fast and well-conditioned spectral method for singular integral equations

Abstract: We develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing Chebyshev and ultraspherical polynomials to reformulate the equations as almost-banded infinite-dimensional systems. This is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels. The resulting system can be solved in ${\cal O}(m^2n)$ operations using an adaptive QR factorization, where $m$ is the bandwidth and $n$ is the optimal number of unknowns needed to resolve the true solution. The complexity is reduced to ${\cal O}(m n)$ operations by pre-caching the QR factorization when the same operator is used for multiple right-hand sides. Stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity. Applications considered include the Faraday cage, and acoustic scattering for the Helmholtz and gravity Helmholtz equations, including spectrally accurate numerical evaluation of the far- and near-field solution. The Julia software package SingularIntegralEquations.jl implements our method with a convenient, user-friendly interface.

A Very Low Complexity QRD-M Algorithm Based on Limited Tree Search for MIMO Systems

Abstract: We present a very low complexity QRD-M algorithm for MIMO systems. The original QRD-M algorithm decomposes the MIMO channel matrix into upper triangular matrix and applies a limited tree search. To accomplish near- MLD(Maximum Likelihood Detection) performance for QRD-M algorithm, number of search points at each layer must be the modulation size. In the proposed scheme, each of survival branches are extended only to the corresponding QR decomposition (QRD)-based detection symbol in the next layer and its neighboring symbols in the constellation. Using this approach, we can significantly decrease the complexity of conventional QRD-M algorithm. Simulation results show that the proposed algorithm scheme achieves the detection performance near to that of the MLD with negligibly low complexity.

A Fixed-Point Implementation for QR Decomposition

Abstract: Matrix triangularization and orthogonalization are prerequisites to solving least square problems and find applications in a wide variety of communication systems and signal processing applications such as MIMO systems and matrix inversion. QR decomposition using modified Gram-Schmidt (MGS) orthogonalization is one of the numerically stable techniques used in this regard. This paper presents a fixed point implementation of QR decomposition based on MGS algorithm using a novel LUT based approach. The proposed architecture is based on log-domain arithmetic operations. The error performance of various fixed-point arithmetic operations has been discussed and optimum LUT sizes are presented based on simulation results for various fractional-precisions. The proposed architecture also paves way for an efficient parallel VLSI implementation of QR decomposition using MGS

Tiled QR factorization algorithms

Abstract: This work revisits existing algorithms for the QR factorization of rectangular matrices composed of p-by-q tiles, where p >= q. Within this framework, we study the critical paths and performance of algorithms such as Sameh and Kuck, Modi and Clarke, Greedy, and those found within PLASMA. Although neither Modi and Clarke nor Greedy is optimal, both are shown to be asymptotically optimal for all matrices of size p = q^2 f(q), where f is any function such that \lim_{+\infty} f= 0. This novel and important complexity result applies to all matrices where p and q are proportional, p = \lambda q, with \lambda >= 1, thereby encompassing many important situations in practice (least squares). We provide an extensive set of experiments that show the superiority of the new algorithms for tall matrices.