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16a. While each of those diamonds is considered as one block, their back transformation (application to the matrix Z) needs to follow the dependency order. For example, applying block 4 and block 5 of the V2 ’s in Fig. 16a modifies block row 4 and block row 5, respectively, of the eigenvector matrix Z drawn in Fig. 16b where one can easily observe the overlapped region. The order dictates that block 4 needs to be applied before block 5. It is possible to compute this phase efficiently by splitting Z by blocks of columns over both the CPUs and the GPU as shown in Fig.

Some unique properties, such as numerical stability, of a GPU tridiagonal solver for the application are also very critical. So far, only few previous works [4, 23] recognized the numerical stability issue of current GPU tridiagonal solvers, and even fewer ones [4] investigated it. Numerical stability becomes the most important future work for the research of GPU tridiagonal solvers. Acknowledgements This project was partly supported by the STARnet Center for Future Architecture Research (C-FAR), the DoE Vancouver Project (DE-FC02-10ER26004/DE-SC0005515), and the UIUC CUDA Center of Excellence.

Appl. Math. 27(1–2), 215–227 (1989) 4. : Multi-sweep algorithms for the symmetric eigenproblem. In: Vector and Parallel Processing - VECPAR’98. Lecture Notes in Computer Science, vol. 1573, pp. 20–28. Springer, Berlin (1999) 5. : Matrix Computations, 3rd edn. Johns Hopkins, Baltimore (1996) 6. : Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In: Proceedings of SC ’11, pp. 8:1–8:11. ACM, New York (2011) 7. : A comprehensive study of task coalescing for selecting parallelism granularity in a two-stage bidiagonal reduction.

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14th International Symposium on Mathematical Programming

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