Relative Perturbations for s.d.d. Matrices with applications

1. Relative Perturbations for s.d.d. Matrices with applications J. Matejaš, V. Hari

2. Motivation Intention to make a sound accuracy proof for • Kogbetliantz method • two-sided Jacobi method for symmetric indefinite s.d.d. matrices We need an appropriate relative perturbation result.

3. Perturbation results for s.d.d. Hermitian matrices nonsingular. Let be Hermitian matrices with Let and be the eigenvalues respectively. of and

4. is positive definite and if If where Let Note that, if i.e. is s.d.d., then and thus Note also that the assumption holds if that is

5. Let What is known for symmetric indefinite s.d.d. matrices? be symmetric and let be s.d.d. for all If then We seek for the estimate s.d.d. where is

6. Let New perturbation result be such that where then If is replaced with If the assumption then we have even simpler result, implies

7. What is new ? What is better ? Weaker and simpler assumptions and somewhat better estimate ! In detail : ■ the assumption: s.d.d. for is is now removed ■ the new assumption is: s.d.d. ; is is easy to compute and ■ compatibility with the existing result for positive definite matrices (the same assumptions and the same type of estimate -- no exponential function) the new result ■ and for all yields the better estimate ■ for larger the new result is even if realistic, while the existing one is useless

8. Corollaries Definite matrices If is definite (positive or negative), then implies i.e. the new result reduces to the existing one.

9. Skew – Hermitian matrices Let be skew-Hermitian matrices and let and be the eigenvalues of and respectively, such that and hold. Note that and are Hermitian. Let be nonsingular and let Let be such that If then

10. Hidden Hermitian matrices Let and where are Hermitian and are diagonal such that is positive definite. Let and be the eigenvalues of and respectively. Let be nonsingular and let Let be such that If then We construct where etc.

11. Relative perturbation of the singular values Let and and let be the singular values of and respectively. Let be nonsingular and let Let be such that and and are nonsingular and then If

12. Proof We apply the eigenvalue perturbation result to the indefinite Hermitian matrices where Here: and thus has nonnegative diagonal elements ; is a permutation matrix such that has nonincreasingly ordered diagonal elements ; The eigenvalues of are The eigenvalues of are

13. Jacobi algorithm. Let Applications be a symmetric matrix. A single Jacobi step annihilates the pivot element at position by the similarity transformation where On the level of is a rotation in the plane 2 by 2 submatrices, we have

14. Algorithm

15. Exact error analysis Assumptions Let denotes the round-off unit, i.e. We use the general model of arithmetic where the floating point result is given by where

16. Technique • We derive error estimates without rounding or neglecting the higher (nonlinear) terms of the errorsand we take into account the signs of the errors. • Using such aproach we can recognize when and why we have • suppression of the initial errors • cancellation of the initial errors

17. Suppression of the initial errors Let us estimate the error in evaluation of the expression Suppose that we have an approximation of Now, we have Note that

18. Cancellation of the initial errors

19. We obtain where Note that

20. We apply such technique to Jacobi algorithm. We consider paralel strategy (Sameh) which is equivalent to the row- and the column-cyclic strategy.

21. Let Perturbation result (revisited) be such that where If then

22. Accuracy result Let be the backward error which is caused by applying to one step, one batch or one sweep of Jacobi method in finite arithmetic. Then we have For example, if then

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