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Computing accurate eigenvectors with the SSVD Algorithm

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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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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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