3,745 research outputs found
HOUSEHOLDER REDUCTION
This tutorial discusses Householder reduction of n linear equations to a triangular form which can be solved by back substitution. The main strengths of the method are its numerical stability and suitability for parallel computing. We explain how Householder reduction can be derived from elementary matrix algebra. The method is illustrated by a numerical example and a Pascal algorithm. We assume that the reader has a general knowledge of vector and matrix algebra but is less familiar with linear transformation of a vector space
A Householder-based algorithm for Hessenberg-triangular reduction
The QZ algorithm for computing eigenvalues and eigenvectors of a matrix
pencil requires that the matrices first be reduced to
Hessenberg-triangular (HT) form. The current method of choice for HT reduction
relies entirely on Givens rotations regrouped and accumulated into small dense
matrices which are subsequently applied using matrix multiplication routines. A
non-vanishing fraction of the total flop count must nevertheless still be
performed as sequences of overlapping Givens rotations alternately applied from
the left and from the right. The many data dependencies associated with this
computational pattern leads to inefficient use of the processor and poor
scalability.
In this paper, we therefore introduce a fundamentally different approach that
relies entirely on (large) Householder reflectors partially accumulated into
block reflectors, by using (compact) WY representations. Even though the new
algorithm requires more floating point operations than the state of the art
algorithm, extensive experiments on both real and synthetic data indicate that
it is still competitive, even in a sequential setting. The new algorithm is
conjectured to have better parallel scalability, an idea which is partially
supported by early small-scale experiments using multi-threaded BLAS. The
design and evaluation of a parallel formulation is future work
An experimental comparison of several approaches to the linear least squares problem
Algorithms and original data matrix approaches compared for linear least squares proble
Quantum mechanical study of molecules - Eigenvalues and eigenvectors of real symmetric matrices
Computer methods for calculating eigenvalue and eigenvectors of real symmetric matrices arising in problems of molecular quantum mechanic
Minimizing Communication for Eigenproblems and the Singular Value Decomposition
Algorithms have two costs: arithmetic and communication. The latter
represents the cost of moving data, either between levels of a memory
hierarchy, or between processors over a network. Communication often dominates
arithmetic and represents a rapidly increasing proportion of the total cost, so
we seek algorithms that minimize communication. In \cite{BDHS10} lower bounds
were presented on the amount of communication required for essentially all
-like algorithms for linear algebra, including eigenvalue problems and
the SVD. Conventional algorithms, including those currently implemented in
(Sca)LAPACK, perform asymptotically more communication than these lower bounds
require. In this paper we present parallel and sequential eigenvalue algorithms
(for pencils, nonsymmetric matrices, and symmetric matrices) and SVD algorithms
that do attain these lower bounds, and analyze their convergence and
communication costs.Comment: 43 pages, 11 figure
A modal parameter extraction procedure applicable to linear time-invariant dynamic systems
Modal analysis has emerged as a valuable tool in many phases of the engineering design process. Complex vibration and acoustic problems in new designs can often be remedied through use of the method. Moreover, the technique has been used to enhance the conceptual understanding of structures by serving to verify analytical models. A new modal parameter estimation procedure is presented. The technique is applicable to linear, time-invariant systems and accommodates multiple input excitations. In order to provide a background for the derivation of the method, some modal parameter extraction procedures currently in use are described. Key features implemented in the new technique are elaborated upon
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