Mixed precision bisection

We discuss the implementation of the bisection algorithm for the computation of the eigenvalues of symmetric tridiagonal matrices in a context of mixed precision arithmetic. This approach is motivated by the emergence of processors which carry out floating-point operations much faster in single precision than they do in double precision. Perturbation theory results are used to decide when to switch from single to double precision. Numerical examples are presente

Reliable eigenvalues of symmetric tridiagonals

For the eigenvalues of a symmetric tridiagonal matrix T, the most accurate algorithms deliver approximations which are the exact eigenvalues of a matrix whose entries differ from the corresponding entries of T by small relative perturbations. However, for matrices with eigenvalues of different magnitudes, the number of correct digits in the computed approximations for eigenvalues of size smaller than ‖T‖₂ depends on how well such eigenvalues are defined by the data. Some classes of matrices are known to define their eigenvalues to high relative accuracy but, in general, there is no simple way to estimate well the number of correct digits in the approximations. To remedy this, we propose a method that provides sharp bounds for the eigenvalues of T. We present some numerical examples to illustrate the usefulness of our method.FEDER (Programa Operacional Factores de Competitividade)FCT (Projecto PEst-C/MAT/UI0013/201

Performance degradation in the presence of subnormal floating-point values

Abstrac

Proposed Consistent Exception Handling for the BLAS and LAPACK

Numerical exceptions, which may be caused by overflow, operations like division by 0 or sqrt(-1), or convergence failures, are unavoidable in many cases, in particular when software is used on unforeseen and difficult inputs. As more aspects of society become automated, e.g., self-driving cars, health monitors, and cyber-physical systems more generally, it is becoming increasingly important to design software that is resilient to exceptions, and that responds to them in a consistent way. Consistency is needed to allow users to build higher-level software that is also resilient and consistent (and so on recursively). In this paper we explore the design space of consistent exception handling for the widely used BLAS and LAPACK linear algebra libraries, pointing out a variety of instances of inconsistent exception handling in the current versions, and propose a new design that balances consistency, complexity, ease of use, and performance. Some compromises are needed, because there are preexisting inconsistencies that are outside our control, including in or between existing vendor BLAS implementations, different programming languages, and even compilers for the same programming language. And user requests from our surveys are quite diverse. We also propose our design as a possible model for other numerical software, and welcome comments on our design choices