computing accurate eigenvectors with the ssvd algorithm
DESCRIPTION
more. Computing accurate eigenvectors with the SSVD Algorithm. Juan Manuel Molera (joint work with Froilán M. Dopico) Departamento de Matemáticas, Universidad Carlos III de Madrid [email protected]. Outline. High Relative Accuracy Algorithms for the symmetric eigenvalue problem - PowerPoint PPT PresentationTRANSCRIPT
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 1
Juan Manuel Molera(joint work with Froilán M. Dopico)
Departamento de Matemáticas,
Universidad Carlos III de [email protected]
Computing accurate eigenvectors Computing accurate eigenvectors with the SSVD Algorithmwith the SSVD Algorithm
moremore
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 2
OutlineOutline
1. High Relative Accuracy Algorithms for the symmetric eigenvalue problem
2. SSVD Algorithm3. Computing (more) accurate eigenvectors
with the SSVD Algorithm4. Conclusions
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 3
eig(A)eig(A)double(eig(sym(A)))double(eig(sym(A)))
Sometimes, QR Algorithm can failSometimes, QR Algorithm can fail
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 4
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 …
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 5
…….. ……..++ -- ++ ++--
…….. ……..
SSVD AlgorithmSSVD Algorithm
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 6
If the SVD is computed with small multiplicative errorsIf the SVD is computed with small multiplicative errors
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 7
Step 2.1: ClustersStep 2.1: Clusters
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 8
Steps 2.2, 2.3Steps 2.2, 2.3
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 9
Step 2.2: Putting the signsStep 2.2: Putting the signs
-- --
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 10
Step 2.2: Putting the signsStep 2.2: Putting the signs
++++++ -- --
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 11
Step 2.3: Getting the eigenvectorsStep 2.3: Getting the eigenvectors
++++++ -- --
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 12
If the SVD is computed with small multiplicative errorsIf the SVD is computed with small multiplicative errors
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 13
++++++ -- --
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 14
The precision of the eigenvalues is The precision of the eigenvalues is always that of the singular valuesalways that of the singular values
……..++--++-- --
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 15
……..++--++-- --
The precision of the eigenvectors is The precision of the eigenvectors is determined by the singular values relgapdetermined by the singular values relgap
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 16
What can go wrong?What can go wrong?
++--++ -- --++---- ++ --
++++ ++++- - -- - - - - -- - -
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 17
++--++ -- --++---- ++ --
++++ ++++- - -- - - - - -- - -
It can be fixed!It can be fixed!
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 18
++--++ -- --++
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 19
++--++ -- -- ------ -- --
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 20
How is it done?How is it done?
++--++ -- --++
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 21
How is it done?How is it done?
++--++ -- --++
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 22
How is it done?How is it done?
++--++ -- --++
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 23
How is it done?How is it done?
++--++ -- --++
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 24
Algorithm to get PAlgorithm to get P
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 25
++--++ -- --++
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 26
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
IWASEP VI, May 2006 SSVD Accurate Eigenvectors 27
Conclusions
Algorithm SSVDAlgorithm SSVD