Bounds on eigenvalues of complex interval matrices

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1 Bounds on eigenvalues of complex interval matrices Milan Hladík Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic SWIM 20, Bourges, France June 4 5 M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 / 8

2 Introduction Interval matrix A := [A,A] = {A R n n A A A}, The corresponding center and radius matrices A c := (A + A), 2 A := (A A). 2 M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 2 / 8

3 Introduction Interval matrix A := [A,A] = {A R n n A A A}, The corresponding center and radius matrices A c := (A + A), 2 A := (A A). 2 Complex interval matrix A + ib. M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 2 / 8

4 Introduction Interval matrix A := [A,A] = {A R n n A A A}, The corresponding center and radius matrices A c := (A + A), 2 A := (A A). 2 Complex interval matrix A + ib. Eigenvalue set Λ(A + ib) = {λ + iµ A A B B x + iy 0 : (A + ib)(x + iy) = (λ + iµ)(x + iy)}. M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 2 / 8

5 Introduction Selected publications Deif, 99: exact bounds under strong assumptions Mayer, 994: enclosure for the complex case based on Taylor expansion Kolev and Petrakieva, 2005: enclosure for real parts by solving nonlinear system of equations Rohn, 998: a cheap formula for an enclosure Hertz, 2009: extension of Rohn s formula to the complex case M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 3 / 8

6 Introduction Selected publications Deif, 99: exact bounds under strong assumptions Mayer, 994: enclosure for the complex case based on Taylor expansion Kolev and Petrakieva, 2005: enclosure for real parts by solving nonlinear system of equations Rohn, 998: a cheap formula for an enclosure Hertz, 2009: extension of Rohn s formula to the complex case Other directions approximations Hurwitz / Schur stability checking M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 3 / 8

7 Introduction Notation ρ( ) spectral radius λ ( ) λ n ( ) eigenvalues of a symmetric matrix Theorem (Rohn, 998) For each eigenvalue λ + iµ Λ(A) we have ( ) ( ) λ λ 2 (A c + A T c ) + ρ 2 (A + A T ), ( ) ( ) λ λ n 2 (A c + A T c ) ρ 2 (A + A T ), ( 0 2 µ λ (A c A T c ) ) ( ρ (A + A T ) ) 2 (AT c A c ) 0 2 (AT + A, ) 0 ( 0 2 µ λ (A c A T c ) ) ( 0 2 n ρ (A + A T ) ) 2 (AT c A c ) 0 2 (AT + A. ) 0 M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 4 / 8

8 Introduction Theorem (Hertz, 2009) For each eigenvalue λ + iµ Λ(A + ib) we have ««λ λ 2 (Ac + AT c ) + ρ 2 (A + A T ) 0 + λ 2 (BT c Bc) «0 2 (Bc BT c ) 0 + ρ 2 (BT + B «) 2 (B + B T) 0, ««λ λ n 2 (Ac + AT c ) ρ 2 (A + A T ) «0 + λ n 2 (BT c B c) 0 2 (Bc ρ 2 (BT + B «) BT c ) 0 2 (B + B T ) 0, ««µ λ 2 (Bc + BT c ) + ρ 2 (B + B T ) 0 + λ 2 (Ac AT c ) «0 2 (AT c Ac) 0 + ρ 2 (A + A T ) «2 (AT + A, ) 0 ««µ λ n 2 (Bc + BT c ) ρ 2 (B + B T ) 0 + λ n 2 (Ac «AT c ) 0 ρ 2 (A + A T ) «2 (AT c A c) 0 2 (AT + A. ) 0 M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 5 / 8

9 Symmetric case Our approach Reduction to the symmetric case. M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 6 / 8

10 Symmetric case Our approach Reduction to the symmetric case. Symmetric interval matrix M S = {M M M = M T }. M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 6 / 8

11 Symmetric case Our approach Reduction to the symmetric case. Symmetric interval matrix M S = {M M M = M T }. Eigenvalue sets Let λ (M) λ 2 (M) λ n (M). be eigenvalues of a symmetric M R n n. Then λ i (M S ) = [λ i (M S ),λ i (M S )] := { λ i (M) M M S}, i =,...,n. M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 6 / 8

12 Enclosures for the symmetric case Theorem (Rohn, 2005) λ i (A S ) [λ i (A c ) ρ(a ),λ i (A c ) + ρ(a )], i =,...,n. M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 7 / 8

13 Enclosures for the symmetric case Theorem (Rohn, 2005) λ i (A S ) [λ i (A c ) ρ(a ),λ i (A c ) + ρ(a )], i =,...,n. Theorem (Hertz, 992) Define Z := {} {±} n = {(, ±,..., ±)} and for a z Z define A z,a z AS in this way: { { a ij if z i = z j, (a z ) ij = a ij if z i z j,, (a z ) a ij = ij if z i = z j, a ij if z i z j. Then λ (A S ) = max z Z λ (A z ), λ n (A S ) = min z Z λ n(a z ). M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 7 / 8

14 Enclosures for the symmetric case Theorem (Rohn, 2005) λ i (A S ) [λ i (A c ) ρ(a ),λ i (A c ) + ρ(a )], i =,...,n. Theorem (Hertz, 992) Define Z := {} {±} n = {(, ±,..., ±)} and for a z Z define A z,a z AS in this way: { { a ij if z i = z j, (a z ) ij = a ij if z i z j,, (a z ) a ij = ij if z i = z j, a ij if z i z j. Then λ (A S ) = max z Z λ (A z ), λ n (A S ) = min z Z λ n(a z ). Others Hladík, Daney and Tsigaridas, 200, 20: filtering,... M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 7 / 8

