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

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multiple rank updates to matrix factorizations
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Multiple-Rank Updates to Matrix Factorizations Zack 8/30/2013 - - PowerPoint PPT Presentation

Feb 17, 2024 460 likes •898 views

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

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Multiple-Rank Updates to Matrix Factorizations

Zack 8/30/2013

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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

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Introduction

u Dirext method

u Left-looking method u Right-looking method

u Computation effort

u O(n3) for dense matrix u O(n1.05-2) for sparse matrix

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Introduction

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Multiple-rank Update Theory

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Multiple-rank Update Theory

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Multiple-rank Update Theory

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Multiple-rank Update Theory

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Multiple-rank Update Theory

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Multiple-rank Update Theory

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Multiple-rank Update Theory

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Multiple-rank Update Theory

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Multiple-rank Update Theory

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Multiple-rank Update Theory

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Multiple-rank Update Theory

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Multiple-rank Update Theory

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Multiple-rank Update Theory

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Multiple-rank Update Theory

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Multiple-rank Update Theory

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Multiple-rank Update Theory

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Multiple-rank Update Theory

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Multiple Rank-1s Update

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Group Update

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Group Update

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Multiple-rank Update Theory

Methods Flops Comments

Left-looking LU O(n2) memory write, bad in parallel Right-looking LU O(n3) memory write, good in parallel QR O(n3) memory write, good in parallel n rank-1 updates O(n3) memory write, good in parallel

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Multiple-rank Update Theory

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Sparse Matrix Factorization by Multiple- rank Update

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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.

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Sparse Matrix Factorization by Multiple- rank Update

u Base-camp Theory

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Symmetric Factorization by Multiple- rank Update

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Symmetric Factorization by Multiple- rank Update

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Symmetric Factorization by Multiple- rank Update

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Symmetric Factorization by Multiple- rank Update

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Symmetric Factorization by Multiple- rank Update

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Symmetric Factorization by Multiple- rank Update

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Symmetric Factorization by Multiple- rank Update

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Symmetric Factorization by Multiple- rank Update

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Symmetric Factorization by Multiple- rank Update

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Symmetric Factorization by Multiple- rank Update

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Conclusion

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

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References

u Deng L. Multiple-rank Updates to Matrix Factorizations for Nonlinear Analysis

and Circuit Design[D]. Stanford University, 2010.

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THANK YOU