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- Typically, the eigenvalue problem is solved in
Parallel Solution of Symmetric Eigenvalue Problems Zack 2/21/2014 - - PowerPoint PPT Presentation
Parallel Solution of Symmetric Eigenvalue Problems Zack 2/21/2014 - - PowerPoint PPT Presentation
Parallel Solution of Symmetric Eigenvalue Problems Zack 2/21/2014 Typically, the eigenvalue problem is solved in three successive steps: Reduction to tridiagonal form Solution of the tridiagonal eigenproblem Back transformation
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- (2005) Compute all eigenpairs of a
15,000*15,000 matrix on 16 processor:
– 546s for reduction – 22.2s for tridiagonal solution – 160s for backtransformation
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Reduction to banded (full matrix)
- Householder transformation
- Original one-step tridiagonalization
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If we only reduce the matrix to banded form, the update can be done with a block orthogonal transformation, such as WY representation 𝑅 = 𝐽 + 𝑋𝑍𝑈 with width-b matrices W and Y, then the vast majority of the operations can be done with highly efficient BLAS-3 routines instead
- f memory bandwidth-limited BLAS-2 or BLAS-1.
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Band-to-tridiagonal
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Band-to-tridiagonal
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Band-to-tridiagonal
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Band-to-tridiagonal
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Band-to-tridiagonal
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Band-to-tridiagonal
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- Algorithm
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Reduction to banded (sparse matrix)
- Cuthill-McKee(CMK) ordering
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1 𝑦 2 … 𝑦 3 4 𝑦 𝑦 𝑦 𝑦 5 6 7 → 1 𝑦 2 … 𝑦 5 𝑦 3 𝑦 6 𝑦 4 𝑦 7 Band 3 band 1 With CMK method, we can reduce the band without flop computation
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1 2 … 𝑦 𝑦 3 4 𝑦 𝑦 5 6 𝑦 𝑦 7 → 1 2 … 𝑦 𝑦 3 𝑦 7 𝑦 𝑦 𝑦 5 6 4 Band 6 band 3 if some node has large degree, it is hard to reduce band to a small number with permutation.
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- A possible way to deal with the nodes which
has large degree:
- 1. move these nodes to bottom;
- 2. solve the eigen-update
𝐵 𝑐 𝑐𝑈 𝑑 𝑦 𝑧 = 𝜇 𝑦 𝑧 where A is banded matrix, which is easy to be decomposed, b is dense vector.
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- 𝐵
𝑐 𝑐𝑈 𝑑 𝑦 𝑧 = 𝜇 𝑦 𝑧 𝐵𝑦 + 𝑐𝑧 = 𝜇𝑦 𝑐𝑈𝑦 + 𝑑𝑧 = 𝜇𝑧 𝐵𝑦 +
𝑐𝑐𝑈 𝜇−𝑑 𝑦 = 𝜇𝑦
𝐵 +
𝑐𝑐𝑈 𝜇−𝑑 − 𝜇𝐽 𝑦 = 0
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- 𝐵 +
𝑐𝑐𝑈 𝜇−𝑑 − 𝜇𝐽 𝑦 = 0
Suppose that 𝐵 = 𝐹𝐸𝐹𝑈 det 𝐸 + 𝑨𝑨𝑈 𝜇 − 𝑑 − 𝜇𝐽 = 0
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- det 𝐸 +
𝑨𝑨𝑈 𝜇−𝑑 − 𝜇𝐽
= 0
- Secular equation:
1 + 𝑗
𝑨𝑗
2
𝑒𝑗−𝜇 𝜇−𝑑 = 0
which is similar to rank-1 update
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Band-to-tridiagonal (non-uniform)
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Tridiagonal eigensolver
Combination of bisection and inverse iteration(B&I) Able to compute a subset of k eigenpairs at reduced cost Lose orthogonality O(k2n) MRRR O(kn) QR/QL Designed to compute either all
- r no eigenvectors
Good eigensystem O(n3) Divide-and- conquer(D&C)
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Divide-and-conquer algorithm
- Step 1, Split
the original tridiagonal matrix could be split into two half sized submatrices:
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- Step 2, solve subproblems
when subproblems are small enough, call QR/QL method, then we have where,
w
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- Step 3, deflate
(1) some components zi of z are (almost) zero, (2) two elements in diagonal matrix are identical (or close), then zero elments can be generated in z
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- Step 4, rank-1 update eigendecomposition
– Solve secular equation 1 + σ 𝑗=1
𝑜 𝜂𝑗
2
𝑒𝑗−𝜇 = 0
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Back transformation to eigenvectors
- Eigenvectors of T eigenvectors of A
- 𝐹(𝐵) = 𝐼1𝐼2 … 𝐼𝑞𝐹(𝑈)
- -- O(n2p)
– 1D data layout – 2D data layout
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