Pseudospectra of Matrix Pencils and some Distance Problems

Size: px

Start display at page:

Download "Pseudospectra of Matrix Pencils and some Distance Problems"

Josephine Osborne
5 years ago
Views:

1 Pseudospectra of Matrix Pencils and some Distance Problems Rafikul Alam Department of Mathematics Indian Institute of Technology Guwahati Guwahati , INDIA Many thanks to Safique Ahmad and Ralph Byers

2 Outline General framework for pseudospectra of matrix pencils Critical points of backward errors and their significance Distance problems for matrix pencils: distance to nearest defective pencil distance to nearest nonunimodular pencil distance to nearest right nonprime pencil distance to nearest unobservable pair distance to nearest uncontrollable pair distance to nearest unstable pencil

3 Pseudospectrum of a matrix Let A C n n and Λ(A) be spectrum of A. Choose a norm on C n n. Define Λ ɛ (A) := Λ(A + A) A ɛ = {z C : η(z, A) ɛ}, where η(z, A) := inf{ A : z Λ(A + A)}.

4 Pseudospectra of matrix pencils Pencils: L(λ) := A λb, L(λ) := A λ B Spectrum of L : Λ(L) Λ ɛ (L) := {z : (L(z) + L(z))x = 0 A ɛα 1, B ɛα 2 } = {z C : η(z, L) ɛ}, η(z, L) := min x =1 L(z)x α 1 + z α 2. [SIMAX]

5 Pseudospectra of matrix pencils Pencils: L(λ) := A λb, L(λ) := A λ B Frobenius norm: F Λ ɛ (L) := η F (z, L) := σ min(l(z)) 1 + z 2. [ A, B] F ɛ Λ(L + L) = {z C : η F (z, L) ɛ}

6 The BIG picture Space of pencils: L {A(C, C n n ), L(C 2, C n n )}. C n C n n L Pencil norm/seminorm: Let L, L L. Define Λ ɛ (L) := Λ(L + L) L ɛ = {z : η(z, L) ɛ} η(z, L) := inf{ L : z Λ(L + L), L L}.

7 The space L p L(λ) := A λb or L(c, s) := ca sb. Pencil p-norm/seminorm: p 1 + q 1 = 1 Weight vector: w := (w 1, w 2 ), w j 0 w 1 := (w1 1, w 1 2 ), w 1 j = 0 if w j = 0. L w,p := ( A, B ) w,p = (w 1 A, w 2 B ) p. L(λ) L w,p (1, λ) w 1,q L(c, s) L w,p (c, s) w 1,q.

8 Λ ɛ,w,p (L) := Pseudospectra L w,p ɛ ), obtain pseu- Setting p := and w := (α 1 dospectra [HT] Λ(L + L). 1, α 1 2 Setting p := 2, w := (1, 1) and 2, / F obtain pseudospectra [LS]/[SIMAX] Which norm to choose?

9 Backward error η w,p (z, L) = min x =1 L(z)x (1, z) w 1,q L w,p. For 2-norm and Frobenius norm on C n n, η w,p (z, L) = σ min(l(z)). (1, z) w 1,q Henceforth, we consider only 2-norm on C n n. Fact: z log η w,p (z, L) is subharmonic.

10 Gradient of p-norms Consider N p (λ) := (1, λ) p for λ C, 1 p. (a) If 1 < p < then N p (λ) = λ λ p 2 N p 1. p (b) If p = 1 then N p (λ) = λ for λ 0. λ (c) If p = then N p (λ) = 0, when λ < 1, λ, when λ > 1. λ

11 Gradient of backward error Theorem 1 Let L(z) = A zb C m n. Suppose σ min (L(λ)) simple. If ( (1, λ) w 1,q) exists then η w,p (λ, L) exists and η w,p (λ, L) = u Bv + η w,p (λ, L) ( (1, λ) w 1,q ), (1, λ) w 1,q where u and v unit left and right singular vectors of L(λ) corresponding to σ min (L(λ)).

12 Gradient of backward error Theorem 2 Consider L(z) := l i=0 A i z i C m n. Suppose σ min (L(λ)) simple. Set N q (λ) := (1, λ,, λ l ) w 1,q. Then η w,p (λ, L) = u z L(λ)v η w,p (λ, L) N q (λ), N q (λ) where u and v unit left and right singular vectors of L(λ) corresponding to σ min (L(λ)).

