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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20 Mar 2003
TL;DR: The Cholesky and QR decompositions are reformulated to avoid the many square-root and division operations required in their natural form by decoupling the numerator and denominator calculations and applying scaling by powers of 2 to ensure numerical stability in the recursions.
Abstract: Motivated by the need for the Cholesky factorization in implementing a spherical MIMO detector, this paper considers Cholesky and QR decompositions suitable for fixed-point implementation. In particular, we reformulate the decompositions to avoid the many square-root and division operations required in their natural form. This is achieved by decoupling the numerator and denominator calculations and applying scaling by powers of 2 (corresponding to bit shifts) to ensure numerical stability in the recursions. We consider the impact on the spherical detector formulation.
70 citations
TL;DR: In this paper, an improved stochastic subspace identification algorithm is introduced to solve the low computational efficiency problem of the Data-driven Stochastic Subspace Identification (DSSA) problem.
70 citations
TL;DR: The proposed IQRD hardware is constructed by the diagonal and the triangular process with fewer gate counts and lower power consumption than TSAQRD, and the total clock latency is only 10 m - 5 cycles.
Abstract: Implementation of an iterative QR decomposition (QRD) (IQRD) architecture based on the modified Gram-Schmidt (MGS) algorithm is proposed in this paper. A QRD is extensively adopted by the detection of multiple-input-multiple-output systems. In order to achieve computational efficiency with robust numerical stability, a triangular systolic array (TSA) for QRD of large-size matrices is presented. In addition, the TSA architecture can be modified into an iterative architecture that is called IQRD for reducing hardware cost. The IQRD hardware is constructed by the diagonal and the triangular process with fewer gate counts and lower power consumption than TSAQRD. For a 4 t 4 matrix, the hardware area of the proposed IQRD can reduce about 41% of the gate counts in TSAQRD. For a generic square matrix of order m IQRD, the latency required is 2m - 1 time units, which is based on the MGS algorithm. Thus, the total clock latency is only 10 m - 5 cycles.
70 citations
TL;DR: In this paper, the authors extended the Francis QR algorithm to quaternion and antiquaternion matrices, and they used unitary similarity transformations to calculate the eigenvalues of nearby quaternions despite rounding errors.
Abstract: This paper extends the Francis QR algorithm to quaternion and antiquaternion matrices. It calculates a quaternion version of the Schur decomposition using quaternion unitary similarity transformations. Following a finite step reduction to a Hessenberg-like condensed form, a sequence of implicit QR steps reduces the matrix to triangular form. Eigenvalues may be read off the diagonal. Eigenvectors may be obtained from simple back substitutions. For serial computation, the algorithm uses only half the work and storage of the unstructured Francis QR iteration. By preserving quaternion structure, the algorithm calculates the eigenvalues of a nearby quaternion matrix despite rounding errors.
69 citations
05 Mar 2006
TL;DR: A class of simple Jacobi-type algorithms for non-orthogonal matrix joint diagonalization based on the LU or QR factorization is introduced, which replaces a high dimensional minimization problem by a sequence of simple onedimensional minimization problems.
Abstract: A class of simple Jacobi-type algorithms for non-orthogonal matrix joint diagonalization based on the LU or QR factorization is introduced. By appropriate parametrization of the underlying manifolds, i.e. using triangular and orthogonal Jacobi matrices we replace a high dimensional minimization problem by a sequence of simple one dimensional minimization problems. In addition, a new scale-invariant cost function for non-orthogonal joint diagonalization is employed. These algorithms are step-size free. Numerical simulations demonstrate the efficiency of the methods.
68 citations