George J. Davis

Six factors which affect the condition number of matrices associated with kriging

Determining kriging weights to estimate some variable of interest at a given point in the field involves solving a system of linear equations. The matrix of this linear system is subject to numerical instability, and this instability is measured by the matrix condition number. Six parameters in the kriging process have been identified which directly affect this condition number. Analysis of a series of 648 experiments gives some insight on these parameters, and how the condition number relates to kriging variance.

Finding eigenvalues and eigenvectors of unsymmetric matrices using a hypercube multiprocessor

Distributed-memory algorithms for finding the eigenvalues and eigenvectors of a dense unsymmetric matrix are given. While several parallel algorithms have been developed for symmetric systems, little work has been done on the unsymmetric case. Our parallel implementation proceeds in three major steps: reduction of the original matrix to Hessenberg form, application of the implicit double-shift QR algorithm to compute the eigenvalues, and back transformations to compute the eigenvectors. Several modifications to our parallel QR algorithm, including ring communication and pipelining, are discussed and compared. Results and timings are given.

Algorithm 598: an algorithm to compute solvent of the matrix equation AX 2 + BX + C = 0

a n d we s eek a m a t r i x S such t h a t F ( S ) = O. S u c h a m a t r i x S is c a l l ed a solvent . M u c h o f w h a t is k n o w n a b o u t s o l v e n t s a n d t h e i r c a l c u l a t i o n c a n b e f o u n d in L a n c a s t e r [7], a n d t h e w o r k s o f Denn i s , T r a u b , a n d W e b e r [2, 3]. T h i s F O R T R A N p a c k a g e i m p l e m e n t s t h e i d e a s in [1], w h e r e s o m e p e r t u r b a t i o n t h e o r y a n d a n e r r o r ana lys i s a r e p r e s e n t e d . F i n d i n g a s o l v e n t o f F is c lose ly r e l a t e d to t h e q u a d r a t i c e i genva lue p r o b l e m of f ind ing sca l a r s ~ a n d n o n z e r o v e c t o r s x s u c h t h a t

The rank of a graph after vertex addition

Abstract Let r(G) denote the rank of the adjacency matrix of a graph G. When a vertex and its incident edges are deleted from G, the rank of the resultant graph cannot exceed r(G) and can decrease by at most 2. The problem of determining the exact effect of adding a single vertex to a graph is more difficult, since the number of edges that can be added with this vertex is variable. The rank of the new graph cannot decrease and it can increase by at most 2. We obtain results examining several cases of vertex addition.

Adaptive quadrature on a message-passing multiprocessor

Abstract We present an adaptive quadrature algorithm for message-passing multiprocessors. One node processor is specially designated to collect partial sums and dynamically balance the work-load. Four different redistribution strategies are considered. Results and timings for several different functions are given.

Assessment of Linear Dependencies in Multivariate Data

A procedure to identify the linear dependency structure in multivariate data is presented. The linear dependency analysis (LDA) provides a method for assessing the number of dependencies using the eigenvalues of the sample correlation matrix. The dependency structure is then identified from the right singular vectors from a singular value decomposition of the centered and scaled data matrix. An algorithm to identify competing dependencies is given along with procedures for estimating and testing the dependency coefficients. Example data sets from regression, factor analysis, discriminant analysis, and principal component analysis are analyzed.

Algorithm 633: An algorithm for linear dependency analysis of multivariate data

The importance of detecting statistical dependencies in multivariate data has been discussed many times (e.g., Belsley et al. [l]). Recently, Kane et al. [5] have developed a procedure called Linear Dependency Analysis (LDA), which assesses the existence of linear dependencies in a multivariate data matrix X. This paper describes the algorithm implementing the LDA procedure. A brief description of some of the statistical and linear algebra theory behind the procedure is given below for notational purposes. Kane et al. [5] should be consulted for additional details and discussion of the procedure’s theoretical foundations. LDA examines the potential for partitioning the R. X p matrix X into the n X p1 matrix X1 containing the predictor variables and the n x p2 matrix X2 containing the estimated variables, and the appropriateness of using the linear relationship

Opportunities in international teacher exchanges (abstract)

Teaching in a foreign country is an exating opportunity which contains many interesting challenges and rewards. Exchange teachers must understand cliff erences in culture and educational systems which directly effect how students learn. In the process, teachers often rethink their entire teaching strategy, posing important questions about themselves and their disciplines. The resulting perspectives gained by exchangers adds new dimensions to both their professional and personal lives.