13 Critical points Let λ C and η w,p (λ, L) 0. Then λ said to be nongeneric critical point of η w,p if σ min (L(λ)) is multiple. generic critical point of η w,p if η w,p (λ, L) = 0. Theorem 3 Let L(λ) = A λb C n n and µ Λ(L). Then µ multiple y Bx = 0 for some left and right eigenvectors y and x, respectively.

14 Critical points are multiple eigenvalues Theorem 4 Let L(λ) := A λb. Let u and v be normalized left and right singular vectors of L(µ) corresponding to σ min (L(µ)). Set N q (c, s) := (c, s) w 1,q. Define A := η w,p (µ, L) c N q (1, µ) uv, B := η w,p (µ, L) s N q (1, µ) uv and consider L(λ) = A λ B. Then (a) L w,p = η w,p (µ, L), (b) (L(µ) + L(µ))v = 0 and u (L(µ) + L(µ)) = 0,

15 Critical points are multiple eigenvalues Theorem 4 (Cont.) (c) u (B + B)v = u Bv + η w,p (µ, L) s N q (1, µ). (d) If σ min (L(µ)) is simple then u (B + B)v = u Bv + η w,p (µ, L) s N q (1, µ) = N q (1, µ) η w,p (µ, L). So, µ generic critical point u (B + B)v = 0 µ defective. Also, µ nongeneric critical point µ multiple.

16 Critical points are multiple eigenvalues Theorem 5 Consider L(z) := l i=0 A i z i C n n. Let u and v be unit left and right singular vectors of L(λ) corresponding to σ min (L(λ)). Consider N q : C l+1 R, z z w 1,q. Define A i := η w,p (λ, L) i N q (1, λ,, λ l ) uv. (a) L w,p = η w,p (λ, L) (b) L(λ) = η w,p (λ, L)N q (1, λ,..., λ l )uv

17 Critical points are multiple eigenvalues Theorem 5 (cont.) (c) z L(λ) = η w,p (λ, L) N q (1, λ,..., λ l )uv (d) (L(λ) + L(λ))v = 0 and u (L(λ) + L(λ)) = 0 (e) u ( z L(λ) + z L(λ))v = u z L(λ)v η w,p (λ, L) N q (1, λ,..., λ l ) If σ min (L(λ)) simple and N q exists then u ( z L(λ)+ z L(λ))v = N q (1, λ,..., λ l ) η w,p (λ, L)

18 Distance problem For U C closed, η U w,p restriction of η w,p on U. S U be set of critical points of η U w,p. Consider the distance problem d(l) := inf z S U η U w,p(z, L). Then z min S U minimal critical point of η U w,p if d(l) = η U w,p(z min, L).

19 Minimal pencil Let L(z min ) = UΣV. If Σ(n, n) simple then set u := U(:, n) and v := V (:.n) else u := U(:, n 1 : n) and v := V (:, n 1 : n). Define ( recall N q (c, s) := (c, s) w 1,q ) A := η w,p (z min, L) c N q (1, z min ) uv, B := η w,p (z min, L) s N q (1, z min ) uv. Then L + L, where L(λ) = A λ B, called minimal pencil.

20 Nearest defective pencil Let L(λ) := A λb be regular and simple. Then for U = C, we have d(l) = η w,p (z min, L) = min{ L w,p : L + L has multiple eigenvalue}. Further, z min is a common boundary point of two components of Λ ɛ,w,p (L) for smallest ɛ. Also, z min is a minimal saddle point of η w,p.

21 Nearest defective pencil (cont.) WLOG, consider w := (1, 1) and set L := L w,2, η(z, L) := η w,2 (z, L). Theorem 6 Let L(z min ) = UΣV and σ n := Σ(n, n). Set u := U(:, n), v := V (:, n) if σ n simple, else u := U(:, n 1 : n), v := V (:, n 1 : n). Define A := σ n uv 1 + z min 2, B := z min σ n uv 1 + z min 2. Then d(l) = L = σ n 1+ zmin 2 and z min defective/multiple eigenvalue value of L + L.

22 Diagonal pencils Theorem 7 Let A be normal and simple. Suppose λ i λ j = min k l λ k λ l. Set z w := (λ i + λ j )/2. Define A := (λ i λ j ) (x 4 i x j )(x i + x j ), where x js are normalized eigenvectors. Then d(a) = A 2 = λ i λ j /2 and z w defective eigenvalue of A + A.

