In honor to Professor Francisco J. Lisbona. ZARAGOZA, September 2012

Transcription

1 A MATRIX FACTORIZATION BASED ON NUMERICAL RADII Michel Crouzeix Université de Rennes 1 Workshop on Numerical Methods for Ordinary and Partial Di erential Equations and Applications In honor to Professor Francisco J. Lisbona ZARAGOZA, September 212

2 Motivation : Stability of time discretization.

3 Motivation : Stability of time discretization. Test problem y (t) = y(t), y() = y. Solution y(t) =e t y.

4 Motivation : Stability of time discretization. Test problem y (t) = y(t), y() = y. Solution y(t) =e t y. Properties lim t1 y(t) =, if Re <, y(.) is bounded if Re apple.

5 Motivation : Stability of time discretization. Test problem y (t) = y(t), y() = y. Discretization, with a time step t>, and times t n = n t. y n = R( t) n y. R(z) is a rational (polynomial) approximation of e z R(z) =e z + O(z p+1 ).

6 Motivation : Stability of time discretization. y n = R( t) n y. Stability domain S = {z 2 C ; R(z) < 1}, S = {z 2 C ; R(z) apple1}.

7 Motivation : Stability of time discretization. y n = R( t) n y. Stability domain S = {z 2 C ; R(z) < 1}, S = {z 2 C ; R(z) apple1}. Properties lim n1 y n =, if t 2S. y n is bounded if t 2 S.

8 Approximation of parabolic equations. After discretization in space by a FEM, we get d dt (u h(t),v h ) h + a h (u h (t),v h )=, 8v h 2 V h, u h () = u h,

9 Approximation of parabolic equations. After discretization in space by a FEM, we get d dt (u h(t),v h ) h + a h (u h (t),v h )=, 8v h 2 V h, Defining A h 2L(V h ) by (A h w h,v h ) h = a h (w h,v h ), 8w h,v h 2 V h, this may be also written u h (t)+a hu h (t) =, u h () = u h,

10 Approximation of parabolic equations. After discretization in space by a FEM, we get d dt (u h(t),v h ) h + a h (u h (t),v h )=, 8v h 2 V h, Defining A h 2L(V h ) by (A h w h,v h ) h = a h (w h,v h ), 8w h,v h 2 V h, this may be also written u h (t)+a hu h (t) =, u h () = u h, or also, in the node basis used for the computation, MU (t)+ku(t) =, U() = U, with M mass matrix, K sti ness matrix.

11 Approximation of parabolic equations. After a full discretization, the approximation of u(t n ) is given by u n h = R( ta h) n u h.

12 Approximation of parabolic equations. After a full discretization, the approximation of u(t n ) is given by u n h = R( ta h) n u h. von Neumann necessary stability condition for all 2 (A h ), t 2 S. (vn) (A h )=denoting the spectrum of A h.

13 Approximation of parabolic equations. After a full discretization, the approximation of u(t n ) is given by u n h = R( ta h) n u h. von Neumann necessary stability condition If for all 2 (A h ), t 2 S. (vn) A h is self-adjoint, (or normal) then (vn) =) kr( ta h ) n k h apple 1 =) stability. Stability means kr( ta h ) n k h is bounded.

14 Approximation of parabolic equations. After a full discretization, the approximation of u(t n ) is given by u n h = R( ta h) n u h. von Neumann necessary stability condition for all 2 (A h ), t 2 S. (vn) In general, in the non-normal case, the von Neumann condition is not su cient. The analysis is much more di cult. The numerical range W (A h ) plays an interesting role W (A h )={(A h v h,v h ) h = a h (v h,v h );v h 2 V h, (v h,v h ) h =1}. Here, V h is considered as a vectorial space on C.

15 Approximation of parabolic equations. After a full discretization, the approximation of u(t n ) is given by u n h = R( ta h) n u h. The numerical range W (A h ) plays an interesting role W (A h )={(A h v h,v h ) h = a h (v h,v h );v h 2 V h, (v h,v h ) h =1}. I have succeeded to obtain if tw(a h ) S, then kr( ta h ) n k h apple 12.1

16 Approximation of parabolic equations. After a full discretization, the approximation of u(t n ) is given by u n h = R( ta h) n u h. The numerical range W (A h ) plays an interesting role W (A h )={(A h v h,v h ) h = a h (v h,v h );v h 2 V h, (v h,v h ) h =1}. I have succeeded to obtain if tw(a h ) S, then kr( ta h ) n k h apple 12.1 A nice result, due to Okubo and Ando, entails that, if D is a disk such that tw(a h ) D S, then kr( ta h ) n k h apple 2.

