IWASEP VI, May 2006 SSVD Accurate Eigenvectors 1

Juan Manuel Molera(joint work with Froilán M. Dopico)

Departamento de Matemáticas,

Universidad Carlos III de Madridmolera@math.uc3m.es

Computing accurate eigenvectors Computing accurate eigenvectors with the SSVD Algorithmwith the SSVD Algorithm

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OutlineOutline

1. High Relative Accuracy Algorithms for the symmetric eigenvalue problem

2. SSVD Algorithm3. Computing (more) accurate eigenvectors

with the SSVD Algorithm4. Conclusions

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eig(A)eig(A)double(eig(sym(A)))double(eig(sym(A)))

Sometimes, QR Algorithm can failSometimes, QR Algorithm can fail

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eig(A)eig(A)double(eig(sym(A)))double(eig(sym(A)))

……can provide High Relative Accuracycan provide High Relative Accuracy

SSVD and J-ORTHOGONAL Algorithms …SSVD and J-ORTHOGONAL Algorithms …

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…….. ……..++ -- ++ ++--

…….. ……..

SSVD AlgorithmSSVD Algorithm

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If the SVD is computed with small multiplicative errorsIf the SVD is computed with small multiplicative errors

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Step 2.1: ClustersStep 2.1: Clusters

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Steps 2.2, 2.3Steps 2.2, 2.3

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Step 2.2: Putting the signsStep 2.2: Putting the signs

-- --

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Step 2.2: Putting the signsStep 2.2: Putting the signs

++++++ -- --

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Step 2.3: Getting the eigenvectorsStep 2.3: Getting the eigenvectors

++++++ -- --

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If the SVD is computed with small multiplicative errorsIf the SVD is computed with small multiplicative errors

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++++++ -- --

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The precision of the eigenvalues is The precision of the eigenvalues is always that of the singular valuesalways that of the singular values

……..++--++-- --

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……..++--++-- --

The precision of the eigenvectors is The precision of the eigenvectors is determined by the singular values relgapdetermined by the singular values relgap

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What can go wrong?What can go wrong?

++--++ -- --++---- ++ --

++++ ++++- - -- - - - - -- - -

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++--++ -- --++---- ++ --

++++ ++++- - -- - - - - -- - -

It can be fixed!It can be fixed!

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++--++ -- --++

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++--++ -- -- ------ -- --

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How is it done?How is it done?

++--++ -- --++

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How is it done?How is it done?

++--++ -- --++

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How is it done?How is it done?

++--++ -- --++

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How is it done?How is it done?

++--++ -- --++

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Algorithm to get PAlgorithm to get P

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++--++ -- --++

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Conclusions

• It uses an SVD as starting point• It uses orthogonal rotations• It delivers the same precision for evalues as the

precision provided for singular values• For the evectors, it is able to provide “relgap()-

governed” errors, surpassing the “relgap()-sing. vectors” delivered by the SVD

Algorithm SSVDAlgorithm SSVD

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Conclusions

Algorithm SSVDAlgorithm SSVD