Jacobi method for small matrices

Transcription

1 Jacobi method for small matrices Erna Begović University of Zagreb Joint work with Vjeran Hari CIME-EMS Summer School

2 OUTLINE Why small matrices? Jacobi method and pivot strategies Parallel strategies on 4 4 matrices Conclusion E. Begović (University of Zagreb) Jacobi - small matrices CIME Summer School 2 / 14

3 JACOBI METHOD Goal: diagonal matrix. One step: A (k+1) = U T k A(k) U k, k 0, A (0) = A, where U k = R(i(k), j(k), φ(k)) are orthogonal, R(i, j, φ) = cos φ sin φ sin φ cos φ i j. E. Begović (University of Zagreb) Jacobi - small matrices CIME Summer School 3 / 14

4 PIVOT STRATEGIES Pivot strategy is the function k (i(k), j(k)). Cyclic strategies Serial strategies (row and column) Strategies with permutations inside columns / rows Examples: C 1, C 3, C 2, C 4. E. Begović (University of Zagreb) Jacobi - small matrices CIME Summer School 4 / 14

5 REFERENCES Jacobi method: J. Demmel, K. Veselić: Jacobi s method is more accurate than QR. SIAM J. Matrix Anal. Appl. 13 (4) (1992) Z. Drmač: A global convergence proof of cyclic Jacobi methods with block rotations. SIAM J. Matrix Anal. Appl. 31 (3) (2009) V. Hari: Convergence to Diagonal Form of Block Jacobi-type Methods. Numer. Math. 129 (3) (2015) C. G. J. Jacobi: Über ein Leichtes Verfahren die in der Theorie der Seculr Störungen Vorkommenden Gleichungen Numerisch Aufzulösen. Crelle s J. 30 (1846) Pivot strategies: E. R. Hansen: On Cyclic Jacobi Methods. SIAM J. Appl. Math. 11 (2) (1963) W. F. Mascarenhas: On the Convergence of the Jacobi Methods for Arbitrary Orderings. SIAM J. Matrix Anal. Appl. 16 (4) (1995) G. Shroff, R. Schreiber: On the Convergence of the Cyclyc Jacobi Method for Parallel Block Orderings. SIAM J. Matrix Anal. Appl. 10 (3) (1989) E. Begović (University of Zagreb) Jacobi - small matrices CIME Summer School 5 / 14

6 PARALLEL STRATEGIES ON 4 4 MATRICES We restrict our attention to those cyclic pivot strategies (16 strategies) that enable full parallelization of the method. It will be sufficient to study pivot strategy , which corresponds to parallel strategy E. Begović (University of Zagreb) Jacobi - small matrices CIME Summer School 6 / 14

7 CONVERGENCE Off-norm of a symmetric matrix S(A) = off(a) = 2 A diag(a) F = n 1 n i=1 j=i+1 a 2 ij, A = AT. We find the constant γ, 0 γ < 1, such that S 2 (A [3] ) γs 2 (A), where A [3] is obtained from A after three full cycles. E. Begović (University of Zagreb) Jacobi - small matrices CIME Summer School 7 / 14

8 SLOW CONVERGENCE - EXAMPLE A = A (0) = d + p 1 + p 2 0 ɛ + p 1 a + p 1 0 d + p 2 a ɛ ɛ + p 1 a d + p 1 0 a + p 1 ɛ 0 d where a = 1, ɛ = 10 k 10, d = 1 + ɛ, p 2 = k, p 3 = k 10 and, S(A) = E. Begović (University of Zagreb) Jacobi - small matrices CIME Summer School 8 / 14

9 SLOW CONVERGENCE - EXAMPLE k (i(k), j(k)) S(A (k) ) 1 (1, 3) (2, 4) (1, 4) (2, 3) (1, 2) (3, 4) (1, 3) (2, 4) (1, 4) (2, 3) e 89 E. Begović (University of Zagreb) Jacobi - small matrices CIME Summer School 9 / 14

10 SLOW CONVERGENCE - EXAMPLE As long as the rotation angle is close enough to ± π 4, the off-norm will not change in the next two steps. k (i, j) sin φ(i, j) 1 (1, 3) (2, 4) (1, 4) (2, 3) (1, 2) (3, 4) (1, 3) e 15 8 (2, 4) e 15 E. Begović (University of Zagreb) Jacobi - small matrices CIME Summer School 10 / 14

11 CHANGING THE APPROACH E. Begović (University of Zagreb) Jacobi - small matrices CIME Summer School 11 / 14

12 CHANGING THE APPROACH Pivot strategy O = (1, 3), (2, 4), (1, 4), (2, 3), (1, 2), (3, 4) on matrix Let Q = A = a 11 0 a 13 a 14 0 a 22 a 23 a 24 a 13 a 23 a 33 0 a 14 a 24 0 a 44, QT XQ =. x 11 x 13 x 14 x 12 x 31 x 33 x 34 x 32 x 41 x 43 x 44 x 42 x 21 x 23 x 24 x 22. E. Begović (University of Zagreb) Jacobi - small matrices CIME Summer School 11 / 14

13 OPERATOR T Definition For A S 4 we define T = ( R(1, 3, φ)r(2, 4, ψ)q ) T A ( R(1, 3, φ)r(2, 4, ψ)q ), where R(1, 3, φ) and R(2, 4, ψ) are Jacobi rotations which annihilate the elements a 13 and a 24 of A, respectively, and φ, ψ [ π 4, π 4 ]. Properties of T : T k (A) = (Q k ) T A (2k) Q k, k 0, S(T k (A)) = S(A (2k) ), k 0. E. Begović (University of Zagreb) Jacobi - small matrices CIME Summer School 12 / 14

14 MAIN THEOREM Proposition Let A S 4 be such that a 12 = 0, a 34 = 0 and ɛ = Then S(T 6 (A)) (1 ɛ)s(a). Theorem Let A S 4 be such that a 12 = 0, a 34 = 0 and let A (12) be obtained by applying 12 steps of the Jacobi method under the strategy O = (1, 3), (2, 4), (1, 4), (2, 3), (1, 2), (3, 4) to A. Then S(A (12) ) ( )S(A). E. Begović: Convergence of Block Jacobi Methods. Ph.D. thesis, University of Zagreb, E. Begović, V. Hari: On the Global Convergence of the Jacobi Method for Symmetric Matrices of order 4 under Parallel Strategies. in preparation E. Begović (University of Zagreb) Jacobi - small matrices CIME Summer School 13 / 14

15 CONCLUSION Theorem Let A S 4 and let A [3] be obtained by applying 3 full cycles of the Jacobi method under any cyclic strategy on A. Then there is a constant γ such that S 2 (A [3] ) γs 2 (A), 0 γ < 1. E. Begović (University of Zagreb) Jacobi - small matrices CIME Summer School 14 / 14

16 CONCLUSION Theorem Let A S 4 and let A [3] be obtained by applying 3 full cycles of the Jacobi method under any cyclic strategy on A. Then there is a constant γ such that S 2 (A [3] ) γs 2 (A), 0 γ < 1. Thank you! E. Begović (University of Zagreb) Jacobi - small matrices CIME Summer School 14 / 14