17 It is this result of Okubo and Ando which has motivated the following developments

18 Notations. We denote by C d the space of column vectors with d complex components. It will be equipped with the usual inner product (u, v) =v u.

19 Notations. We denote by C d the space of column vectors with d complex components. It will be equipped with the usual inner product (u, v) =v u. If A 2 C d,d is a complex d d square matrix, we denote by W (A) its numerical range and by w(a) its numerical radius W (A) ={u Au ; u u =1}, w(a) =max{ u Au ; u u =1}.

20 Notations. We denote by C d the space of column vectors with d complex components. It will be equipped with the usual inner product (u, v) =v u. If A 2 C d,d is a complex d d square matrix, we denote by W (A) its numerical range and by w(a) its numerical radius W (A) ={u Au ; u u =1}, w(a) =max{ u Au ; u u =1}. We will say that a matrix M =(m ij ) is k-invariant if m ij = ij, for all i apple k and for all j apple k.

21 We introduce a family of triangular matrices T (k, ) T (2, )= 1... sin cos ,T(3, )= C A sin cos C A,

22 We introduce a family of triangular matrices T (k, ) T (2, )= 1... sin cos ,T(3, )= C A sin cos and more generally defined by t k,k 1 (k, ) =sin, t kk (k, ) = cos, t ij (k, ) = ij otherwise. 1 C A,

23 We introduce a family of triangular matrices T (k, ) T (2, )= 1... sin cos ,T(3, )= C A sin cos and more generally defined by t k,k 1 (k, ) =sin, t kk (k, ) = cos, t ij (k, ) = ij otherwise. Note that the matrix T (k, ) is k 2-invariant. 1 C A,

24 Theorem 1. Let A 2 C d,d be a given square matrix. Then we can write with A = M diag( 1,..., d ) N, M = U 2 T (2, 2 ) U 3 T (3, 3 ) U d T (d, d ) N = U 2 T (2, 2 )U 3 T (3, 3 ) U d T (d, d ), w(a) = 1 2 d, the j being complex numbers, the matrices U j being unitary and j 2 invariant, and the real numbers j belonging to [, 2 ].

25 Theorem 1. Let A 2 C d,d be a given square matrix. Then we can write with A = M diag( 1,..., d ) N, M = U 2 T (2, 2 ) U 3 T (3, 3 ) U d T (d, d ) N = U 2 T (2, 2 )U 3 T (3, 3 ) U d T (d, d ), w(a) = 1 2 d, the j being complex numbers, the matrices U j being unitary and j 2 invariant, and the real numbers j belonging to [, 2 ]. This Theorem essentially is a translation in a matrix formulation of a decomposition result due to Dritschel and Woederman.

26 Converse part. If we have with A = M diag( 1,..., d ) N, M = U 2 T (2, 2 ) U 3 T (3, 3 ) U d T (d, d ) N = U 2 T (2, 2 )U 3 T (3, 3 ) U d T (d, d ), 1 2 d, the j being complex numbers, the matrices U j being unitary and j 2 invariant, and the real numbers j belonging to [, 2 ]. Then, it holds w(a) = 1 and 1 2 W (A).

27 Remark. We note that, if a + b = 2, 1 sin cos + 1 a 1 sin cos 1 sin cos 1 b 1 sin cos = 2. 2

28 Remark. We note that, if a + b = 2, 1 sin cos + 1 a 1 sin cos 1 sin cos 1 b 1 sin cos = 2. 2 This shows that T (d, d )T (d, d )+T (d, d )T (d, d )=2,

29 Remark. We note that, if a + b = 2, 1 sin cos + 1 a 1 sin cos 1 sin cos 1 b 1 sin cos = 2. 2 This shows that T (d, d )T (d, d )+T(d, d )T (d, d )=2, therefore U d T (d, d )T (d, d )Ud + U dt (d, d )T (d, d )Ud =2.

30 Remark. We note that, if a + b = 2, 1 sin cos + 1 a 1 sin cos 1 sin cos 1 b 1 sin cos = 2. 2 This shows that T (d, d )T (d, d )+T(d, d )T (d, d )=2, therefore U d T (d, d )T (d, d )Ud + U dt (d, d )T (d, d )Ud =2. Recall that M = U 2 T (2, 2 ) U 3 T (3, 3 ) U d T (d, d ) N = U 2 T (2, 2 )U 3 T (3, 3 ) U d T (d, d ) We obtain by induction MM + NN = 2.

31 Corollary 2. The condition w(a) apple 1 is equivalent to : there exists a self-adjoint matrix B with apple B apple 2 and a contraction C such that A =2sinBCcosB.

