A PARALLEL BISECTION ALGORITHM (WITHOUT COMMUNICATION). Rui Ralha DMAT, CMAT Univ. do Minho Portugal [email protected] Acknowledgements. CMAT FCT POCTI (European Union contribution) Prof. B. Parlett. Outline. Counting eigenvalues of symmetric tridiagonals
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Univ. do Minho
POCTI (European Union contribution)
Prof. B. Parlett
In  the authors wrote
“…Ideally, we would like a bracketing algorithm that was simultaneously parallel, load balanced, devoid of communication, and correct in the face of nonmonotonicity. We still do not know how to achieve this completely; in the most general case, when different parallel processors do not even possess the same floating point format, we do not know how to implement a correct and reasonably fast algorithm at all. Even when floating point formats are the same, we do not know how to avoid some global communication…”
and considered a bracketing algorithm to be correct if
(1) every eigenvalue is computed exactly once,
(2) the computed eigenvalues are correct to within the user
specified error tolerance,
(3) the computed eigenvalues are in sorted order.
To partition the initial Gerschgorin interval into p subintervals of equal width
and assign to processor i the task of finding all the eigenvalues in the
ith subinterval . But, even with processors with the same arithmetic
(nonmonotonic) the algorithm may be incorrect.
For example, with n=p=3, it may happen 
Therefore, the second eigenvalue will be computed twice (processors 1 and 3)
For the Wilkinson’s matrix of order 21 we have
With single precision in Matlab we get
With double precision we get
We assume that eigenvalues are to be gathered in a “master”
processor (this is a standard feature of ScaLAPACK). Supose that the
“master” receives (out of order) and knows that the
processor that computed has better accuracy. Then, it keeps
and, if required, it corrects to be smaller than .
Conclusions -1] of order 10^4