Multishift variants of the QZ algorithm with aggressive early deflation

New variants of the QZ algorithm for solving the generalized eigenvalue problem are proposed. An extension of the small-bulge multishift QR algorithm is developed, which chases chains of many small bulges instead of only one bulge in each QZ iteration. This allows the effective use of level 3 BLAS operations, which in turn can provide efficient utilization of high performance computing systems with deep memory hierarchies. Moreover, an extension of the aggressive early deflation strategy is proposed, which can identify and de. ate converged eigenvalues long before classic deflation strategies would. Consequently, the number of overall QZ iterations needed until convergence is considerably reduced. As a third ingredient, we reconsider the deflation of infinite eigenvalues and present a new deflation algorithm, which is particularly effective in the presence of a large number of infinite eigenvalues. Combining all these developments, our implementation significantly improves existing implementations of the QZ algorithm. This is demonstrated by numerical experiments with random matrix pairs as well as with matrix pairs arising from various applications

A new deflation criterion for the QZ algorithm

The QZ algorithm computes the Schur form of a matrix pencil. It is an iterative algorithm and at some point, it must decide that an eigenvalue has converged and move on with another one. Choosing a criterion that makes this decision is nontrivial. If it is too strict, the algorithm might waste iterations on already converged eigenvalues. If it is not strict enough, the computed eigenvalues might be inaccurate. Additionally, the criterion should not be computationally expensive to evaluate. This paper introduces a new criterion based on the size of and the gap between the eigenvalues. This is similar to the work of Ahues and Tissuer for the QR algorithm. Theoretical arguments and numerical experiments suggest that it outperforms the most popular criteria in terms of accuracy. Additionally, this paper evaluates some commonly used criteria for infinite eigenvalues.Comment: 10 pages, 6 figure

A multishift, multipole rational QZ method with aggressive early deflation

The rational QZ method generalizes the QZ method by implicitly supporting rational subspace iteration. In this paper we extend the rational QZ method by introducing shifts and poles of higher multiplicity in the Hessenberg pencil, which is a pencil consisting of two Hessenberg matrices. The result is a multishift, multipole iteration on block Hessenberg pencils which allows one to stick to real arithmetic for a real input pencil. In combination with optimally packed shifts and aggressive early deflation as an advanced deflation technique we obtain an efficient method for the dense generalized eigenvalue problem. In the numerical experiments we compare the results with state-of-the-art routines for the generalized eigenvalue problem and show that we are competitive in terms of speed and accuracy

A Novel Parallel QR Algorithm For Hybrid Distributed Memory HPC Systems

A novel variant of the parallel QR algorithm for solving dense nonsymmetric eigenvalue problems on hybrid distributed high performance computing systems is presented. For this purpose, we introduce the concept of multiwindow bulge chain chasing and parallelize aggressive early deflation. The multiwindow approach ensures that most computations when chasing chains of bulges are performed in level 3 BLAS operations, while the aim of aggressive early deflation is to speed up the convergence of the QR algorithm. Mixed MPI-OpenMP coding techniques are utilized for porting the codes to distributed memory platforms with multithreaded nodes, such as multicore processors. Numerous numerical experiments confirm the superior performance of our parallel QR algorithm in comparison with the existing ScaLAPACK code, leading to an implementation that is one to two orders of magnitude faster for sufficiently large problems, including a number of examples from applications

A parallel Schur method for solving continuous-time algebraic Riccati equations

Numerical algorithms for solving the continuous-time algebraic Riccati matrix equation on a distributed memory parallel computer are considered. In particular, it is shown that the Schur method, based on computing the stable invariant subspace of a Hamiltonian matrix, can be parallelized in an efficient and scalable way. Our implementation employs the state-of-the-art library ScaLAPACK as well as recently developed parallel methods for reordering the eigenvalues in a real Schur form. Some experimental results are presented, confirming the scalability of our implementation and comparing it with an existing implementation of the matrix sign iteration from the PLiCOC library

Parallel eigenvalue reordering in real Schur forms

A parallel algorithm for reordering the eigenvalues in the real Schur form of a matrix is presented and discussed. Our novel approach adopts computational windows and delays multiple outside-window updates until each window has been completely reordered locally. By using multiple concurrent windows the parallel algorithm has a high level of concurrency, and most work is level 3 BLAS operations. The presented algorithm is also extended to the generalized real Schur form. Experimental results for ScaLAPACK-style Fortran 77 implementations on a Linux cluster confirm the efficiency and scalability of our algorithms in terms of more than 16 times of parallel speedup using 64 processors for large-scale problems. Even on a single processor our implementation is demonstrated to perform significantly better compared with the state-of-the-art serial implementation. Copyright (C) 2009 John Wiley & Sons, Ltd