32 Corollary 2. The condition w(a) apple 1 is equivalent to : there exists a self-adjoint matrix B with apple B apple 2 and a contraction C such that A =2sinBCcosB. Proof. (=) ) We have A = M diag( i ) N and MM + NN = 2. We can find B such that MM =2sin 2 B and NN = 2 cos 2 B, then M = p 2sinBU and N = p 2 cosbv, with unitary matrices U, V.

33 Corollary 2. The condition w(a) apple 1 is equivalent to : there exists a self-adjoint matrix B with apple B apple 2 and a contraction C such that A =2sinBCcosB. Proof. (=) ) We have A = M diag( i ) N and MM + NN = 2. We can find B such that MM =2sin 2 B and NN = 2 cos 2 B, then M = p 2sinBU and N = p 2 cosbv, with unitary matrices U, V. We set C = U diag( i ) V...

34 Corollary 2. The condition w(a) apple 1 is equivalent to : there exists a self-adjoint matrix B with apple B apple 2 and a contraction C such that A =2sinBCcosB. Proof. (=) ) We have A = M diag( i ) N and MM + NN = 2. We can find B such that MM =2sin 2 B and NN = 2 cos 2 B, then M = p 2sinBU and N = p 2 cosbv, with unitary matrices U, V. We set C = U diag( i ) V... This factorization is a particular case of a factorization due to T. Ando, in his famous paper Structure of operators with numerical radius 1, (1973). It generally di ers of the maximal or minimal one since, here C is a contraction, while Ando obtains a unitary operator.

35 Corollary 3. Given a matrix A, there exists an invertible matrix X such that kx 1 AXk applew(a) and kx 1 kkxk apple2.

36 Corollary 3. Given a matrix A, there exists an invertible matrix X such that kx 1 AXk applew(a) and kx 1 kkxk apple2. Proof. (In the case w(a) = 1). We use A = 2sinBCcosB with apple B apple 2. We set X = g(b), with g(x) =max(1, 2sinx). Then we have kxk apple2, kx 1 kapple1, kx 1 (2 sin B)k apple1, k(cos B)Xk apple1,

37 Corollary 3. Given a matrix A, there exists an invertible matrix X such that kx 1 AXk applew(a) and kx 1 kkxk apple2. Proof. (In the case w(a) = 1). We use A = 2sinBCcosB with apple B apple 2. We set X = g(b), with g(x) =max(1, 2sinx). Then we have kxk apple2, kx 1 kapple1, kx 1 (2 sin B)k apple1, k(cos B)Xk apple1, Therefore kx 1 AXk = kx 1 (2 sin B) C (cos B)Xk apple1.

38 Corollary 3. Given a matrix A, there exists an invertible matrix X such that kx 1 AXk applew(a) and kx 1 kkxk apple2. Proof. (In the case w(a) = 1). We use A = 2sinBCcosB with apple B apple 2. We set X = g(b), with g(x) =max(1, 2sinx). Then we have kxk apple2, kx 1 kapple1, kx 1 (2 sin B)k apple1, k(cos B)Xk apple1, Therefore kx 1 AXk = kx 1 (2 sin B) C (cos B)Xk apple1. This result was first noticed by Okubo and Ando (1975).

39 Corollary 3. Given a matrix A, there exists an invertible matrix X such that kx 1 AXk applew(a) and kx 1 kkxk apple2. Proof. (In the case w(a) = 1). We use A = 2sinBCcosB with apple B apple 2. We set X = g(b), with g(x) =max(1, 2sinx). Then we have kxk apple2, kx 1 kapple1, kx 1 (2 sin B)k apple1, k(cos B)Xk apple1, Therefore kx 1 AXk = kx 1 (2 sin B) C (cos B)Xk apple1. Corollary 4. If w(a) apple 1, then, for all rational functions R, kr(a)k apple2 sup{ R(z) ; z 2 D}, where D = {z 2 C, z apple1}.

40 The proof of Theorem 1 is based on Lemma. Let 2 W (A) such that = w(a). Then, there exist a unitary matrix U, a real number, and a matrix B such that or U AU = T (2, ) U AU = T (2, 2 )T (3, ) 1... C. B A T (2, ), with 2 [, /2[, B 1 C A T (3, ) T (2, 2 ).

41 The proof of Theorem 1 is based on Lemma. Let 2 W (A) such that = w(a). Then, there exist a unitary matrix U, a real number, and a matrix B such that or U AU = T (2, ) U AU = T (2, 2 )T (3, ) 1... C. B A T (2, ), with 2 [, /2[, B Furthermore, we have W (B) W (A). 1 C A T (3, ) T (2, 2 ).

