Gary W. Howell

Cache efficient bidiagonalization using BLAS 2.5 operators

On cache based computer architectures using current standard algorithms, Householder bidiagonalization requires a significant portion of the execution time for computing matrix singular values and vectors. In this paper we reorganize the sequence of operations for Householder bidiagonalization of a general m × n matrix, so that two (_GEMV) vector-matrix multiplications can be done with one pass of the unreduced trailing part of the matrix through cache. Two new BLAS operations approximately cut in half the transfer of data from main memory to cache, reducing execution times by up to 25 per cent. We give detailed algorithm descriptions and compare timings with the current LAPACK bidiagonalization algorithm.

Enhanced dispersion in oscillatory flows

Measurements of the dispersion coefficient in oscillating pipe flows for a wide range of Womersley numbers, tidal displacements, and tube radii are presented. The results are shown to be in good agreement with those of a recent laminar dispersion theory for oscillatory flows.

The BR Eigenvalue Algorithm

The BR algorithm, a new method for calculating the eigenvalues of an upper Hessenberg matrix, is introduced. It is a bulge-chasing algorithm like the QR algorithm, but, unlike the QR algorithm, it is well adapted to computing the eigenvalues of the narrow-band, nearly tridiagonal matrices generated by the look-ahead Lanczos process. This paper describes the BR algorithm and gives numerical evidence that it works well in conjunction with the Lanczos process. On the biggest problems run so far, the BR algorithm beats the

ChemModLab: A Web-Based Cheminformatics Modeling Laboratory

QR

Algorithm 841: BHESS: Gaussian reduction to a similar banded Hessenberg form

algorithm by a factor of 30--60 in computing time and a factor of over 100 in matrix storage space.

Separation of isotopes by oscillatory flow

ChemModLab, written by the ECCR @ NCSU consortium under NIH support, is a toolbox for fitting and assessing quantitative structure-activity relationships (QSARs). Its elements are: a cheminformatic front end used to supply molecular descriptors for use in modeling; a set of methods for fitting models; and methods for validating the resulting model. Compounds may be input as structures from which standard descriptors will be calculated using the freely available cheminformatic front end PowerMV; PowerMV also supports compound visualization. In addition, the user can directly input their own choices of descriptors, so the capability for comparing descriptors is effectively unlimited. The statistical methodologies comprise a comprehensive collection of approaches whose validity and utility have been accepted by experts in the fields. As far as possible, these tools are implemented in open-source software linked into the flexible R platform, giving the user the capability of applying many different QSAR modeling methods in a seamless way. As promising new QSAR methodologies emerge from the statistical and data-mining communities, they will be incorporated in the laboratory. The web site also incorporates links to public-domain data sets that can be used as test cases for proposed new modeling methods. The capabilities of ChemModLab are illustrated using a variety of biological responses, with different modeling methodologies being applied to each. These show clear differences in quality of the fitted QSAR model, and in computational requirements. The laboratory is web-based, and use is free. Researchers with new assay data, a new descriptor set, or a new modeling method may readily build QSAR models and benchmark their results against other findings. Users may also examine the diversity of the molecules identified by a QSAR model. Moreover, users have the choice of placing their data sets in a public area to facilitate communication with other researchers; or can keep them hidden to preserve confidentiality.

Two point boundary value problems associated with a system of first order nonlinear impulse differential equations

BHESS uses Gaussian similarity transformations to reduce a general real square matrix to similar upper Hessenberg form. Multipliers are bounded in root mean square by a user-supplied parameter. If the input matrix is not highly nonnormal and the user-supplied tolerance on multipliers is of a size greater than ten, the returned matrix usually has small upper bandwidth. In such a case, eigenvalues of the returned matrix can be determined by the bulge-chasing BR iteration or by Rayleigh quotient iteration. BHESS followed by BR iteration determines a complete spectrum in about one-fifth the time required for orthogonal reduction to Hessenberg form followed by QR iterations. The FORTRAN 77 code provided for BHESS runs efficiently on a cache-based architecture.

Derivative error bounds for Lagrange interpolation: an extension of Cauchy's bound for the error of Lagrange interpolation

In 1978, Paiva [Nature 271, 434 (1978)] accomplished separation of gases by use of the ‘‘Taylor diffusion’’ which occurs in slow steady pipe flow. Results of Chatwin [J. Fluid Mech. 71, 513 (1975)], Watson [J. Fluid Mech. 133, 233 (1983)], and Kurzweg, Howell, and Jaeger [Phys. Fluids 27, 1046 (1984)] demonstrate that Taylor diffusion occurs also in the case of oscillatory pipe flow. A generalization of Paiva’s separation process to oscillatory flow is presented and possible advantages of such an implementation are explored.

LU Preconditioning for Overdetermined Sparse Least Squares Problems

We show assumptions of boundedness and Lipschitz conditions that guarantee the existence of a unique solution to a nonlinear differential boundary value problem. The result utilizes a variation of parameters of formula due to V. Lakshmikantham, et. al. and depends upon a contraction mapping.

Direct fail-proof triangularization algorithms for AX + XB = C with Error-Free and parallel implementations

Abstract It is shown that for any n + 1 times continuously differentiable function f and any choice of n + 1 knots, the Lagrange interpolation polynomial L of degree n satisfies ∥f (n) − L (n) ∥ ⩽ ∥ω (n) ∥ (n + 1)! ∥tf (n + 1) ∥ , where ∥ ∥ denotes the supremum norm. Further, this bound is the best possible. Applications of the above bound to the differencing formula are suggested. It is also shown that for j = 1, 2, …, n − 1, ∥f (j) − L (j) ∥ ⩽ ∥ω (j) ∥ j!(n + 1 − j)! ∥tf (n + 1) ∥ . This formula may be considered as a generalization of a formula due to Ciarlet, Schultz, and Varga (Numerical methods of high-order accuracy, Numer. Math. 9 (1967), 394–430) and may be compared to the conjectured best bound ∥f (j) − L (j) ∥ ⩽ ∥ω (j) ∥ (n + 1)! ∥tf (n + 1) ∥ .