LAPACK is often mentioned as a positive example of a software library that encapsulates complex, robust, and widely used numerical algorithms for a wide range of applications. At installation time, the user has the option of running a (limited) number of test cases to verify the integrity of the installation process. On the algorithm developer's side, however, more exhaustive tests are usually performed to study algorithm behavior on a variety of problem settings and also computer architectures. In this process, difficult test cases need to be found that reflect particular challenges of an application or push algorithms to extreme behavior. These tests are then assembled into a comprehensive collection, therefore making it possible for any new or competing algorithm to be stressed in a similar way. This article describes an infrastructure for exhaustively testing the symmetric tridiagonal eigensolvers implemented in LAPACK. It consists of two parts: a selection of carefully chosen test matrices with particular idiosyncrasies and a portable testing framework that allows for easy testing and data processing. The tester facilitates experiments with algorithmic choices, parameter and threshold studies, and performance comparisons on different architectures.

In the 1990's, Dhillon and Parlett devised the algorithm of multiple relatively robust representations (MRRR) for computing numerically orthogonal eigenvectors of a symmetric tridiagonal matrix T with O(n 2 ) cost. While previous publications related to MRRR focused on theoretical aspects of the algorithm, a documentation of software issues has been missing. In this article, we discuss the design and implementation of the new MRRR version STEGR that will be included in the next LAPACK release. By giving an algorithmic description of MRRR and identifying governing parameters, we hope to make STEGR more easily accessible and suitable for future performance tuning. Furthermore, this should help users understand design choices and tradeoffs when using the code.

Tensor topology is useful in providing a simplified and yet detailed representation of a tensor field. Recently the field of 3D tensor topology is advanced by the discovery that degenerate tensors usually form lines in their most basic configurations. These lines form the backbone for further topological analysis. A number of ways for extracting and tracing the degenerate tensor lines have also been proposed. In this paper, we complete the previous work by studying the behavior and extracting the separating surfaces emanating from these degenerate lines. First, we show that analysis of eigenvectors around a 3D degenerate tensor can be reduced to 2D. That is, in most instances, the 3D separating surfaces are just the trajectory of the individual 2D separatrices which includes trisectors and wedges. But the proof is by no means trivial since it is closely related to perturbation theory around a pair of singular slate. Such analysis naturally breaks down at the tangential points where the degenerate lines pass through the plane spanned by the eigenvectors associated with the repeated eigenvalues. Second, we show that the separatrices along a degenerate line may switch types (e.g. trisectors to wedges) exactly at the points where the eigenplane is tangential to the degenerate curve. This property leads to interesting and yet complicated configuration of surfaces around such transition points. Finally, we apply the technique to several common data sets to verify its correctness.

This paper addresses several issues related to topological analysis of 3D second order symmetric tensor fields. First, we show that the degenerate features in such data sets form stable topological lines rather than points, as previously thought. Second, the paper presents two different methods for extracting these features by identifying the individual points on these lines and connecting them. Third, this paper proposes an analytical form of obtaining tangents at the degenerate points along these topological lines. The tangents are derived from a Hessian factorization technique on the tensor discriminant and leads to a fast and stable solution. Together, these three advances allow us to extract the backbone topological lines that form the basis for topological analysis of tensor fields.

Background1.1 The Sensitivity of Eigenvalues.Several good programs are available for the computation of the eigenvalues of real and complex matrices [2, 3,7].Because of the limitations of finite precision arithmetic, these programs cannot produce, in general, the exact eigenvalues of the given matrix A. However the computed numbers are always (very close to) the eigenvalues of a matrix A -k E which is very close to A. This matrix E is not unique and error analyses ~6] have shown the existence of E's with satisfactorily small upper bounds on IIE II / II A If.Here I1" II denotes an appropriate matrix norm.It follows from these remarks that a good program will not always deliver accurate approximations to the eigenvalues of A. It can happen that some, or all, of the eigenvalues are very sensitive to changes in the matrix elements; so some, or all, of the eigenvalues of A ~-E may differ sharply from those of A. Actually this is true only for non-normal matrices.Real symmetric matrices--indeed all normal matrices--determine their eigenvalues very well; the change induced in an eigenvalue of such an A cannot exceed the spectral norm of E (which is defined below).Two questions arise: How can this sensitivity be measured, and how cheaply can it be computed?Simple eigenvalues.To any simple eigenvalue ~ of A there correspond both a

article The Influence of the Compiler on the Cost of Mathematical Software—in Particular on the Cost of Triangular Factorization Share on Authors: B. N. Parlett Computer Science Division, University of California, Berkeley CA Computer Science Division, University of California, Berkeley CAView Profile , Y. Wang Department of Mathematms, Umverslty of California, Berkeley, CA Department of Mathematms, Umverslty of California, Berkeley, CAView Profile Authors Info & Claims ACM Transactions on Mathematical SoftwareVolume 1Issue 1March 1975 pp 35–46https://doi.org/10.1145/355626.355633Online:01 March 1975Publication History 14citation288DownloadsMetricsTotal Citations14Total Downloads288Last 12 Months4Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

Manipulative algebra obtain the next approximation to y.The process may then be repeated.It will be obvious that the examples given above make extensive use of the restricted forms of the multiplication and substitution programs, and in all the work that has been undertaken with the scheme these restricted forms have been more frequently used than the actual exact routines themselves.