Algoritmos paralelos para la reducción de sistemas lineales de control estables

[EN] In the field of control theory, sometimes system models of big size (with many state variables) appear. When one of these high order systems needs to be simulated, studied or controlled, it is convenient to perform a previous work of model reduction in order to reduce the necessary (economic and temporal) costs. This process of obtaining a low order adequate representation of the original system usually has a high cost, because it has to be done with the original unreduced system. Thus, it is important to have high performance implementations for the problem of reducing linear control systems. In this thesis high performance implementations for some methods of model reduction have been developed. Current algorithms for model reduction of stable linear control systems and their implementation in the control library SLICOT have been analysed. New parallel algorithms for the methods strongly based on solving Lyapunov equations have been proposed. The new developed routines are incorporated in the high performance library for control PSLICOT. Apart from the main functions in charge of model reduction, all operations appearing in the problem and not having a high performance version yet have also been parallelised. One of these operations is the solution of Lyapunov equations in standard form. Parallel routines for solving these equations have been developed. These routines solve the equation obtaining directly the Cholesky factor of the solution, which fits better their application in model reduction. For this, Hammarling's method has been parallelised. The new routines solve in parallel and for dense matrices the four possible variants of standard Lyapunov equations: discrete and continuous versions, both transposed and not transposed. Interfaces offered by all the parallelised routines are similar to that of the existing routines in ScaLAPACK library, so they are easy to use from a user of this kind of libraries. The new routines work with the same data distribution used in this library: 2D block cyclic distribution, which allows many other distributions. Thanks to the developed work, now there are available high performance parallel routines to reduce linear control systems by using different variants of the Square-Root Balance & Truncate model reduction method: with or without balancing and with or without using the singular perturbation approximation formulas. They are parallel implementations of the same algorithms and methods used in the sequential routines of the SLICOT library. This allows to efficiently reduce models of linear control systems of big size. Moreover, existing software in ScaLAPACK for the eigenvalue problem has been improved by covering cases not treated there: the solution of the generalised problem (by transforming it into standard form, which is not always possible) and the computation of the eigenvectors. This part of the work has been applied to a real problem of simulation of oceanic flows. Here, it is necessary to compute all the eigenvalues and eigenvectors of a generalised eigenvalue problem with a very big dimension. As a consequence, enormous eigenvalue problems have been solved (with matrices of order greater than 400000), that could not be solved previously. Solving them allows to improve the precision in the studies of the behaviour of oceanic flows.[ES] En el campo de la teoría de control en ocasiones aparecen modelos de sistemas con un tamaño elevado (muchas variables de estado). Cuando se pretende simular, estudiar o controlar uno de estos sistemas de orden elevado, conviene realizar un trabajo previo de reducción del modelo del sistema con el propósito de reducir los costes (económicos/temporales) necesarios en un tratamiento posterior. El proceso de obtención de un sistema de orden reducido que represente adecuadamente el sistema original suele ser costoso, ya que necesariamente se tiene que hacer con el sistema original sin reducir. Por esto, resulta conveniente disponer de implementaciones de altas prestaciones para el problema de reducción de sistemas lineales de control. En esta tesis se han desarrollado implementaciones de altas prestaciones para algunos métodos de reducción de modelos. Se han analizado los algoritmos existentes para la reducción de modelos de sistemas lineales de control estables y sus implementaciones en la librería de control SLICOT. Se han propuesto nuevos algoritmos paralelos para los métodos cuyo núcleo principal es la resolución de ecuaciones de Lyapunov. Las nuevas rutinas desarrolladas se incorporan a la librería de computación de altas prestaciones para control PSLICOT. Aparte de las funciones principales a cargo de la reducción de modelos, se han tenido que paralelizar también todas aquellas operaciones numéricas que aparecen en este problema y de las que no se disponía de versiones de altas prestaciones. De estas operaciones, cabe destacar rutinas paralelas para la resolución de la ecuación de Lyapunov en su forma estándar obteniendo directamente el factor de Cholesky de la solución, que es lo que se necesita para la reducción de modelos. El método utilizado es una versión paralela del método de Hammarling. Las rutinas implementadas resuelven en paralelo y para matrices densas las cuatro variantes posibles de la ecuación de Lyapunov: en su forma discreta y continua, traspuestas y sin trasponer. Todas las rutinas paralelizadas ofrecen una interfaz como la de las rutinas de la librería ScaLAPACK, para que puedan