Performance Modeling and Optimal Block Size Selection for the Small-Bulge Multishift QR Algorithm

The small-bulge multishift QR algorithm proposed by Braman, Byers and Mathias is one of the most efficient algorithms for computing the eigenvalues of nonsymmetric matrices on processors with hierarchical memory. However, to fully extract its potential performance, it is crucial to choose the block size m properly according to the target architecture and the matrix size n. In this paper, we construct a performance model for this algorithm. The model has a hierarchical structure that reflects the structure of the original algorithm and given n, m and the performance data of the basic components of the algorithm, such as the level-3 BLAS routines and the double implicit shift QR routine, predicts the total execution time. Experiments on SMP machines with PowerPC G5 and Opteron processors show that the variation of the execution time as a function of m predicted by the model agrees well with the measurements. Thus our model can be used to automatically select the optimal value of m for a given matrix size on a given architecture.

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