23 Diagonal pencils (cont.) Consider L(c, s) = c diag(α i ) s diag(β i ). d(l) =?? We define departure Dep : C 2 R by Dep(x, y) := x 1 y 2 x 2 y 1 x y x, y. Set cos(θ) := x, y. Then for unit x and y Dep(x, y) = 1 2 chord(x 1/x 2, y 1 /y 2 ) sec(θ/2). Obviously, Dep(x, y) = 0 x = y in CP 1

24 Diagonal pencils (cont.) Theorem 8 Let L(c, s) := c diag(α i ) s diag(β i ) be simple. Set x j := (α j, β j ), j = 1 : n. Then d(l) = min i j Dep(x i, x j ). Suppose that Dep(x i, x j ) = min k l Dep(x k, x l ). Set w := sign( x j, x i ). Define c w := β i + wβ j x i + wx j 2, s w := α i + wα j x i + wx j 2. Then (c w, s w ) S 1 and η(c w, s w, L) = Dep(x i, x j ).

25 Diagonal pencils (cont.) x j 2 2 Theorem 6 (cont.) Set t := + x i, x j, x i + wx j 2 u := te i + 1 t 2 e j, v := (te i 1 t 2 we j )sign(c w α i s w β i ). A := c w Dep(x i, x j ) uv, B := s w Dep(x i, x j ) uv and consider L(c, s) := c A s B. Then d(l) = L = Dep(x i, x j ) and (c w, s w ) defective eigenvalue of L + L.

26 Example

27 Example

28 Example

29 Example

30 Nearest nonunimodular pencil Pencil L(λ) = A λb C n n unimodular if rank(l(λ)) = n for all λ C. Then for U = C d(l) = min{ L w,p : L + L nonunimodular} = min η w,p(z, L). z C [ ] 1 λ provided w := (1, 0). Consider L(λ) :=. 0 1 Then η(λ, L) 0 as λ but η(λ, L) 0 for λ C. [Here w := (1, 1).]

31 Nearest right nonprime pencil Pencil L(λ) = A λb C m n right prime if rank(l(λ)) = n for all λ C. Then for U = C d(l) = min z C η w,p(z, L) = min{ L w,p : L + L not right prime} when w = (1, 0). [Boutry, Elad, Golub, Milanfar., SIMAX, 2005]: min { [ A, B] F : (L(λ)+ L(λ))x = 0}. A, B,λ, x =1

32 Nearest unobservable pair [ ] A Pair (A, B) observable L(λ) = λ B right prime. Hence for w = (1, 0), d(l) = min η w,p(z, L) z C [ ] A = min{ B 2 : (A + A, B + B) unobservable}. [ ] I 0

33 Nearest uncontrollable pair L(λ) = X λy C m n left prime if rank(l(λ)) = m for all λ C. Pair (A, B) controllable L(λ) = [A, B] λ[i, 0] left prime. Hence for w = (1, 0), d(l) = min z C η w,p(z, L) = min{ [A, B] 2 : (A + A, B + B) uncontrollable}

34 Nearest unstable pencil Let V {H, D} and L(λ) = A λb be stable, i.e., Λ(L) V. Here D open unit disc and H left half plane. Then for U := V c, d(l) = min z U ηu w,p(z, L) = min η w,p(z, L) z U = min{ L w,p : L + L unstable}.

35 References S. Ahmad, PhD thesis to be submitted to IIT Guwahati. Boutry, Elad, Golub, Milanfar, The generalised eigenvalue problem for nonsquare pencils using a minimal perturbation approach, SIAM J. Matrix Anal. Appl. 27(2005),pp N. J. Higham and F. Tisseur, Structured Pseudospectra for Polynomial Eigenvalue Problems, with Applications, SIAM J. Matrix Anal. Appl. 23(2001),pp N. J. Higham and F. Tisseur, More on pseudospectra for polynomial eigenvalue problems and applications in control theory, Linear Algebra Appl., (2002), pp P.-F. Lavallee and M. Sadkane, Pseudospectra of linear matrix pencils by block diagonalization, Computing 60 (1998), pp

36 Thank you

ETNA Kent State University

ETNA Kent State University Electronic Transactions on Numerical Analysis Volume 38, pp 75-30, 011 Copyright 011, ISSN 1068-9613 ETNA PERTURBATION ANALYSIS FOR COMPLEX SYMMETRIC, SKEW SYMMETRIC, EVEN AND ODD MATRIX POLYNOMIALS SK