Quadratic Matrix Polynomials

Transcription

1 Research Triangularization Matters of Quadratic Matrix Polynomials February 25, 2009 Nick Françoise Higham Tisseur Director School of of Research Mathematics The University of Manchester School of Mathematics Joint work with Yuji Nakatsukasa (M/cr), Leo Taslaman (M/cr) and Ion Zaballa (Universidad del País Vasco) SIAM Conference on Applied Linear Algebra, June / 6

2 Reduction to Triangular Forms Let A, B C n n and let T A, T B denote triangular matrices. Single matrices: there exists U C n n such that U AU = T A, U U = I, (Schur decomp.). Pair of matrices: there exist unitary U, V C n n s.t. U AV = T A, U BV = T B, (generalized Schur decomp.). These reductions have many applications (eigenvalue computation, matrix function computation,... ). Can we extend these reductions to matrix triples? MIMS Françoise Tisseur Triangularization 2 / 21

3 Extension to Matrix Triples Let Q(λ) = λ 2 M + λd + K be regular (det Q(λ) 0). Suppose there exist U, V nonsingular s.t. UQ(λ)V = λ 2 T M + λt D + T K = T (λ) is triangular. Roots λ (1) j, λ (2) j of λ 2 (T M ) jj + λ(t D ) jj + (T K ) jj are e vals of Q(λ). Q(λ (k) 1 )v 1 = U 1 T (λ (k) 1 )e 1 = 0, k = 1, 2. So λ (1) 1 and λ (2) 1 must have the same e vec a strong condition! We can forget about simultaneous triangularization... " Charlie Van Loan, CP3, Monday Use U(λ), V (λ) unimodular instead! MIMS Françoise Tisseur Triangularization 3 / 21

4 Equivalences Definition Two matrix polynomials Q(λ), T (λ) are equivalent if there are unimodular U(λ), V (λ) s.t. T (λ) = U(λ)Q(λ)V (λ). Q(λ) and T (λ) have the same finite elementary divisors. Need equivalence transformations preserving the degree (want Q(λ), T (λ) to be quadratic), the elementary divisors at infinity (i.e., the elementary divisors at 0 of revq(λ) = λ 2 Q(1/λ) = λ 2 K + λd + M). MIMS Françoise Tisseur Triangularization 4 / 21

5 Elementary Divisors at Infinity L(λ) = λ has one linear elementary divisors at 0 and two linear elementary at. Now 1 λ L(λ) = λ , }{{} unimodular has one linear elementary divisor at 0 and one elementary divisor at with partial multiplicity 2. Equivalences can modify the partial multiplicities at. MIMS Françoise Tisseur Triangularization 5 / 21

6 Triangularization by Strong Equivalences Theorem (Zaballa & T, 2012) Any regular Q(λ) = λ 2 M + λd + K is strongly equivalent over C[λ] to triangular T (λ) = λ 2 T M + λt D + T K, i.e., there are unimodular U(λ), V (λ) s.t. T (λ) = U(λ)Q(λ)V (λ) has same finite and infinite elementary divisors as Q(λ). Byproduct of solution to quadratic realizability problem, D.S. Mackey (MS68). Proved by Gohberg, Lancaster & Rodman (1982) for monic polynomials of arbitrary degree. Extended to regular/singular polynomials of arbitrary degree by Taslaman, Nakatsukasa, Zaballa & T. (2012). How to numerically compute T (λ)? MIMS Françoise Tisseur Triangularization 6 / 21

7 Structure Preserving Transformation (SPT) Q(λ) = λ 2 M + λd + K, [ ] I 0 C Q (λ) = λ + 0 M [ 0 K I D ]. Q(λ) C Q (λ) (strong) equivalence U(λ)Q(λ)V (λ) SPT S L C Q (λ)s R T (λ) linearization quadratization C T (λ) MIMS Françoise Tisseur Triangularization 7 / 21

8 A Simple MATLAB Code Let Q(λ) = λ 2 I + λd + K be n n. A = [zeros(n) -K; eye(n) -D]; [U,T] = schur(a, complex ); X =(U(:,1:2:2*n-1)+U(:,2:2:2*n))/sqrt(2); S = [X A*X]; At = S\A*S; MIMS Françoise Tisseur Triangularization 8 / 21

9 A Simple MATLAB Code Let Q(λ) = λ 2 I + λd + K be n n. A = [zeros(n) -K; eye(n) -D]; [U,T] = schur(a, complex ); X =(U(:,1:2:2*n-1)+U(:,2:2:2*n))/sqrt(2); S = [X A*X]; At = S\A*S; [ ] 0 TK A T = is in companion form. I T D T (λ) = λ 2 I + λt D + T K : upper triang., Λ(Q) = Λ(T ). S is an SPT. Cols of X C 2n n are orthonormal. MIMS Françoise Tisseur Triangularization 8 / 21

