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Abstract: 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... (Update)
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.... of processors with identical floating point formats (but slightly different floating point operations turn out to be acceptable) See [4] for further discussion. Assigning the work by index rather than by range and sorting all the eigenvalues at the end may give the desired...
.... Jaeyoung Choi [3] Step 2 is broken into two parts, bisection and inverse iteration, parts of which were written by Inderjit Dhillon [5]. Both bisection and inverse iteration do O(1) communication, with each processor responsible for a subset of eigenvalues and eigenvectors....
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2.1: On the Correctness of Parallel Bisection in Floating Point - Demmel, Dhillon, Ren (1994) (Correct)
0.3: Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector.. - Dhillon (1997) (Correct)
0.3: On Designing Portable High Performance . . . - Demmel, al. (1991) (Correct)
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3: Scalapack: A scalable linear algebra library for distributed memory concurrent c.. (context) - Choi - 1992
3: IEEE standard for binary floating-point arithmetic (context) - IEEE - 1985
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BibTeX entry: (Update)
J. Demmel, I. Dhillon, and H. Ren. "On the correctness of parallel bisection in floating point". Technical Report UCB//CSD-94-805, University of California, Berkeley Computer Science Division, 1994. available via anonymous ftp from tr-ftp.cs.berkeley.edu, in directory pub/tech-reports/csd/csd-94-805, file all.ps. http://citeseer.ist.psu.edu/article/demmel94correctness.html More
@techreport{ demmel94correctness,
author = "James Demmel and Inderjit Dhillon and Huan Ren",
title = "On the correctness of parallel bisection in floating point",
number = "UCB/CSD 94/805",
address = "Berkeley, CA, USA",
pages = "38",
year = "1994",
url = "citeseer.ist.psu.edu/article/demmel94correctness.html" }
Citations (may not include all citations):624 The Algebraic Eigenvalue Problem (context) - Wilkinson - 1965
532 LAPACK Users' Guide (context) - Anderson, Bai et al. - 1992
441 The Symmetric Eigenvalue Problem (context) - Parlett - 1980
175 Matrix Eigensystem Routines -- EISPACK Guide (context) - Smith, Boyle et al. - 1976
133 IEEE Standard for Binary Floating Point Arithmetic (context) - IEEE, York - 1985
81 Accurate singular values of bidiagonal matrices - Demmel, Kahan - 1990
78 Computing accurate eigensystems of scaled diagonally dominan.. - Barlow, Demmel - 1990
76 Parallel numerical linear algebra - Demmel, Heath et al. - 1993
28 On computing accurate singular values and eigenvalues of acy.. - Demmel, Gragg - 1993
24 IEEE Standard for Radix Independent Floating Point Arithmeti.. (context) - IEEE, York - 1987
21 Underflow and the reliability of numerical software (context) - Demmel - 1984
13 Accurate eigenvalues of a symmetric tridiagonal matrix (context) - Kahan - 1966
8 massively parallel eigenvalue computations (context) - Huo, Schreiber - 1993
8 Parallel implementation of bisection for the calculation of .. (context) - Bernstein, Goldstein - 1985
8 On finding eigenvalues and singular values by bisection (context) - Assadullah, Demmel et al.
6 A multiprocessor algorithm for the symmetric eigenproblem (context) - Lo, Phillipe et al. - 1987
6 Analysi and refutation International Standard ISOIEC Languag.. (context) - Analysis, the et al. - 1991
5 A parallel algorithm for the symmetric tridiagonal eigenprob.. (context) - Dhillon, Demmel - 1993
4 The instability and nonmonotonicity of the parallel prefix a.. (context) - Demmel, Ren - 1994
3 When to neglect offdiagonal elements of symmetric tridiagona.. (context) - Kahan - 1966
2 The stability of parallel prefix matrix multiplication with .. (context) - Mathias - 1993
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