ser usadas con facilidad por el usuario habituado a trabajar con este tipo de librerías. Se permiten las mismas distribuciones de datos que en esta librería: una distribución cíclica 2D por bloques, que engloba muchas otras distribuciones. Gracias al trabajo desarrollado, ahora se dispone de versiones paralelas de altas prestaciones para reducir sistemas lineales de control mediante diferentes variantes del método de balanceado y truncamiento de la raíz cuadrada (the Square-Root Balance & Truncate model reduction method): con o sin balanceado y con o sin usar las fórmulas de perturbación singular. Se trata de versiones paralelas de los mismos algoritmos y métodos que se utilizan en las versiones secuenciales de la librería SLICOT. Esto permitirá reducir de forma eficiente modelos de sistemas lineales de control de gran tamaño. También se ha mejorado la aplicabilidad del software existente en ScaLAPACK para el problema de valores propios cubriendo casos que no se contemplaban. Se ha trabajado en la solución del problema generalizado (mediante su transformación a forma estándar, lo que no es aplicable en todos los casos) y también en el cálculo de los vectores propios. Ambas operaciones se han utilizado en un problema real de simulación de flujos oceánicos. En esta aplicación se requiere el cálculo de todos los valores y vectores propios de un problema generalizado de gran dimensión. Como consecuencia, ha sido posible resolver problemas de valores propios generalizados enormes (con matrices de más de 400000 filas y columnas) que no habían podido resolverse con anterioridad, permitiendo así un estudio más preciso del comportamiento de las corrientes oceánicas.[CA] En el camp de la teoria de control de vegades apareixen models de sistemes amb un tamany elevat (moltes variables d'estat). Quan es pretén simular, estudiar o controlar un d'aquests sistemes d'ordre elevat, convé realitzar un treball previ de reducció del model del sistema amb el propòsit de reduir els costos (econòmics/temporals) necesaris en un tractament posterior. El procés d'obtenció d'un sistema d'ordre reduit que represente adequadament el sistema original sol ser costós, perque necessàriament ha de fer-se amb el sistema original sense reduir. Per aquest motiu, resulta convenient disposar d'implementacions d'altes prestacions per al problema de reducció de sistemes lineals de control. En aquesta tesis s'han desenvolupat implementacions d'altes prestacions per a alguns mètodes de reducció de models. S'han anal·litzat els algoritmes existents per a la reducció de models de sistemes lineals de control estables i les seues implementacions en la llibreria de control SLICOT. S'han proposat nous algoritmes paral·lels per als mètodes basats en la resolució d'equacions de Lyapunov. Les noves rutines desenvolupades s'incorporen a la llibreria de computació d'altes prestacions per a control PSLICOT. Apart de les funcions principals a càrrec de la reducció de models, s'han hagut de paral·le\-lit\-zar també totes aquelles operacions numèriques que apareixen en aquest problema i per a les que no es disposava de versions d'altes prestacions. D'aquestes operacions, destaquen rutines paral·leles per a la resolució de l'equació de Lyapunov en la seua forma estàndard obtenint directament el factor de Cholesky de la solució, que és el que es necessita per a la reducció de models. El mètode emprat és una versió paral·lela del mètode de Hammarling. Les rutines implementades resolen en paral·lel i per a matrius denses les quatre variants possibles de l'equació de Lyapunov: en la seua forma discreta i contínua, traspostes i sense trasposar. Totes les rutines paral·lelitzades ofereixen una interfaç com la de les rutines de la llibreria ScaLAPACK, per a que puguen ser usades fàcilment per l'usuari acostumat a treballar amb aquest tipus de llibreries. Es permeten les mateixes distribucions de dades que en aquesta llibreria: distribució cíclica 2D per blocs, que engloba moltes altres distribucions. Gràcies al treball desenvolupat, ara es disposa de versions paral·leles d'altes prestacions per a reduir sistemes lineals de control mitjançant diferents variants del mètode de balancejat i truncament de l'arrel quadrada (the Square-Root Balance & Truncate model reduction method): amb o sense balancejat i amb o sense usar les fórmules de perturbació singular. Son versions paral·leles dels mateixos algoritmes i mètodes que s'utilitzen en les versions sequencials de la llibreria SLICOT. Això permetrà reduir de forma eficient models de sistemes lineals de control de gran tamany. També s'ha mitjorat l'aplicabilitat del software existent en ScaLAPACK per al problema de valors propis cobrint casos que no es contemplaven. S'ha treballat en la solució del problema generalitzat (mitjançant la seua transformació a forma estàndard, cosa que no es pot fer sempre) i també en el càlcul dels vectors propis. Ambdues operacions s'han utilitzat en un problema real de simulació de fluxos oceànics. En aquesta aplicació es requereix el càlcul de tots els valors i vectors propis d'un problema generalitzat de gran dimensió. Com a conseqüència, ha sigut possible resoldre problemes de valors propis generalitzats molt grans (amb matrius de més de 400000 files i columnes) que no s'havien pogut resoldre anteriorment, permetent així un estudi més precís del comportament de les corrents oceàniques.Guerrero López, D. (2015). Algoritmos paralelos para la reducción de sistemas lineales de control estables [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/59415TESI