Handling floating-point exceptions in numeric programs
ACM Transactions on Programming Languages and Systems (TOPLAS)
Practical experience in the numerical dangers of heterogeneous computing
ACM Transactions on Mathematical Software (TOMS)
Peer to peer large scale linear algebra programming and experimentations
LSSC'05 Proceedings of the 5th international conference on Large-Scale Scientific Computing
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Bisection is an easily parallelizable method for finding the eigenvalues of real symmetric tridiagonal matrices, or more generally symmetric acyclic matrices. It requires a function Count(x) which counts the number of eigenvalues less than x. In exact arithmetic Count(x) is an increasing function of x, but this is not necessarily the case with roundoff. Our first result is that as long as the floating point arithmetic is monotonic, the computed function Count(x) implemented appropriately will also be monotonic; this extends an unpublished 1966 result of Kahan to the larger class of symmetric acyclic matrices. Second, we analyze the impact of non\-monotonicity of Count(x) on the serial and parallel implementations of bisection. We present simple and natural implementations which can fail because of nonmonotonicity; this includes the routine bisect in EISPACK. We also show how to implement bisection correctly despite nonmonotonicity; this is important because the fastest known parallel implementation of Count(x) is nonmonotonic even if the floating point is not.