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Computing accurate eigenvectors with the SSVD Algorithm

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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Computing accurate eigenvectors with the SSVD Algorithm

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  1. 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

  2. Outline • High Relative Accuracy Algorithms for the symmetric eigenvalue problem • SSVD Algorithm • Computing (more) accurate eigenvectors with the SSVD Algorithm • Conclusions SSVD Accurate Eigenvectors

  3. Sometimes, QR Algorithm can fail double(eig(sym(A))) eig(A) SSVD Accurate Eigenvectors

  4. SSVD and J-ORTHOGONAL Algorithms … double(eig(sym(A))) eig(A) …can provide High Relative Accuracy SSVD Accurate Eigenvectors

  5. …. …. - - + + + …. …. SSVD Algorithm SSVD Accurate Eigenvectors

  6. If the SVD is computed with small multiplicative errors SSVD Accurate Eigenvectors

  7. Step 2.1: Clusters SSVD Accurate Eigenvectors

  8. Steps 2.2, 2.3 SSVD Accurate Eigenvectors

  9. Step 2.2: Putting the signs -- SSVD Accurate Eigenvectors

  10. Step 2.2: Putting the signs -- +++ SSVD Accurate Eigenvectors

  11. -- +++ Step 2.3: Getting the eigenvectors SSVD Accurate Eigenvectors

  12. If the SVD is computed with small multiplicative errors SSVD Accurate Eigenvectors

  13. -- +++ SSVD Accurate Eigenvectors

  14. - - - + + …. The precision of the eigenvalues is always that of the singular values SSVD Accurate Eigenvectors

  15. The precision of the eigenvectors is determined by the singular values relgap - - - + + …. SSVD Accurate Eigenvectors

  16. - - - - - - + + + + - - - - - - + + + + What can go wrong? SSVD Accurate Eigenvectors

  17. - - - - - - + + + + - - - - - - + + + + It can be fixed! SSVD Accurate Eigenvectors

  18. - - - + + + SSVD Accurate Eigenvectors

  19. - - - - - - - - + + SSVD Accurate Eigenvectors

  20. How is it done? - - - + + + SSVD Accurate Eigenvectors

  21. How is it done? - - - + + + SSVD Accurate Eigenvectors

  22. How is it done? - - - + + + SSVD Accurate Eigenvectors

  23. How is it done? - - - + + + SSVD Accurate Eigenvectors

  24. Algorithm to get P SSVD Accurate Eigenvectors

  25. - - - + + + SSVD Accurate Eigenvectors

  26. 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

  27. Conclusions Algorithm SSVD SSVD Accurate Eigenvectors

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