Accurate and Efficient Expression Evaluation and Linear Algebra

We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean that the computed answer has relative error less than 1, i.e., has some correct leading digits. We also address efficiency, by which we mean algorithms that run in polynomial time in the size of the input. Our results will depend strongly on the model of arithmetic: Most of our results will use the so-called Traditional Model (TM). We give a set of necessary and sufficient conditions to decide whether a high accuracy algorithm exists in the TM, and describe progress toward a decision procedure that will take any problem and provide either a high accuracy algorithm or a proof that none exists. When no accurate algorithm exists in the TM, it is natural to extend the set of available accurate operations by a library of additional operations, such as $x+y+z$, dot products, or indeed any enumerable set which could then be used to build further accurate algorithms. We show how our accurate algorithms and decision procedure for finding them extend to this case. Finally, we address other models of arithmetic, and the relationship between (im)possibility in the TM and (in)efficient algorithms operating on numbers represented as bit strings.Comment: 49 pages, 6 figures, 1 tabl

Computing the singular value decomposition with high relative accuracy

AbstractWe analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy. This means that each computed singular value is guaranteed to have some correct digits, even if the singular values have widely varying magnitudes. This is in contrast to the absolute accuracy provided by conventional backward stable algorithms, which in general only guarantee correct digits in the singular values with large enough magnitudes. It is of interest to compute the tiniest singular values with several correct digits, because in some cases, such as finite element problems and quantum mechanics, it is the smallest singular values that have physical meaning, and should be determined accurately by the data. Many recent papers have identified special classes of matrices where high relative accuracy is possible, since it is not possible in general. The perturbation theory and algorithms for these matrix classes have been quite different, motivating us to seek a common perturbation theory and common algorithm. We provide these in this paper, and show that high relative accuracy is possible in many new cases as well. The briefest way to describe our results is that we can compute the SVD of G to high relative accuracy provided we can accurately factor G=XDYT where D is diagonal and X and Y are any well-conditioned matrices; furthermore, the LDU factorization frequently does the job. We provide many examples of matrix classes permitting such an LDU decomposition

Condicionamiento y alta precisión en problemas espectrales estructurados

Esta memoria trata dos aspectos relacionados con la precisión de algoritmos espectrales para problemas matriciales estructurados: 1) Se definen números de condición estructurados de autovalores múltiples, eventualmente defectivos, y se obtienen fórmulas explícitas para diversas clases estructuradas de matrices, entre ellas las simétricas y antisimétricas complejas, persimétricas, Toeplitz, Hankel, hamiltonianas y antihamiltonianas reales. Para cada clase se compara el número de condición estructurado con el número de condición usual, a fin de identificar casos en los que un algoritmo estructurado pueda ser mucho más preciso que un algoritmo convencional. También se trata el caso de autovalores múltiples de pares regulares de matrices. 2. Se proponen, analizan e implementan algoritmos para factorizar y calcular con alta precisión relativa autovalores y autovectores de dos clases estructuradas de matrices simétricas: las matrices DSTU (escalamientos diagonales de matrices totalmente unimodulares), y las matrices TSC (definidas por medio de una condición de signos sobre sus menores). Los algoritmos tienen dos etapas: una primera en la que se obtiene una factorización simétrica, y una segunda en la que se aplica un método de autovalores tipo Jacobi a la matriz factorizada. La primera etapa es la que se adapta a cada una de las estructuras, aprovechando propiedades especiales de la clase de matrices que permiten evitar cualquier posible cancelación en las operaciones aritméticas del proceso de factorización. Un análisis de errores detallado de la etapa de factorización muestra que ésta se lleva a cabo con una precisión suficiente para garantizar que la segunda etapa produce alta precisión relativa, esto es, que se calculan con alta precisión no sólo los autovalores de mayor módulo (como hacen los algoritmos convencionales) sino también los autovalores más pequeños de la matri

Toward accurate polynomial evaluation in rounded arithmetic

Given a multivariate real (or complex) polynomial $p$ and a domain $\cal D$, we would like to decide whether an algorithm exists to evaluate $p(x)$ accurately for all $x \in {\cal D}$ using rounded real (or complex) arithmetic. Here ``accurately'' means with relative error less than 1, i.e., with some correct leading digits. The answer depends on the model of rounded arithmetic: We assume that for any arithmetic operator $op(a,b)$, for example $a+b$ or $a \cdot b$, its computed value is $op(a,b) \cdot (1 + \delta)$, where $| \delta |$ is bounded by some constant $\epsilon$ where $0 < \epsilon \ll 1$, but $\delta$ is otherwise arbitrary. This model is the traditional one used to analyze the accuracy of floating point algorithms.Our ultimate goal is to establish a decision procedure that, for any $p$ and $\cal D$, either exhibits an accurate algorithm or proves that none exists. In contrast to the case where numbers are stored and manipulated as finite bit strings (e.g., as floating point numbers or rational numbers) we show that some polynomials $p$ are impossible to evaluate accurately. The existence of an accurate algorithm will depend not just on $p$ and $\cal D$, but on which arithmetic operators and which constants are are available and whether branching is permitted. Toward this goal, we present necessary conditions on $p$ for it to be accurately evaluable on open real or complex domains ${\cal D}$. We also give sufficient conditions, and describe progress toward a complete decision procedure. We do present a complete decision procedure for homogeneous polynomials $p$ with integer coefficients, {\cal D} = \C^n, and using only the arithmetic operations $+$, $-$ and $\cdot$.Comment: 54 pages, 6 figures; refereed version; to appear in Foundations of Computational Mathematics: Santander 2005, Cambridge University Press, March 200

Alta precisión relativa en problemas de álgebra lineal numérica en matrices con estructura

Esta tesis se enmarca dentro del campo de la Alta Precisión Relativa (HRA) en Álgebra Lineal Numérica (ALN). Sus líneas maestras son dos. Por un lado, el diseño y análisis de algoritmos que permitan resolver problemas de Álgebra Lineal con más precisión de la habitual para matrices con estructura. Y por otro el estudio de la teoría específica de perturbaciones necesaria para tratar los problemas que nos ocupan. En nuestra investigación hemos tratado dos: La obtención de soluciones precisas del problema de mínimos cuadrados para matrices con estructura (Capítulo 3). La obtención de autovalores y autovectores precisos para matrices simétricas graduadas (Capítulo 4)...