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Comparative Study of One-Sided Factorizations with Multiple Software Packages on Multi-Core Hardware Emmanuel A GULLO Jack D ONGARRA Bilel H ADRI Jakub K URZAK Hatem L TAIEF Piotr L USZCZEK Scheduling for Large-Scale Systems, Knoxville, TN, May


  1. Comparative Study of One-Sided Factorizations with Multiple Software Packages on Multi-Core Hardware Emmanuel A GULLO Jack D ONGARRA Bilel H ADRI Jakub K URZAK Hatem L TAIEF Piotr L USZCZEK Scheduling for Large-Scale Systems, Knoxville, TN, May 13-15, 2009 P LASMA group Comparative Study of One-Sided Factorizations 1

  2. Outline 1. Tile Algorithms Cholesky Factorization QR (&LU) Factorizations 2. Experimental environment Libraries Hardware Metrics 3. Tuning PLASMA 4. Comparison against other libraries Experiments on few cores Experiments on a large number of cores PLASMA scalability 5. Conclusion and current work P LASMA group Comparative Study of One-Sided Factorizations 2

  3. Tile Algorithms Outline 1. Tile Algorithms Cholesky Factorization QR (&LU) Factorizations 2. Experimental environment Libraries Hardware Metrics 3. Tuning PLASMA 4. Comparison against other libraries Experiments on few cores Experiments on a large number of cores PLASMA scalability 5. Conclusion and current work P LASMA group Comparative Study of One-Sided Factorizations 3

  4. Tile Algorithms Cholesky Outline 1. Tile Algorithms Cholesky Factorization QR (&LU) Factorizations 2. Experimental environment Libraries Hardware Metrics 3. Tuning PLASMA 4. Comparison against other libraries Experiments on few cores Experiments on a large number of cores PLASMA scalability 5. Conclusion and current work P LASMA group Comparative Study of One-Sided Factorizations 4

  5. Tile Algorithms Cholesky Tile Cholesky Factorization ��� ��������������� ���� ��� ����������� ���������������� � ������������������������ ������������ � ���������������� ���� ��� ����������������� ������� ��� ����������� �������������������� � ��������������������������������� ���������������� � ������������������������ ⋆ Basically identical to the block algorithm ( LAPACK ). ⋆ Input matrix stored and processed by square tiles. ⋆ Complex DAG. P LASMA group Comparative Study of One-Sided Factorizations 5

  6. Tile Algorithms Cholesky Tile Cholesky Factorization - Static pipeline ⋆ Work partitioned in one dimension (by block-rows). ⋆ Cyclic assignment of work across all steps of the factorization (pipelining of factorization steps). ⋆ Process tracking by a global progress table. ⋆ Stall on dependencies (busy waiting). P LASMA group Comparative Study of One-Sided Factorizations 6

  7. Tile Algorithms QR & LU Outline 1. Tile Algorithms Cholesky Factorization QR (&LU) Factorizations 2. Experimental environment Libraries Hardware Metrics 3. Tuning PLASMA 4. Comparison against other libraries Experiments on few cores Experiments on a large number of cores PLASMA scalability 5. Conclusion and current work P LASMA group Comparative Study of One-Sided Factorizations 7

  8. Tile Algorithms QR & LU Tile QR (&LU) Factorization ��� ��������������� ���� ����������������� � ���������������� ���� ��� ����������������� ���������������������������������� � ���������������������������������� ���� ��� ����������������� ���������������� � ���������������������������������� �������� ��� � ���������������� ����������������������������� � ������������������������������������������� ⋆ Different from the block algorithm. ⋆ Derived from out-of-core algorithm. ⋆ Input matrix stored and processed by square tiles. ⋆ Complex DAG. P LASMA group Comparative Study of One-Sided Factorizations 8

  9. Tile Algorithms QR & LU Tile QR Factorization - Static pipeline ������������������������������������ ���������������������������������������������� ����������������������������������������������� ⋆ Work partitioned in one dimension (by ����������������������������������������������������������� ����������� !"#���"��� ����� ����$���%&'(��) block-rows). �����**�������+�%&'(*�� ,� ����� ����� ����-��%&'(�..���-��%&'(��) ⋆ Cyclic assignment of work across all steps of the ������/�"���������/�" ��� ����/�"������ ������/�" **� ���� �� ����/�" �����%&'(��) factorization (pipelining of factorization steps). ����������/�"��*��#����"�� � �������� ����� ����/�"��$���%&'(�..���/�"��-��%&'(��) ��������������/�"�**����/�"������/�"�+�%&'(*��/�"�� ��������,���/�" �����/�"�� ⋆ Process tracking by a global progress table. ����, ���� �� ����������) �������� �� �� �������) ⋆ Stall on dependencies (busy waiting). ������������ ����� �0�������1�21�2�3���+��� �������������������41�21�2���1�21�2�� ������������0�������1�21�2����� ��������, �������� ���� ) ������������ ����� �0�������1 21�2�3���+��� �������������������41�21�2��41 21�2���1 21�2�� ������������0�������1 21�2����� ��������, ����, ���� ���� �) �������� �� �� �������) ������������ ����� �0�������1�21�2�3����� ������������ ����� �0�������1�21�2�3���+��� �������������������41�21�2���1�21�2��41�21�2�� ��������, �������� ���� �) ������������ ����� �0�������1 21�2�3����� ������������ ����� �0�������1 21�2�3���+��� �������������������41 21�2���1 21�2��41�21�2��41 21�2�� ������������0�������1 21�2����� ��������, ����, ����������/�"��� �����/�" ��������/�"�� , P LASMA group Comparative Study of One-Sided Factorizations 9

  10. Experimental environment Outline 1. Tile Algorithms Cholesky Factorization QR (&LU) Factorizations 2. Experimental environment Libraries Hardware Metrics 3. Tuning PLASMA 4. Comparison against other libraries Experiments on few cores Experiments on a large number of cores PLASMA scalability 5. Conclusion and current work P LASMA group Comparative Study of One-Sided Factorizations 10

  11. Experimental environment Libraries Outline 1. Tile Algorithms Cholesky Factorization QR (&LU) Factorizations 2. Experimental environment Libraries Hardware Metrics 3. Tuning PLASMA 4. Comparison against other libraries Experiments on few cores Experiments on a large number of cores PLASMA scalability 5. Conclusion and current work P LASMA group Comparative Study of One-Sided Factorizations 11

  12. Experimental environment Libraries Libraries ⋆ LAPACK : ◮ LAPACK 3.2 on Intel machine; ◮ LAPACK 3.1.1 on IBM machine; ⋆ SCALAPACK : ◮ SCALAPACK 1.8.0; ⋆ Vendor libraries: ◮ Intel MKL 10.1; ◮ IBM ESSL 4.3; ◮ IBM PESSL 3.3; ⋆ Tile algorithms: ◮ PLASMA ; ◮ TBLAS . P LASMA group Comparative Study of One-Sided Factorizations 12

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