Constructing a Unitary Hessenberg Matrix from Spectral ... - CiteSeerX

Constructing a Unitary Hessenberg Matrix from Spectral Data Gregory Ammar1 , William Gragg2, Lothar Reichel3

In memory of Peter Henrici

Abstract We consider the numerical construction of a unitary Hessenberg matrix from spectral data using an inverse QR algorithm. Any unitary upper Hessenberg matrix H with nonnegative subdiagonal elements can be represented by 2n ? 1 real parameters. This representation, which we refer to as the Schur parameterization of H; facilitates the development of ecient algorithms for this class of matrices. We show that a unitary upper Hessenberg matrix H with positive subdiagonal elements is determined by its eigenvalues and the eigenvalues of a rank-one unitary perturbation of H: The eigenvalues of the perturbation strictly interlace the eigenvalues of H on the unit circle. AMS(MOS) Subject Classi cation: 15A18, 65F15. Keywords: inverse eigenvalue problem, unitary matrix, orthogonal polynomial.

Presented at the NATO Advanced Study Institute on Numerical Linear Algebra, Digital Signal Processing, and Parallel Algorithms, Leuven, Belgium, August, 1988. This research was supported in part by the National Science Foundation under grant DMS-8704196, and by the Foundation Research Program of the Naval Postgraduate School. This paper appears in: Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms, G.H. Golub and P. Van Dooren, eds., NATO ASI Series, Vol. F70, Springer-Verlag, Berlin, 1991, pp. 385{395. Minor corrections incorporated, October 1993.

1 2

3

Northern Illinois University, Department of Mathematical Sciences, DeKalb, IL 60115, USA Naval Postgraduate School, Department of Mathematics, Monterey, CA 93943, USA, on leave from University of Kentucky, Department of Mathematics, Lexington, KY 40506, USA Bergen Scienti c Centre IBM, Allegaten 36, N-5007 Bergen, Norway, on leave from University of Kentucky, Department of Mathematics, Lexington, KY 40506, USA 1

1. Introduction

In this paper we focus on an inverse eigenvalue problem for unitary Hessenberg matrices with positive subdiagonal elements. Throughout this paper all Hessenberg matrices are upper Hessenberg matrices. This class of matrices bears many similarities with the class of Jacobi matrices, i.e. real symmetric tridiagonal matrices with positive subdiagonal elements. Any matrix in either class is normal and has distinct eigenvalues. Both n  n Jacobi matrices and n  n unitary Hessenberg matrices with positive subdiagonal elements can be parameterized by 2n ? 1 real parameters. This is obvious for Jacobi matrices; for unitary Hessenberg matrices this parameterization is described below. Since unitary Hessenberg matrices with positive subdiagonal elements are determined by O(n) parameters, one can develop ecient algorithms for this class of matrices. These algorithms are analogous with algorithms for Jacobi matrices. For example, the unitary QR algorithm, described in [Gr2], has many similarities with the QR algorithm for Jacobi matrices. Another example is provided by the divide-and-conquer methods that have been developed for both the tridiagonal and unitary eigenproblems [Cu], [DS], [GR1], [GR2]. In this paper we show another analogy of unitary Hessenberg matrices with Jacobi matrices, namely, that a unitary Hessenberg matrix H with positive subdiagonal elements is uniquely determined by its eigenvalues and the eigenvalues of a unitary rank-one perturbation of H . The matrix H can be constructed using O(n2 ) arithmetic operations using an inverse unitary QR algorithm. Similar results for Jacobi matrices are well-established [BG1], [BG2], [GH]. The inverse unitary QR algorithm is analogous with the algorithm described in [GH] for Jacobi matrices.

2. Unitary Hessenberg Matrices and Szego Polynomials

We refer to a nite Schur parameter sequence of length n as a sequence of complex numbers f j gnj=1 ?1 with j j j 0 is uniquely determined by a nite Schur parameter sequence of length n. In fact, the Schur parameters ?1 can be determined from H by f j gnj=1 and the complementary Schur parameters fj gnj=1

j = j+1;j ;

1  j < n;

j = ?1;j =1 2


j?1 ;

1  j  n:

