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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 molera@math.uc3m.es. Outline. High Relative Accuracy Algorithms for the symmetric eigenvalue problem
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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 molera@math.uc3m.es SSVD Accurate Eigenvectors
Outline • High Relative Accuracy Algorithms for the symmetric eigenvalue problem • SSVD Algorithm • Computing (more) accurate eigenvectors with the SSVD Algorithm • Conclusions SSVD Accurate Eigenvectors
Sometimes, QR Algorithm can fail double(eig(sym(A))) eig(A) SSVD Accurate Eigenvectors
SSVD and J-ORTHOGONAL Algorithms … double(eig(sym(A))) eig(A) …can provide High Relative Accuracy SSVD Accurate Eigenvectors
…. …. - - + + + …. …. SSVD Algorithm SSVD Accurate Eigenvectors
If the SVD is computed with small multiplicative errors SSVD Accurate Eigenvectors
Step 2.1: Clusters SSVD Accurate Eigenvectors
Steps 2.2, 2.3 SSVD Accurate Eigenvectors
Step 2.2: Putting the signs -- SSVD Accurate Eigenvectors
Step 2.2: Putting the signs -- +++ SSVD Accurate Eigenvectors
-- +++ Step 2.3: Getting the eigenvectors SSVD Accurate Eigenvectors
If the SVD is computed with small multiplicative errors SSVD Accurate Eigenvectors
-- +++ SSVD Accurate Eigenvectors
- - - + + …. The precision of the eigenvalues is always that of the singular values SSVD Accurate Eigenvectors
The precision of the eigenvectors is determined by the singular values relgap - - - + + …. SSVD Accurate Eigenvectors
- - - - - - + + + + - - - - - - + + + + What can go wrong? SSVD Accurate Eigenvectors
- - - - - - + + + + - - - - - - + + + + It can be fixed! SSVD Accurate Eigenvectors
- - - + + + SSVD Accurate Eigenvectors
- - - - - - - - + + SSVD Accurate Eigenvectors
How is it done? - - - + + + SSVD Accurate Eigenvectors
How is it done? - - - + + + SSVD Accurate Eigenvectors
How is it done? - - - + + + SSVD Accurate Eigenvectors
How is it done? - - - + + + SSVD Accurate Eigenvectors
Algorithm to get P SSVD Accurate Eigenvectors
- - - + + + SSVD Accurate Eigenvectors
Conclusions Algorithm SSVD • 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 SSVD Accurate Eigenvectors
Conclusions Algorithm SSVD SSVD Accurate Eigenvectors