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Multiple-Rank Updates to Matrix Factorizations Zack 8/30/2013 - PowerPoint PPT Presentation

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Multiple-Rank Updates to Matrix Factorizations Zack 8/30/2013 Outline u Introduction u Multiple-rank Update Theory u Sparse Matrix Factorization by Multiple-rank Update u Symmetric Factorization by Multiple-rank Update u Conclusion Introduction

  1. Multiple-Rank Updates to Matrix Factorizations Zack 8/30/2013

  2. Outline u Introduction u Multiple-rank Update Theory u Sparse Matrix Factorization by Multiple-rank Update u Symmetric Factorization by Multiple-rank Update u Conclusion

  3. Introduction u Dirext method u Left-looking method u Right-looking method u Computation effort u O(n 3 ) for dense matrix u O(n 1.05-2 ) for sparse matrix

  4. Introduction u

  5. Multiple-rank Update Theory u

  6. Multiple-rank Update Theory u

  7. Multiple-rank Update Theory u

  8. Multiple-rank Update Theory u

  9. Multiple-rank Update Theory u

  10. Multiple-rank Update Theory u

  11. Multiple-rank Update Theory u

  12. Multiple-rank Update Theory u

  13. Multiple-rank Update Theory u

  14. Multiple-rank Update Theory u

  15. Multiple-rank Update Theory u

  16. Multiple-rank Update Theory u

  17. Multiple-rank Update Theory u

  18. Multiple-rank Update Theory u

  19. Multiple-rank Update Theory u

  20. Multiple-rank Update Theory u

  21. Multiple-rank Update Theory u

  22. Multiple Rank-1s Update

  23. Group Update

  24. Group Update

  25. Multiple-rank Update Theory Methods Flops Comments Left-looking LU O(n 2 ) memory write, bad in parallel Right-looking LU O(n 3 ) memory write, good in parallel QR O(n 3 ) memory write, good in parallel n rank-1 updates O(n 3 ) memory write, good in parallel

  26. Multiple-rank Update Theory u

  27. Sparse Matrix Factorization by Multiple- rank Update u

  28. Sparse Matrix Factorization by Multiple- rank Update u The path of a rank-1 update follow exactly the path of the elimination tree. u Tridiagonal matrix’s elimination tree.

  29. Sparse Matrix Factorization by Multiple- rank Update u Base-camp Theory

  30. Symmetric Factorization by Multiple- rank Update u

  31. Symmetric Factorization by Multiple- rank Update u

  32. Symmetric Factorization by Multiple- rank Update u

  33. Symmetric Factorization by Multiple- rank Update u

  34. Symmetric Factorization by Multiple- rank Update u

  35. Symmetric Factorization by Multiple- rank Update u

  36. Symmetric Factorization by Multiple- rank Update u

  37. Symmetric Factorization by Multiple- rank Update u

  38. Symmetric Factorization by Multiple- rank Update u

  39. Symmetric Factorization by Multiple- rank Update u

  40. Conclusion u Introduced multiple 1-rank update method u Optimized the method when matrix is sparse. u Optimized the method when matrix is symmetry.

  41. References u Deng L. Multiple-rank Updates to Matrix Factorizations for Nonlinear Analysis and Circuit Design[D]. Stanford University, 2010.

  42. THANK YOU

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