Hence, we have a one-to-one correspondence between Schur parameter sequences of length n and n  n unitary Hessenberg matrices with positive subdiagonal elements. This Schur parameterization of unitary Hessenberg matrices with positive subdiagonal elements shows that these matrices are determined by 2n ? 1 real parameters: the real and imaginary parts of j for 1  j < n, and the 2

argument of n. To avoid numerical instability, however, we also retain the complementary Schur parameters. Let H = Hn := H ( 1; : : :; n?1; n), and let Hk := G1 ( 1 )G2 ( 2) : : :Gk?1 ( k?1 )G~ k ( k ) be the leading principal submatrix of H = Hn of order k. Introduce the functions

k () := eT1 (I ? Hk )?1 e1 ; k () := det(I ? Hk ); k () := det(I ? Hk00?1 );

where e1 := [1; 0; : : :; 0]T , and Hk00?1 denotes the trailing principal submatrix of Hk of order k ? 1. Thus, k () = k ()= k (). The following proposition can be veri ed by induction. Proposition 2.1. The polynomials k () and k (), k > 0, satisfy "

#

"

 k k () k () ~k () ~k () = k  1

#"

# "

#

"

#

1 1= 0 () 0 () k?1 () k?1 () ~k?1 () ~k?1 () ; ~0 () ~0 () = 1 0 :

It follows that for each k, the polynomials ~k () := k k (1=) are obtained by reversing and conjugating the coecients of k (). From the initial conditions we recognize k () to be the kth Szego polynomial determined by the Schur parameter sequence f j gnj=1 : The Szego polynomials f k gnk=0 are orthogonal with respect to a discrete measure on the unit circle. This measure assigns a positive weight !k to each zero k of n(). These weights are the numerators in the partial fraction decomposition

n () =:

n X

!k ; k=1  ? k

see [Gr1] for details. Let H = U U , with  = diag[1; 2; : : :; n] and U unitary, be the spectral resolution of H = H ( 1; : : :; n?1; n), where  denotes transposition and complex conjugation. Let u := U T e1 be the vector containing the rst components of the eigenvectors of H . Since H has nonzero subdiagonal elements, every entry of u = [1; 2; : : :; n]T is nonzero, and we normalize U so that each k > 0. Then n 2 X k : n() = u (I ? )?1 u = k=1  ? k P Hence, the weights !k = k2 are guaranteed to be positive, and nk=1 !k = 1.

3. The Inverse Unitary QR Algorithm

Given n distinct unimodular complex numbers fk gnk=1 and associated positive weights fk2gnk=1 , we can construct a unitary Hessenberg matrix H with the k and k equal to the eigenvalues and rst components of the corresponding eigenvectors of H , respectively. This construction is achieved using an inverse QR algorithm, which is analogous with the procedure of [GH] for real symmetric tridiagonal matrices. The required Hessenberg matrix is obtained byperforming a sequence of elementary unitary    simi  larity transformations to transform the matrix u u to a Hessenberg matrix e eH1 without 1

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using or changing the arbitrary entry  . Then H = H ( 1; : : :; n?1; n) has the desired eigenvalues and associated eigenvectors. The idea is to build the Hessenberg matrix by adding weight-abscissa pairs one at a time. Suppose that we have constructed the unitary Hessenberg matrix Hm := H ( 1; : : :; m?P 1 ; m ), for some m m < n, corresponding with the weight-abscissa pairs f(!k ; k )gk=1 . Let 0 := ( mk=1 !k )1=2 and assume that the rst components of the eigenvectors of Hm are f(k =0 )gm k=1 . In order to add the weight-abscissa pair ( 2; ) and construct the corresponding (m + 1)  (m + 1) unitary Hessenberg matrix Hm0 +1 := H (01 ; 02; : : :; 0m+1 ), we perform a sequence of unitary similarity transformations to put the (m + 2)  (m + 2) matrix 2

  0   6 2 3 6   0 e1     6 0 6 (1)  ~ 4 5 H :=   0 = 66     0 e1 0 Hm    6 4    p

3 7

 777  77  75  

into Hessenberg form without changing  . Let 00 := 02 +  2 and 0 := ?=00 . Then 2

3   0 0 6 0       7 6 7 6 0      7 6 G2 (0 )H~ (1) =: H~ (2) = 66      777     75 6 4      is a Hessenberg matrix with a trailing principal (m + 1)  (m + 1) submatrix, which is both unitary and of Hessenberg form. On the completion of the similarity transformation of H~ (1) , we obtain 2  0 3 0 6 00       7 6       777 6 6 H~ (2)G2 (0 ) = 66

