A Householder-based algorithm for Hessenberg-triangular reduction

The QZ algorithm for computing eigenvalues and eigenvectors of a matrix pencil $A - \lambda B$ 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

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

Dynamic analysis of modified composite helicopter blade

U ovom radu izvršena je modalna analiza modifikovane lopatice helikoptera 'Gazela'. Modifikovana lopatica je kompletno kompozitna sa saćastom ispunom. Prikazan je metod određivanja modova oscilovanja i sopstvenih frekvencija. Modifikovana lopatica sastoji se od saćaste ispune, ramenjače od 3D usmerenog kompozita i tankih karbonskih ploča kao oplate. Da bi se odredila matrica krutosti ispune korišćen je metod ekvivalentnih masa. U cilju nalaženja optimalnog metoda za određivanje sopstvenih frekvencija ispitano je nekoliko poznatih metoda. Metod Lancosa pokazao je najtačnije rezultate kroz umereno procesorsko vreme kada je u pitanju određivanje sopstvenih frekvencija i modova oscilovanja kod struktura od kompozitnih materijala sa saćastim ispunama. Ovom metodom izračunata su prva četiri moda oscilovanja modifikovane kompozitne lopatice, i prikazani su rezultati modova oscilovanja i deformacione energije lopatice.In the present study, modal analysis has been performed on modified Gazelle helicopter blade. The construction of the blade is fully composite with the honeycomb core. The approach to determining structure mode shapes and natural frequencies is presented. Modified blade consists of core material, 3D unidirectional composite spar and thin carbon composite facesheets as blade skin. To determine the stiffness of the honeycomb core, the equivalent mass approach was used. Several methods of eigenvalue extraction have been investigated in order to find optimal method which can be used in dynamic analysis of composite structures containing honeycomb cores. Among all extraction methods investigated, it was found that combined Lanczos method is most effective in terms of accuracy and CPU time for eigenvalue extraction in composite structures with honeycomb core having large number of degrees of freedom. Strain energies for first four mode shapes of modified helicopter blade have been calculated using numerical approach and results are presented

Dynamic analysis of modified composite helicopter blade

U ovom radu izvršena je modalna analiza modifikovane lopatice helikoptera 'Gazela'. Modifikovana lopatica je kompletno kompozitna sa saćastom ispunom. Prikazan je metod određivanja modova oscilovanja i sopstvenih frekvencija. Modifikovana lopatica sastoji se od saćaste ispune, ramenjače od 3D usmerenog kompozita i tankih karbonskih ploča kao oplate. Da bi se odredila matrica krutosti ispune korišćen je metod ekvivalentnih masa. U cilju nalaženja optimalnog metoda za određivanje sopstvenih frekvencija ispitano je nekoliko poznatih metoda. Metod Lancosa pokazao je najtačnije rezultate kroz umereno procesorsko vreme kada je u pitanju određivanje sopstvenih frekvencija i modova oscilovanja kod struktura od kompozitnih materijala sa saćastim ispunama. Ovom metodom izračunata su prva četiri moda oscilovanja modifikovane kompozitne lopatice, i prikazani su rezultati modova oscilovanja i deformacione energije lopatice.In the present study, modal analysis has been performed on modified Gazelle helicopter blade. The construction of the blade is fully composite with the honeycomb core. The approach to determining structure mode shapes and natural frequencies is presented. Modified blade consists of core material, 3D unidirectional composite spar and thin carbon composite facesheets as blade skin. To determine the stiffness of the honeycomb core, the equivalent mass approach was used. Several methods of eigenvalue extraction have been investigated in order to find optimal method which can be used in dynamic analysis of composite structures containing honeycomb cores. Among all extraction methods investigated, it was found that combined Lanczos method is most effective in terms of accuracy and CPU time for eigenvalue extraction in composite structures with honeycomb core having large number of degrees of freedom. Strain energies for first four mode shapes of modified helicopter blade have been calculated using numerical approach and results are presented

Using reconfigurable computing technology to accelerate matrix decomposition and applications

Matrix decomposition plays an increasingly significant role in many scientific and engineering applications. Among numerous techniques, Singular Value Decomposition (SVD) and Eigenvalue Decomposition (EVD) are widely used as factorization tools to perform Principal Component Analysis for dimensionality reduction and pattern recognition in image processing, text mining and wireless communications, while QR Decomposition (QRD) and sparse LU Decomposition (LUD) are employed to solve the dense or sparse linear system of equations in bioinformatics, power system and computer vision. Matrix decompositions are computationally expensive and their sequential implementations often fail to meet the requirements of many time-sensitive applications. The emergence of reconfigurable computing has provided a flexible and low-cost opportunity to pursue high-performance parallel designs, and the use of FPGAs has shown promise in accelerating this class of computation. In this research, we have proposed and implemented several highly parallel FPGA-based architectures to accelerate matrix decompositions and their applications in data mining and signal processing. Specifically, in this dissertation we describe the following contributions: • We propose an efficient FPGA-based double-precision floating-point architecture for EVD, which can efficiently analyze large-scale matrices. • We implement a floating-point Hestenes-Jacobi architecture for SVD, which is capable of analyzing arbitrary sized matrices. • We introduce a novel deeply pipelined reconfigurable architecture for QRD, which can be dynamically configured to perform either Householder transformation or Givens rotation in a manner that takes advantage of the strengths of each. • We design a configurable architecture for sparse LUD that supports both symmetric and asymmetric sparse matrices with arbitrary sparsity patterns. • By further extending the proposed hardware solution for SVD, we parallelize a popular text mining tool-Latent Semantic Indexing with an FPGA-based architecture. • We present a configurable architecture to accelerate Homotopy l1-minimization, in which the modification of the proposed FPGA architecture for sparse LUD is used at its core to parallelize both Cholesky decomposition and rank-1 update. Our experimental results using an FPGA-based acceleration system indicate the efficiency of our proposed novel architectures, with application and dimension-dependent speedups over an optimized software implementation that range from 1.5ÃÂ to 43.6ÃÂ in terms of computation time

Covariance Estimation: The GLM and Regularization Perspectives

Finding an unconstrained and statistically interpretable reparameterization of a covariance matrix is still an open problem in statistics. Its solution is of central importance in covariance estimation, particularly in the recent high-dimensional data environment where enforcing the positive-definiteness constraint could be computationally expensive. We provide a survey of the progress made in modeling covariance matrices from two relatively complementary perspectives: (1) generalized linear models (GLM) or parsimony and use of covariates in low dimensions, and (2) regularization or sparsity for high-dimensional data. An emerging, unifying and powerful trend in both perspectives is that of reducing a covariance estimation problem to that of estimating a sequence of regression problems. We point out several instances of the regression-based formulation. A notable case is in sparse estimation of a precision matrix or a Gaussian graphical model leading to the fast graphical LASSO algorithm. Some advantages and limitations of the regression-based Cholesky decomposition relative to the classical spectral (eigenvalue) and variance-correlation decompositions are highlighted. The former provides an unconstrained and statistically interpretable reparameterization, and guarantees the positive-definiteness of the estimated covariance matrix. It reduces the unintuitive task of covariance estimation to that of modeling a sequence of regressions at the cost of imposing an a priori order among the variables. Elementwise regularization of the sample covariance matrix such as banding, tapering and thresholding has desirable asymptotic properties and the sparse estimated covariance matrix is positive definite with probability tending to one for large samples and dimensions.Comment: Published in at http://dx.doi.org/10.1214/11-STS358 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org