42 Sketch of proof. It su ces to consider the case =12 W (A) D. Then there exists u 1 with u 1 Au 1 = 1 = u 1 u 1. Since we have v (Re A)v =Re(v Av) apple v v, for all v, we deduce (Re A)u 1 = u 1. I denote by Re A = 1 2 (A+A ) the self-adjoint part of A.

43 Sketch of proof. It su ces to consider the case =12 W (A) D. Then there exists u 1 with u 1 Au 1 = 1 = u 1 u 1. Since we have v (Re A)v =Re(v Av) apple v v, for all v, we deduce (Re A)u 1 = u 1. We can write Au 1 = u 1 + u 2, with u 1 u 2 =, u 2 u 2 = 1,, then we also have Au 1 = u 1 + u 2 and A u 1 = u 1 u 2.

44 Sketch of proof. It su ces to consider the case =12 W (A) D. Then there exists u 1 with u 1 Au 1 = 1 = u 1 u 1. Since we have v (Re A)v =Re(v Av) apple v v, for all v, we deduce (Re A)u 1 = u 1. We can write Au 1 = u 1 + u 2, with u 1 u 2 =, u 2 u 2 = 1,, then we also have Au 1 = u 1 + u 2 and A u 1 = u 1 u 2. Now, we choose a unitary matrix U with u 1,u 2 as two first columns, then U AU = 1... M. 1 C A.

45 Sketch of proof. It su ces to consider the case =12 W (A) D. Then there exists u 1 with u 1 Au 1 = 1 = u 1 u 1. Since we have v (Re A)v =Re(v Av) apple v v, for all v, we deduce (Re A)u 1 = u 1. We can write Au 1 = u 1 + u 2, with u 1 u 2 =, u 2 u 2 = 1,, then we also have Au 1 = u 1 + u 2 and A u 1 = u 1 u 2. Now, we choose a unitary matrix U with u 1,u 2 as two first columns, then U AU = 1... M. Writing v Av applev v, with v = u 1 + iy u 2, y 2 R, we get 1+cy 2 2 iy apple1+y 2, 8y, with c = u 2 Au 2. 1 C A.

46 Sketch of proof. It su ces to consider the case =12 W (A) D. Then there exists u 1 with u 1 Au 1 = 1 = u 1 u 1. Since we have v (Re A)v =Re(v Av) apple v v, for all v, we deduce (Re A)u 1 = u 1. We can write Au 1 = u 1 + u 2, with u 1 u 2 =, u 2 u 2 = 1,, then we also have Au 1 = u 1 + u 2 and A u 1 = u 1 u 2. Now, we choose a unitary matrix U with u 1,u 2 as two first columns, then U AU = 1... M. Writing v Av applev v, with v = u 1 + iy u 2, y 2 R, we get 1+cy 2 2 iy apple1+y 2, 8y, with c = u 2 Au 2. This implies 2 apple (1 Re c)/2 apple 1 and c = 1 if = 1. This allows us to set =sin. 1 C A.

47 Sketch of proof. We have U AU = 1... M. Thus we can write, if cos 6=, U AU = 1... sin cos C B 1 C A, with =sin, B This corresponds to the desired factorization. 1 C B 1 sin... cos C A.

48 Sketch of proof. We have U AU = 1... M. Thus we can write, if cos 6=, U AU = 1... sin cos C B 1 C A, with =sin, B 1 C B This corresponds to the desired factorization. It remains to look at the case = 1. 1 sin... cos C A.

49 Sketch of proof. If = 1, U AU = g... h M. 1 C A, with g, h 2 C d 2.

50 Sketch of proof. If = 1, U AU = g... h M. 1 C A, with g, h 2 C d 2. With v =(1,i,w )U, we get, for all w, 2i + i(g w w h)+w M w = v Av applev v =2+w w. This yields Re(g w w h)=for all w, thus g = h.

51 Sketch of proof. If = 1, U AU = g... h M. With v =(1,i,w )U, we get, for all w, 1 C A, with g, h 2 C d 2. 2i + i(g w w h)+w M w = v Av applev v =2+w w. This yields Re(g w w h)=for all w, thus g = h. Now with v =(, 1,w )U, we get 1+2Re(g w)+w M w = v Av applev v =1+w w. This yields Re(g w)=for all w, thus g = h =.

52 Sketch of proof. Finally, if = 1, U AU = M 1 C A = T (2, 2 ) which is the desired factorization M 1 C A T (2, 2 ),

53 T. Ando, Structure of operators with numerical radius 1, ActaSci. Math. (Szeged) 34 (1973) M.A. Dritschel, H.J. Woederman, Model Theory and Linear Extreme Points in the numerical Radius Unit Ball, Memoirs of the A.M.S., 129, n. 615, K. Okubo, T. Ando, Constants related to operators of class C, Manuscripta Math., 16 (1975),