A PARALLEL BISECTION ALGORITHM (WITHOUT COMMUNICATION) Rui Ralha DMAT, CMAT Univ. do Minho Portugal r_ralha@math.uminho.pt

CMAT FCT POCTI (European Union contribution) Prof. B. Parlett Acknowledgements CMAT FCT POCTI (European Union contribution) Prof. B. Parlett

Outline Counting eigenvalues of symmetric tridiagonals The ScaLAPACK’s routine A parallel algorithm without communication An alternative algorithm Some conclusions

Counting eigenvalues

Nonmonotonicity of Count(x)

The ScaLAPACK’s implementation (1)

The ScaLAPACK’s implementation (2) In [1] 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.

The ScaLAPACK’s implementation (3)

The ScaLAPACK’s implementation (4)

Drawbacks of the ScaLAPACK’s implementation

A simple and incorrect parallel algorithm (without communication) 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 [1] Therefore, the second eigenvalue will be computed twice (processors 1 and 3)

Parallel bisection for computing the eigenvalues of [-1 2 -1] with 100 processors

Our proposal (1)

Our proposal (2)

Our proposal (3)

Our proposal (4)

Sorting eigenvalues 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 .

An alternative algorithm (1) Phase 1(equal for every processor): carry out a (not too large) number of bisection steps in a breadth first search to get a “good picture” of the spectrum. Produces a number of intervals (at least p = number of processors). Phase 2: distributes intervals to processors trying to achieve load- balance (the same number of eigenvalues to each processor) Phase 3: each processor computes the assigned eigenvalues to some prescribed accuracy

An alternative algorithm (2)

An alternative algorithm (3) Preliminar implementation (in Matlab) Finishes Phase 1 when enough intervals have been produced such that, for each k=1,…,p-1, an end point x of one of those intervals satisfies This may affect the speedup by 10%. This termination criteria for Phase 1 may be hard (i.e, take too many bisection steps) to satisfy in some cases.

Parallel bisection for computing the eigenvalues of [-1 2 -1] of order 10^4

Conclusions Parallel bracketing in ScaLAPACK’s requires global communication; We have proposed an algorithm that is communication free and is load balanced in the sense that each processor computes the same number of eigenvalues (if p divides n); In homogeneous systems, our algorithm produces sorted eigenvalues even when the arithmetic is nonmonotonic; In heterogeneous systems, eigenvalues may be unsorted (they may be sorted by the “master” if required);