     77 : (3:1)     6 7 4   5  

The circled element in (3.1) forms a \bulge", which is to be chased down along the subdiagonal in order to obtain a matrix of Hessenberg form. De ne G3 (1 ) so that G3 (1 )H~ (2) G2 (0 ) =: H~ (3) is a Hessenberg matrix. Then H~ (3) G3 (1 ) has a bulge, which we annihilate by multiplying from the left with G4 (2 ). Proceeding in this manner, we ultimately chase the bulge o the bottom of the matrix, and obtain the Hessenberg matrix H~ (m?1) Gm?1 ( m?3 ), which is unitarily similar to H~ (1) . The trailing principal (m + 1)  (m + 1) submatrix of H~ (m?1) Gm?1 ( m?3 ) is unitarily similar to a unitary Hessenberg matrix with positive subdiagonal elements. The latter matrix is the desired Hessenberg matrix Hm0 +1 = H ( 10 ; 20 ; : : :; m0 +1). This procedure for adding a weight-abscissa pair to Hm , if implemented by directly manipulating the elements of the matrices H~ (k) , for 1  k < m, would require O(m2 ) arithmetic operations. However, 4

we note that for each k the trailing (m+1)(m+1) principal submatrix of H~ (k) is unitary and of Hes) , senberg form, and, therefore, is unitarily similar to a unitary Hessenberg matrix, denoted by H^ m(k+1 with positive subdiagonal elements. Hence, we can carry out the similarity transformations by ma+2) nipulating the Schur parameters of the matrices H^ m(k+1 = H ( 10 ; 20 ; : : :; k0 ;  k k ; k+1; : : :; m ). This gives rise to a method that requires only O(m) arithmetic operations in order to add a weightabscissa pair to Hm . We refer to this method as the inverse unitary QR algorithm because of its relationship with the unitary QR algorithm presented in [Gr2]. Inverse Unitary QR Algorithm: adding a weight-abscissa pair (2; ), where jj = 1;  > 0. p 00 := 02 +  2; 0 := ?=00 ; 0 := 0 =00 ; for k := 1; 2; : : :; m 6 q 6 0 6 k := k?1 k2 + jk?1 + k k?2   k?1 j2 ; 6 6 6 k := k?1 (k?1 + k k?2 k?1 )=k0 ; 6 6 :=  = 0 ; k?1 k k 4 k 0

k := k2?1 k ?  k?2 2k?1 ;

m0 +1 := ? m ; Thus, the inverse unitary QR algorithm can be used to construct the unitary n  n Hessenberg matrix H = H ( 1; : : :; n?1; n) from its eigenvalues and the rst components of its normalized eigenvectors in O(n2 ) arithmetic operations.

4. An Inverse Spectral Problem

n X Let H = H ( 1; : : :; n?1; n) and n() = n (()) =  ?!k . Let  := ei for some 0 <  < 2 , n k k=1 and consider the polynomials k () := (1 ? )k() +  k (); 0  k  n: (4:1) Proposition 4.1. The polynomials fk ()gnk=0 are the monic Szego polynomials corresponding with the Schur sequence f k gnk=1 . Proof. The de nition (4.1) and Proposition 2.1 yield          k () =  k k?1 () ; 0() = 1 ; k ()

k  1 k?1 () 0 ()  where k () := (1 ? )~k () +  ~k (). On setting ~k () = k (), we obtain 





k () =   k ~k ()  k  1



 



 

k?1 () 0 () 1 ~k?1 () ; ~0 () = 1 :

An immediate consequence of Proposition 4.1 is that each zero of n () has unit modulus. When  = ?1 the polynomials fk ()gnk=0 are known as the Szego polynomials of the second kind corresponding with f k gnk=1 . 5

Proposition 4.2. TheQzeros fk gnk of n() strictly interlace the zeros fk gnk of n() on the n =0

=0

unit circle. Moreover, k=1 (k =k ) = . Proof. Let  := ei for some 0   < 2 , and let  =: ei for some 0 <  < 2 . Then n X k n () := n (()) = !k ??  n k k=1 n   X = ! 1 ? k + (1 ? k ) k

j ? k j i=  = 2 ei= !k Re(e j ?(1?j k )) : k k Thus, Im(e?i= n(ei ))  0 for 0   < 2 . Let k =: ei . We may assume that 0   <  <


< n < 2. Then n X ? i= i e n (e ) = 2 !k cos(=2) ? cos(=2 + k ? ) (1 ? cos(k ? )) + sin (k ? ) k n X k ? ) + sin(=2) sin(k ? ) = !k cos(=2) ? cos(=2)1cos( ? cos(k ? ) k k=1 n X

2

2

2

2

=1

2

k

2

2

2

2

=1

1

=1

=

n X k=1

!k (cos(=2) + sin(=2)cot((k ? )=2)) :

