On the stability of the Bareiss and related Toeplitz factorization algorithms

This paper contains a numerical stability analysis of factorization algorithms for computing the Cholesky decomposition of symmetric positive definite matrices of displacement rank 2. The algorithms in the class can be expressed as sequences of elementary downdating steps. The stability of the factorization algorithms follows directly from the numerical properties of algorithms for realizing elementary downdating operations. It is shown that the Bareiss algorithm for factorizing a symmetric positive definite Toeplitz matrix is in the class and hence the Bareiss algorithm is stable. Some numerical experiments that compare behavior of the Bareiss algorithm and the Levinson algorithm are presented. These experiments indicate that in general (when the reection coefficients are not all positive) the Levinson algorithm can give much larger residuals than the Bareiss algorithm

Pseudovirgaria, a fungicolous hyphomycete genus

The genus Pseudovirgaria, based on P. hyperparasitica, was recently introduced for a mycoparasite of rust sori of various species of Frommeëlla, Pucciniastrum and Phragmidium in Korea. In the present study, an older name introduced by Saccardo based on European material, Rhinotrichum griseum, is shown to resemble P. hyperparasitica. Morphological study and ITS barcodes from fresh collections of R. griseum from Austria on uredinia and telia of Phragmidium bulbosum on Rubus spp. reveal that it is distinct from P. hyperparasitica. The status of the genus Rhinotrichum, introduced for a fungus occurring on dry wood, remains unclear. Pseudovirgaria grisea comb. nov. is therefore proposed for the mycoparasite occurring on rust fungi in Europe, and an epitype is designated from the recent collections

A weakly stable algorithm for general Toeplitz systems

We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A. Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx = A^Tb, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm

Kernel perturbations for convolution first kind Volterra integral equations

Because of their causal structure, (convolution) Volterra integral equations arise as models in a variety of real-world situations including rheological stress-strain analysis, population dynamics and insurance risk prediction. In such practical situations, often only an approximation for the kernel is available. Consequently, a key aspect in the analysis of such equations is estimating the effect of kernel perturbations on the solutions. In this paper, it is shown how kernel perturbation results derived for the interconversion equation of rheology can be extended to the analysis of kernel perturbations for first kind convolutional integral equations with positive kernels, solutions and forcing terms.MI: Global COE Program Education-and-Research Hub for Mathematics-for-IndustryグローバルCOEプログラム「マス･フォア･インダストリ教育研究拠点

Kernel perturbations for convolution first kind Volterra integral equations

MI: Global COE Program Education-and-Research Hub for Mathematics-for-IndustryグローバルCOEプログラム「マス･フォア･インダストリ教育研究拠点」Because of their causal structure, (convolution) Volterra integral equations arise as models in a variety of real-world situations including rheological stress-strain analysis, population dynamics and insurance risk prediction. In such practical situations, often only an approximation for the kernel is available. Consequently, a key aspect in the analysis of such equations is estimating the effect of kernel perturbations on the solutions. In this paper, it is shown how kernel perturbation results derived for the interconversion equation of rheology can be extended to the analysis of kernel perturbations for first kind convolutional integral equations with positive kernels, solutions and forcing terms

A weakly stable algorithm for general Toeplitz systems

We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R T R is close to A T A. Thus, when the algorithm is used to solve the semi-normal equations R T Rx = A T b, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem min �Ax − b�2

A weakly stable algorithm for general Toeplitz systems

Abstract unaccessible

Leighton's bounds for Sturm-Liouville eigenvalues

AbstractAn error is pointed out in a method of W. Leighton for computing two-sided bounds for the eigenvalues of the Sturm-Liouville problem (ry′)′ + λpy = 0, y(a) = y(b) = 0. The error is corrected, the underlying theory is examined and the method is generalized