15 Main result Theorem For each eigenvalue λ + iµ Λ(A + ib) we have ( λ 2 (A + ) AT S ( ) 2 (BT B) n 2 (B BT ) 2 (A + λ λ 2 (A + ) AT S ) 2 (BT B) AT ) 2 (B BT ) 2 (A +, AT ) ( λ 2 (B + BT ) 2 (A ) S ( AT ) n 2 (AT A) 2 (B + µ λ 2 (B + BT ) 2 (A ) S AT ) BT ) 2 (AT A) 2 (B +. BT ) M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 8 / 8

16 Main result Corollary For each λ + iµ Λ(A + ib) we have 2 λ λ (Ac +! AT c ) 2 (BT c B c) 2 (Bc BT c ) 2 (Ac + + ρ AT c ) 2 λ λ (Ac +! AT c ) 2 (BT c B c) n 2 (Bc BT c ) 2 (Ac + ρ AT c ) 2 µ λ (Bc + BT c ) 2 (Ac! AT c ) 2 (AT c A c) 2 (Bc + + ρ BT c ) 2 µ λ (Bc + BT c ) 2 (Ac! AT c ) n 2 (AT c A c) 2 (Bc + ρ BT c ) (A 2 + A T ) (B 2 + B ) T (A 2 + A T ) (B 2 + B ) T (B 2 + B ) T 2 (AT + A ) (B 2 + B ) T 2 (AT + A )! 2 (BT + B ) (A 2 + A T, )! 2 (BT + B ) (A 2 + A T, ) (A! 2 + A T ) (B, 2 + B ) T (A! 2 + A T ) (B 2 + B ) T. M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 9 / 8

17 Main result Corollary For each λ + iµ Λ(A + ib) we have 2 λ λ (Ac +! AT c ) 2 (BT c B c) 2 (Bc BT c ) 2 (Ac + + ρ AT c ) 2 λ λ (Ac +! AT c ) 2 (BT c B c) n 2 (Bc BT c ) 2 (Ac + ρ AT c ) 2 µ λ (Bc + BT c ) 2 (Ac! AT c ) 2 (AT c A c) 2 (Bc + + ρ BT c ) 2 µ λ (Bc + BT c ) 2 (Ac! AT c ) n 2 (AT c A c) 2 (Bc + ρ BT c ) (A 2 + A T ) (B 2 + B ) T (A 2 + A T ) (B 2 + B ) T (B 2 + B ) T 2 (AT + A ) (B 2 + B ) T 2 (AT + A )! 2 (BT + B ) (A 2 + A T, )! 2 (BT + B ) (A 2 + A T, ) (A! 2 + A T ) (B, 2 + B ) T (A! 2 + A T ) (B 2 + B ) T. Others using simple bounds for the symmetric case, but: for B = 0 we have the same bounds as Rohn, 998, in general, the bouds are as good as Hertz, 2009 M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 9 / 8

18 Examples Example (Seyranian, Kirillov, and Mailybaev, 2005) Let s i s 2 i s 3 B(s) = i i s + i s 2 0 s 3, i s 3 s 3 0 where s s = ([0,0.2],[0.9797,],0). Hertz enclosure: λ [ , 8.534], µ [ 7.43,.8843]. Our approach simple bounds: λ [ , 7.646], µ [ , ]. by filtering: λ [ , 6.742], µ [ 5.407, ]. best bounds: λ [ 4.580, 6.303], µ [ , ]. M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 0 / 8

19 Examples Example (con t) Monter Carlo simulation: M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 / 8

20 Examples Example (Petkovski, 99) Let 0 A = 2 [.399, 0.00] Hertz enclosure: λ [.9068, ], µ [ 2.59, 2.59]. Our approach simple bounds: λ [.9068, ], µ [ 2.59, 2.59]. by filtering: λ [.6474, ], µ [ 2.934, 2.934]. best bounds: λ [.6474, ], µ [ 2.2, 2.2]. M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 2 / 8

21 Examples Example (con t) Monter Carlo simulation: M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 3 / 8

22 Examples Example (Xiao and Unbehauen, 2000) Let 0 [, ] A = 0 [, ] [, ] [, ] 0. Hertz enclosure: λ [ 2.443,.543], µ [.443,.443]. Our approach simple bounds: λ [ 2.443,.543], µ [.443,.443]. by filtering: λ [ ,.532], µ [.443,.443]. best bounds: λ [.9674,.0674], µ [.443,.443]. M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 4 / 8

23 Examples Example (con t) Monter Carlo simulation: M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 5 / 8

24 Examples Example (Wang, Michel and Liu, 994) Let [ 3, 2] [4, 5] [4, 6] [,.5] A = [ 4, 3] [ 4, 3] [ 4, 3] [, 2] [ 5, 4] [2, 3] [ 5, 4] [, 0] [, 0.] [0, ] [, 2] [ 4, 2.5] Hertz enclosure: λ [ 8.822, ], µ [ , ]. Our approach simple bounds: λ [ 8.822, ], µ [ , ]. by filtering: λ [ , 3.384], µ [ , ]. best bounds: λ [ 7.369, ], µ [ , ]. M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 6 / 8

25 Examples Example (con t) Monter Carlo simulation: M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 7 / 8

26 Conclusion Conclusion reduction of the problem to real symmetric one cheap and tight enclosure outperforms Rohn (998) and Hertz (2009) formulae M. Hladík (CUNI) Eigenvalues of complex interval matrices June 4 5, 20 8 / 8