10 Schur s Theorem for Complex Matrices Matrix version: if A C n n then there exists a unitary U such that U AU = T is a triangular matrix. Subspaces version: Theorem Let A C n n. There are subspaces V 1,..., V n of C n satisfying (i) C n = V 1 V 2 V n, (ii) for k = 1: n, V 1 V k, is A-invariant, (iii) for k = 1: n, V k = u k, where u 1,... u n form an orthonormal system of vectors of C n. MIMS Françoise Tisseur Triangularization 9 / 21

11 Schur-like Theorem for Q(λ) Let Q(λ) = λ 2 M + λd + K, det M 0. Theorem (Zaballa, T., 2012) Let λi A C[λ] 2n 2n be a linearization of n n Q(λ). There are subspaces V 1,..., V n of C 2n satisfying (i) C 2n = V 1 V 2 V n, (ii) for k = 1: n, V 1 V k is A-invariant, (iii) for k = 1: n, dim V k = 2 and V k = x k, Ax k, where x 1,... x n form an orthogonal system of vectors of C 2n. V k = x k, Ax k is a Krylov subspace of dimension 2. The x j are generating vectors. If X = [x 1... x n ] then S = [X AX] is nonsingular. MIMS Françoise Tisseur Triangularization 10 / 21

12 Triangularizing SPT for Q(λ) = λ 2 M + λd + K Let S = [X AX], where X contains generating vectors for linearization λi A of Q(λ). [ ] S 1 B11 B AS =: B A[X AX] = [X AX] 12. B 21 B 22 [ ] [ ] B11 0 = and hence B has companion form. I B 21 x 1, Ax 1, x k, Ax k A-invariant B 12 and B 22 are upper triangular. S = [X AX] is a triangularizing SPT. MIMS Françoise Tisseur Triangularization 11 / 21

13 Schur-like Theorem: Matrix Form Let Q(λ) = λ 2 M + λd + K C[λ] n n with det(m) 0. Theorem (Zaballa, T., 2012) For any linearization λi A of Q(λ), there exists U C 2n n with orthonormal columns s.t. [U AU] is nonsingular and [ ] 0 [U AU] 1 T0 A[U AU] =, I n T 1 where I n λ 2 + T 1 λ + T 0 is triangular and equivalent to Q(λ). The columns of U are generating vectors for A. Extends to arbitrary degree matrix polynomials. MIMS Françoise Tisseur Triangularization 12 / 21

14 Generating Vectors from Schur Vectors Theorem (Nakatsukasa, Taslaman, Zaballa & T, 2012) Let A C 2n 2n have Schur decomposition T 11 T 1n U AU =...., T ij C 2 2. T nn [ ] νj1 If there is v j = s.t. T jj v j α j v j, j = 1: n then cols of ν j2 X = Udiag(v 1,..., v n ) = [ν 11 u 1 + ν 12 u 2,... ν n1 u 2n 1 + ν n2 u 2n ] are generating vectors. Hence [ 0 [X AX] 1 TK A[X AX] = I T D where T D, T K are upper triangular. MIMS Françoise Tisseur Triangularization 13 / 21 ],

15 Schur form with Nonderogatory Blocks If T jj αi 2, i.e., T jj is nonderogatory then there exists 0 v j C 2 s.t. v j is not an e vec of T jj. Theorem (Nakatsukasa, Taslaman, Zaballa, T., 2012) Any Schur form T of a linearization A C 2n 2n of Q(λ) C[λ] n n is unitarily similar to a Schur form T with nonderogatory 2 2 diagonal blocks. Proof is constructive and nontrivial. Relies on the property that λ Λ(A) = Λ(Q) has geometric multiplicity less or equal to n. MIMS Françoise Tisseur Triangularization 14 / 21

16 Triangular Matrix Coefficients Q(λ) is equivalent to triangular T (λ) = λ 2 T M + λt D + T K. Have explicit expressions for T M, T D, T K in terms of either the n orthonormal generating vectors for A, or the Schur form T of A, (or Jordan form of A, see Taslaman MS 35) where λi A is a linearization of Q(λ) C[λ] n n with nonsingular leading coeff. MIMS Françoise Tisseur Triangularization 15 / 21

