On the Error in the Product QR Decomposition

This is the published version, also available here: http://dx.doi.org/10.1137/090761562.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

Computation of the inverse Laplace Transform based on a Collocation method which uses only real values

We develop a numerical algorithm for inverting a Laplace transform (LT), based on Laguerre polynomial series expansion of the inverse function under the assumption that the LT is known on the real axis only. The method belongs to the class of Collocation methods (C-methods), and is applicable when the LT function is regular at infinity. Difficulties associated with these problems are due to their intrinsic ill-posedness. The main contribution of this paper is to provide computable estimates of truncation, discretization, conditioning and roundoff errors introduced by numerical computations. Moreover, we introduce the pseudoaccuracy which will be used by the numerical algorithm in order to provide uniform scaled accuracy of the computed approximation for any x with respect to ex . These estimates are then employed to dynamically truncate the series expansion. In other words, the number of the terms of the series acts like the regularization parameter which provides the trade-off between errors. With the aim to validate the reliability and usability of the algorithm experiments were carried out on several test functions

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 et al. (Numer Math 120:671–700, 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 from converging 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. © 2014, Springer-Verlag Berlin Heidelberg

Error analysis and model adaptivity for flows in gas networks

In the simulation and optimization of natural gas flow in a pipeline network, a hierarchy of models is used that employs different formulations of the Euler equations. While the optimization is performed on piecewise linear models, the flow simulation is based on the one to three dimensional Euler equations including the temperature distributions. To decide which model class in the hierarchy is adequate to achieve a desired accuracy, this paper presents an error and perturbation analysis for a two level model hierarchy including the isothermal Euler equations in semilinear form and the stationary Euler equations in purely algebraic form. The focus of the work is on the effect of data uncertainty, discretization, and rounding errors in the numerical simulation of these models and their interaction. Two simple discretization schemes for the semilinear model are compared with respect to their conditioning and temporal stepsizes are determined for which a well-conditioned problem is obtained. The results are based on new componentwise relative condition numbers for the solution of nonlinear systems of equations. More- over, the model error between the semilinear and the algebraic model is computed, the maximum pipeline length is determined for which the algebraic model can be used safely, and a condition is derived for which the isothermal model is adequate.DFG, TRR 154, Mathematische Modellierung, Simulation und Optimierung am Beispiel von Gasnetzwerke

Stability analysis of the Toeplitz-like matrix by vector product (via FFT)

In this paper the numerical stability of the Toeplitz-like matrix by vector product, performed via FFT, is analyzed. The error appears to depend on the magnitude of the generators of the matrix. The numerical?experimentation conrms the theoretical result

Sharp error bounds for complex floating-point inversion

International audienceWe study the accuracy of the classic algorithm for inverting a complex number given by its real and imaginary parts as floating-point numbers. Our analyses are done in binary floating-point arithmetic, with an unbounded exponent range and in precision $p$; we also assume that the basic arithmetic operations ($+$, $-$, $\times$, $/$) are rounded to nearest, so that the roundoff unit is $u = 2^{-p}$. We bound the largest relative error in the computed inverse either in the componentwise or in the normwise sense. We prove the componentwise relative error bound $3u$ for the complex inversion algorithm (assuming $p \ge 4$), and we show that this bound is asymptotically optimal (as $p\to \infty$) when $p$ is even, and sharp when using one of the basic IEEE 754 binary formats with an odd precision ($p=53,113$). This componentwise bound obviously leads to the same bound $3u$ for the normwise relative error. However, we prove that the smaller bound $2.707131u$ holds (assuming $p \ge 24$) for the normwise relative error, and we illustrate the sharpness of this bound for the basic IEEE 754 binary formats ($p=24, 53, 113$) using numerical examples