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

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 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

Introduction u

Multiple-rank Update Theory u

Multiple-rank Update Theory u

Multiple-rank Update Theory u

Multiple-rank Update Theory u

Multiple-rank Update Theory u

Multiple-rank Update Theory u

Multiple-rank Update Theory u

Multiple-rank Update Theory u

Multiple-rank Update Theory u

Multiple-rank Update Theory u

Multiple-rank Update Theory u

Multiple-rank Update Theory u

Multiple-rank Update Theory u

Multiple-rank Update Theory u

Multiple-rank Update Theory u

Multiple-rank Update Theory u

Multiple-rank Update Theory u

Multiple Rank-1s Update

Group Update

Group Update

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

Multiple-rank Update Theory u

Sparse Matrix Factorization by Multiple- rank Update u

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.

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

Symmetric Factorization by Multiple- rank Update u

Symmetric Factorization by Multiple- rank Update u

Symmetric Factorization by Multiple- rank Update u

Symmetric Factorization by Multiple- rank Update u

Symmetric Factorization by Multiple- rank Update u

Symmetric Factorization by Multiple- rank Update u

Symmetric Factorization by Multiple- rank Update u

Symmetric Factorization by Multiple- rank Update u

Symmetric Factorization by Multiple- rank Update u

Symmetric Factorization by Multiple- rank Update u

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

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

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