RKPACK A numerical package for solving large eigenproblems

1. RKPACKA numerical package for solving large eigenproblems Che-Rung Lee

2. Outline • Introduction • RKPACK • Experiments • Conclusion University of Maryland, College Park

3. Introduction • The residual Krylov method • Shift-invert enhancement • Properties and examples University of Maryland, College Park

4. The residual Krylov method • Basic algorithm • Let be a selected eigenpair approximation of A. • Compute the residual . • Use r in subspace expansion. University of Maryland, College Park

5. Properties • The selected approximation (candidate) can converge even with errors. • The allowed error || f || must be less than  ||r||, for a constant  <1. • The residual Krylov method can work with an initial subspace that contains good Ritz approximations. University of Maryland, College Park

6. Example • A 100x100 matrix with eigenvalues 1, 0.95, …,0.9599. University of Maryland, College Park

7. Shift-invert enhancement • Algorithm: (shift value =  ) • Let be a selected eigenpair approximation of A. • Compute the residual . • Solve the equation . • Use v in subspace expansion. • Equation in step 3 can be solved in low accuracy, such as 103. University of Maryland, College Park

8. 100 10-5 10-10 10-15 0 5 10 15 20 25 30 35 40 Example • The same matrix • Shift value is 1.3 • Linear systems are solved to 103. University of Maryland, College Park

9. RKPACK • Features • Computation modes • Memory requirement • Time complexity University of Maryland, College Park

10. Features • Can compute several selected eigenpairs • Allow imprecise computational results with shift-invert enhancement • Can start with an appropriate initial subspace • Use the Krylov-Schur restarting algorithm • Use reverse communication University of Maryland, College Park

11. Computation modes • Two computation modes • The normal mode: • needs matrix vector multiplication only • The imprecise shift-invert mode: • needs matrix vector multiplication and linear system solving (with low accuracy requirement) • can change the shift value • Both can be initialized with a subspace. University of Maryland, College Park

12. Memory requirement • Use the Krylov-Schur restarting algorithm to control the maximum dimension of subspace • Required memory: O(nm)+O(m2) • n: the order of matrix A • m: the maximum dimension of subspace University of Maryland, College Park

13. Time complexity • The normal mode: • kf (n)+kO(nm)+kO(m3) • f (n): the time for matrix vector multiplication. • k: the number of iterations • The imprecise shift-invert mode • kf (n) + kO(nm) + kO(m3) + kg(n, ) • g(n, ) : the time for solving linear system to the precision . University of Maryland, College Park

14. Experiments • Test problem • Performance of RKPACK • The inexact residual Krylov method • The successive inner-outer process University of Maryland, College Park

15. Test problem • Let A be a 1000010000 matrix with first 100 eigenvalues 1, 0.95, …, 0.9599, and the rest randomly distributed in (0.25, 0.75). • Eigenvectors are randomly generated. • Maximum dimension of subspace is 20. • Stopping criterion: when the norm of residual is smaller than 1013. University of Maryland, College Park

16. Performance of the normal mode • Compute six dominant eigenpairs. • Compare to the mode 1 of ARPACK • Etime: elapse time (second) • MVM: number of matrix vector multiplications • Iteration: number of subspace expansions University of Maryland, College Park

17. The imprecise shift-invert mode • Compute six smallest eigenvalues. • Use GMRES to solve linear system. (shift = 0) • Compare to the mode 3 of ARPACK • Prec: precision requirement of solution University of Maryland, College Park

18. Inexact residual Krylov method • Allow increasing errors in the computation • Use the normal mode with matrix A1. • The required precision of solving A1. •  is the desired precision of computed eigenpairs • m is the maximum dimension of subspace University of Maryland, College Park

19. 10-2 10-4 10-6 10-8 10-10 20 30 40 50 60 70 80 Experiment and result • Compute six smallest eigenpairs. • The required precision (using GMRES) • Etime: 910.11 second • MVM: 6282 • Iteration: 67 University of Maryland, College Park

20. Successive inner-outer process • Use the convergence properties of Krylov subspace (superlinear) to minimize total number of MVM. (Golub, Zhang and Zha, 2000) • Divide the process into stages, with increasing precision requirement. • The original algorithm can only compute a single eigenpair University of Maryland, College Park

21. 100 10-2 10-4 10-6 10-8 10-10 10-12 20 40 60 80 100 120 140 160 Experiment and result • Compute six smallest eigenpairs. • Four stages with required precision (GMRES)103,106,109,1012. • Etime : 1188.12 • MVM : 13307 • Iteration : 163 University of Maryland, College Park

22. Conclusion • Summary • Future work University of Maryland, College Park

23. Summary • The residual Krylov method for eigenproblems allows errors in the computation, and can work on an appropriate initial subspace. • RKPACK can solve eigenproblems rapidly when uses the imprecise shift-invert enhancement, and is able to integrate other algorithms easily. University of Maryland, College Park

24. … Future work • Parallelization • Data parallelism • Block version of the residual Krylov method • Other eigenvector approximations • Refine Ritz vector or Harmonic Ritz vector • New algorithms • Inexact methods, residual power method … University of Maryland, College Park