Componentwise Error Analysis Research Articles

Componentwise accurate fluid queue computations using doubling algorithms

Markov-modulated fluid queues are popular stochastic processes frequently used for modelling real-life applications. An important performance measure to evaluate in these applications is their steady-state behaviour, which is determined by the stationary density. Computing it requires solving a (nonsymmetric) M-matrix algebraic Riccati equation, and indeed computing the stationary density is the most important application of this class of equations. Xue, Xu and Li [Numer. Math., 2012] provided a componentwise first-order perturbation analysis of this equation, proving that the solution can be computed to high relative accuracy even in the smallest entries, and suggested several algorithms for computing it. An important step in all proposed algorithms is using so-called triplet representations, which are special representations for M-matrices that allow for a high-accuracy variant of Gaussian elimination, the GTH-like algorithm. However, triplet representations for all the M-matrices needed in the algorithm were not found explicitly. This can lead to an accuracy loss that prevents the algorithms to converge in the componentwise sense. In this paper, we focus on the structured doubling algorithm, the most efficient among the proposed methods in Xue et al., and build upon their results, providing (i) explicit and cancellation-free expressions for the needed triplet representations, allowing the algorithm to be performed in a really cancellation-free fashion; (ii) an algorithm to evaluate the final part of the computation to obtain the stationary density; and (iii) a componentwise error analysis for the resulting algorithm, the first explicit one for this class of algorithms. We also present numerical results to illustrate the accuracy advantage of our method over standard (normwise-accurate) algorithms.

Componentwise error analysis for the block LU factorization of totally nonnegative matrices

For the first time, perturbation bounds including componentwise perturbation bounds for the block LU factorization have been provided by Dopico and Molera (2005) [5]. In this paper, componentwise error analysis is presented for computing the block LU factorization of nonsingular totally nonnegative matrices. We present a componentwise bound on the equivalent perturbation for the computed block LU factorization. Consequently, combining with the componentwise perturbation results we derive componentwise forward error bounds for the computed block factors.

Chebyshev acceleration of iterative refinement

It is well known that the FGMRES algorithm can be used as an alternative to iterative refinement and, in some instances, is successful in computing a backward stable solution when iterative refinement fails to converge. In this study, we analyse how variants of the Chebyshev algorithm can also be used to accelerate iterative refinement without loss of numerical stability and at a computational cost at each iteration that is less than that of FGMRES and only marginally greater than that of iterative refinement. A component-wise error analysis of the procedure is presented and numerical tests on selected sparse test problems are used to corroborate the theory.

Componentwise error analysis for linear systems associated with sign regular matrices

In this paper, we present a componentwise error analysis of solving linear systems whose coefficient matrices are sign regular matrices with the same signature sequences as those of Jacobi sign regular matrices associated with corner cutting algorithms in computer aided geometric design. It is interesting to show that for such sign regular matrices, in finite precision floating point arithmetic, the computed factors by Gaussian elimination with the last two row exchanges satisfy a small componentwise relative backward error. Consequently, the computed solutions of linear systems associated with such matrices admit pleasantly small componentwise relative backward errors.

On the Error in the Product QR Decomposition

We develop both a normwise and a componentwise error analysis for the QR factorization of long products of invertible matrices. We obtain global error bounds for both the orthogonal and upper triangular factors that depend on uniform bounds on the size of the local error, the local degree of nonnormality, and integral separation, a natural condition related to gaps between eigenvalues but for products of matrices. We illustrate our analytical results with numerical results that show the dependence on the degree of nonnormality and the strength of integral separation.

On the Stability of a Partitioning Algorithm for Tridiagonal Systems

Componentwise error analysis for a parallel partitioning algorithm for tridiagonal systems is presented. Bounds on the equivalent perturbations are obtained, depending on three constants. Then bounds on the forward error are presented as well, depending on two types of condition numbers. Estimates on the first constant come directly from the roundoff error analysis of the tridiagonal Gaussian elimination [N. Higham, SIAM J. Matrix Anal. Appl., 11 (1990), pp. 521--530]. In the present paper, the second and third constants are bounded for some special classes of matrices, i.e., diagonally dominant (row or column), symmetric positive definite, M-matrices, and totally nonnegative. One of the features of the analysis is that the exact forward and backward errors are bounded, not just their first order approximations, with respect to the machine precision. In all the bounds, the linear terms are given separately to show that the terms of higher order are small enough.

Componentwise error analysis for FFTs with applications to fast Helmholtz solvers

We analyze the stability of the Cooley-Tukey algorithm for the Fast Fourier Transform of ordern=2 k and of its inverse by using componentwise error analysis. We prove that the components of the roundoff errors are linearly related to the result in exact arithmetic. We describe the structure of the error matrix and we give optimal bounds for the total error in infinity norm and inL 2 norm. The theoretical upper bounds are based on a “worst case” analysis where all the rounding errors work in the same direction. We show by means of a statistical error analysis that in realistic cases the max-norm error grows asymptotically like the logarithm of the sequence length by machine precision. Finally, we use the previous results for introducing tight upper bounds on the algorithmic error for some of the classical fast Helmholtz equation solvers based on the Faster Fourier Transform and for some algorithms used in the study of turbulence.

On the stability of the cyclic reduction without back substitution for tridiagonal systems

Componentwise error analysis for a modification of the cyclic reduction without back substitution for a tridiagonal system is presented. We consider relative roundoff errors and equivalent perturbations, so the main supposition is that all the data is nonzero. First, backward analysis for the computation of each component of the solution in separate is presented. Bounds on the relative equivalent perturbations are obtained depending on two constants. From these bounds it is easy to obtain a componentwise forward error analysis. Then the two constants are defined for some special classes of matrices, i.e. diagonally dominant (row or column), symmetric positive definite, totally nonnegative andM-matrices, and it is shown that the bounds for these classes of matrices are small.

Hip process quality based on combination of numerical and stochastic models

For the first time, perturbation bounds including componentwise perturbation bounds for the block LU factorization have been provided by Dopico and Molera (2005) [5]. In this paper, componentwise error analysis is presented for computing the block LU factorization of nonsingular totally nonnegative matrices. We present a componentwise bound on the equivalent perturbation for the computed block LU factorization. Consequently, combining with the componentwise perturbation results we derive componentwise forward error bounds for the computed block factors.