From 0 <  < 2 and !k > 0 for all k, it follows that n X 1 ? i= 2 d 2 i e d n(e ) = 2 sin(=2) k=1 !k = sin ((k ? )=2) > 0 for  6= k ; 1  k  n:

Thus, n (ei ) ! ?1 as  & k , and n (ei ) ! 1 as  % k+1 . This shows that  ! n (ei ) has precisely one zero in ]k ; k+1 [. Consequently, the zeros j of n () strictly interlace the j on the unit circle. The second statement of the Proposition follows from the fact that n (0)= n(0) = : P Let k =: eik and k =: eik for 0  k ; k < 2 . Then we have nk=1 (k ? k ) =  . We may assume that the arguments have been ordered so that 0  1 < 1 < 2 <


< n?1 < n < n < 1 + 2: (4:2) Proposition 4.3. With the above notation, and the ordering (4.2), the weights !k are given by n Q

sin((j ? k )=2) 1 j =1 : !k = sin(=2) Qn sin(( ?  )=2) j =1 j 6=k

Proof. We have

j

k

(1 ? )k !k = lim ( ? k ) n (()) = n0 ((k )) = ! n n k k

6

n Q

(k ? j )

j =1 n Q

j =1 j 6=k

(k ? j )

since n () and n() are monic. Thus, n Q

(eik ? eij )

n Q

(1 ? ei(j ?k ) )

n Q

sin((j ? k )=2) j =1 j =1 i= 2 j =1 ? i k = Q = ?2ie Q ; (4:3) (1 ? )!k = e n n n Q i i (  ?  ) i j j k k (e ? e ) (1 ? e ) sin((j ? k )=2) j =1 j =1 j =1 j= 6 k j 6=k j 6=k because

1 ? ei = ?2iei =2 sin( =2); (4:4) P and nj=1 (j ? j ) =  . The formula for the weights now follows by substituting (4.4) with :=  into (4.3). We can now state the inverse spectral problem and its solution. Problem: Given two sets of n mutually interlacing points on the unit circle fkgnk=1 and fk gnk=1, determine the unique unitary Hessenberg matrix H = H ( 1; : : :; n?1; n) and jj = 1 such that the spectrum of H is fk gnk=1 and the spectrum of H ( 1; : : :;  n?1;  n) is fk gnk=1 . Solution: Let  := Qnk=1 (k =k ), calculate the weights by Proposition 4.3, and use the inverse unitary QR algorithm to construct H = H ( 1; : : :; n?1; n).

References

[BG1] D. Boley and G.H. Golub, Inverse Eigenvalue Problems for Band Matrices, in Lecture Notes in Mathematics 630, Numerical Analysis, Proceedings of the Biennial Conference Held at Dundee, 1977, G.A. Watson, ed., Springer-Verlag, New York, 1978, pp. 23-31. [BG2] D. Boley and G.H. Golub, A Survey of Matrix Inverse Eigenvalue Problems, Inverse Problems 3, (1987) 595-622. [Cu] J.J.M. Cuppen, A Divide and Conquer Method for the Symmetric Tridiagonal Eigenproblem, Numer. Math. 36, (1981) 177-195. [DS] J.J. Dongarra and D.C. Sorensen, A Fully Parallel Algorithm for the Symmetric Eigenproblem, SIAM J. Sci. Stat. Comput. 8, (1987) s139-s154. [Gr1] W.B. Gragg, Positive De nite Toeplitz Matrices, the Arnoldi Process for Isometric Operators, and Gaussian Quadrature on the Unit Circle (in Russian), in Numerical Methods in Linear Algebra, E.S. Nikolaev, ed., Moscow University Press, Moscow, 1982, pp. 16-32. [Gr2] W.B. Gragg, The QR Algorithm for Unitary Hessenberg Matrices, J. Comput. Appl. Math. 16, (1986) 1-8. [GH] W.B. Gragg and W.J. Harrod, The Numerically Stable Reconstruction of Jacobi Matrices from Spectral Data, Numer. Math. 44, (1984) 317-355. [GR1] W.B. Gragg and L. Reichel, A Divide and Conquer Algorithm for the Unitary Eigenproblem, in Hypercube Multiprocessors 1987, M.T. Heath, ed., SIAM, Philadelphia, 1987, pp. 639-647. [GR2] W.B. Gragg and L. Reichel, A Divide and Conquer Method for the Unitary and Orthogonal Eigenproblems, BSC Report 88/31, Bergen Scienti c Centre, Bergen, 1988.

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