17 Triangular Coefficients from Schur Form Let λi A be a linearization of Q(λ) = λ 2 M + λd + K with T 11 T 1n Schur form T =....,T jj C 2 2 nonderogatory. T nn Take v i, w i C 2 s.t. v i 2 = w i 2 = 1, T ii v i αv i, w i v i = 0, V = diag(v 1,..., v n ) C 2n n, W = diag(w 1,..., w n ) C 2n n. Q(λ) is equivalent to T (λ) = λ 2 T M + λt D + T K, where are triangular. T M = W TV, T D = W T 2 V, T K = T M ((V TV )T D (V T 2 V )) MIMS Françoise Tisseur Triangularization 16 / 21

18 Triangularizing SPT S Let S [ = [X AX], where ] X contains generating [ vectors ] for 0 KM 1 0 A = I DM 1 so that S 1 TK AS = =: A I T T. D S 1 = [ Y A T Y ], Y C 2n n [Garvey et al., 2011]. Can construct Y from generating vectors X and M, D, K. No need to factorize (invert) S. MIMS Françoise Tisseur Triangularization 17 / 21

19 Linear Systems Let S = [X AX] = [Y A T Y ] 1 and X = For every λ / Λ(Q), [ X1 X 2 ], Y = [ Y1 Q(λ) 1 = ( X 1 + λx 2 )T (λ) 1 (Y 1 + λy 2 ) X 2 Y 2, where X 1 = X 1 DX 2 + X 2 T D. Y 2 ]. MIMS Françoise Tisseur Triangularization 18 / 21

20 Linear Systems Let S = [X AX] = [Y A T Y ] 1 and X = For every λ / Λ(Q), [ X1 X 2 ], Y = [ Y1 Q(λ) 1 = ( X 1 + λx 2 )T (λ) 1 (Y 1 + λy 2 ) X 2 Y 2, where X 1 = X 1 DX 2 + X 2 T D. Applications Solving for x, Q(ω)x = b for many values of ω [ω l, ω h ], ω l ω h. Evaluation of transfer function: G(s) = c T Q(s) 1 b. Y 2 ]. MIMS Françoise Tisseur Triangularization 18 / 21

21 Triangularization over R Theorem (Zaballa & T, 2012) Q(λ) R[λ] n n is triangularizable over R[λ] if and only if where p n n c, 2n c is the number of nonreal e vals and p is the largest geometric multiplicity of the real e vals and e vals at infinity. Q(λ) R[λ] 3 3 with elementary divisors in R: (λ 1), (λ 1), (λ 2 + 1) 2 is not triangularizable over R[λ] since p = 2 > n n c = 3 2 = 1. MIMS Françoise Tisseur Triangularization 19 / 21

22 Quasi-triangular Quadratics [Zaballa & T, 2012] Any quadratic Q(λ) R[λ] n n is strongly equivalent to a quadratic of the form T (λ) = 2r 2n 2r [ ] 2r T 1 (λ) T 3 (λ), 2n 2r 0 T 2 (λ) where r = max{0, p + n c n} and T 1 (λ) is quasi-triangular with r 2 2 diag. blocks with elementary divisors (λ λ 1 ), (λ λ 1 ), (λ 2 + d i λ + k i ). Here, λ 1 R has largest geometric multiplicity p. T 2 (λ) is triangular. (Q(λ) has 2n c nonreal e vals.) MIMS Françoise Tisseur Triangularization 20 / 21

23 Summary and Concluding Remarks Any regular quadratic is strongly equivalent to a triangular quadratic. There is a Schur-like theorem for quadratic matrix polynomials. Triangularizing SPTs are defined by n orthonormal (generating) vectors. Can competitive algorithms be designed that compute sets of n orthonormal generating vectors? Subspace iteration, Krylov subspace methods are worth exploring. MIMS Françoise Tisseur Triangularization 21 / 21

24 References I S. D. Garvey, P. Lancaster, A. A. Popov, U. Prells, and I. Zaballa. Filters connecting isospectral quadratic systems. To appear in Linear Algebra Appl., Y. Nakatsukasa, L. Taslaman, and F. Tisseur. Reduction of matrix polynomials to simpler forms. Technical report, Manchester Institute for Mathematical Sciences, The University of Manchester, UK, In preparation. MIMS Françoise Tisseur Triangularization 20 / 21

25 References II L. Taslaman and F. Tisseur. Triangularization of matrix polynomials. Technical report, Manchester Institute for Mathematical Sciences, The University of Manchester, UK, In preparation. F. Tisseur and I. Zaballa. Triangularizing quadratic matrix polynomials. MIMS EPrint , Manchester Institute for Mathematical Sciences, The University of Manchester, UK, MIMS Françoise Tisseur Triangularization 21 / 21