TRIDIAGONAL-DIAGONAL REDUCTION OF SYMMETRIC INDEFINITE PAIRS¤ FRANC» OISE TISSEURy Abstract. We consider the reduction of a symmetric inde¯nite matrix pair (A; B), with B non- singular, to tridiagonal-diagonal form by congruence transformations. This is an important reduction in solving polynomial eigenvalue problems with symmetric coe±cient matrices and in frequency re- sponse computations. First the pair is reduced to symmetric-diagonal form. Then we describe three methods reducing the symmetric-diagonal pair to tridiagonal-diagonal form. Two of them employ more stable versions of Brebner and Grad's pseudosymmetric Givens and pseudosymmetric House- holder reductions, while the third is new and based on a combination of Householder reflectors and hyperbolic rotations. We prove an optimality condition for the transformations used in the third reduction. We present numerical experiments that compare the di®erent approaches and show improvements over Brebner and Grad's reductions. Key words. symmetric inde¯nite generalized eigenvalue problem, tridiagonalization, hyperbolic rotation, uni¯ed rotation, hyperbolic Householder reflector. AMS subject classi¯cations. 65F15 1. Introduction. Motivation for this work comes from the symmetric polyno- mial eigenvalue problem (PEP) m m¡1 (1.1) (¸ A + ¸ A 1 + + A0)u = 0; m m¡ ¢ ¢ ¢ where the Ai, i = 0: m, are n n symmetric matrices. ¸ is called an eigenvalue and u = 0 is the corresponding righ£t eigenvector. The standard way of dealing with the PEP6 in practice is to reformulate it as a generalized eigenvalue problem (GEP) (1.2) Ax = ¸Bx; of size mn. This process is called linearization, as the GEP is linear in ¸. Symmetry in the problem is maintained with an appropriate choice of linearization. For example, we can take 0 0 A0 0 0 A0 0 ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ . 2 . A0 A1 3 2 . A0 A1 . 3 A = . ; B = . 6 . 7 6 0 A0 . 7 6 7 6 7 6 0 A0 Am¡2 7 6 A0 A1 Am¡2 0 7 6 7 6 ¢ ¢ ¢ 7 6 A0 A1 Am¡2 Am¡1 7 6 0 0 Am 7 4 ¢ ¢ ¢ 5 4 ¢ ¢ ¢ ¢ ¢ ¢ ¡ 5 and x = [uT ; ¸uT ; : : : ; ¸m¡1uT ]T . The resulting A and B are symmetric but not de¯nite, and in general the pair (A; B) is inde¯nite. The ¯rst step in most eigensystem computations is the reduction, in a ¯nite number of operations, to a simple form. Only then is an iterative procedure applied. A symmetric inde¯nite pair (A; B) can be reduced to Hessenberg-triangular form and the ¤Numerical Analysis Report 409, Manchester Centre for Computational Mathematics, September 2002. yDepartment of Mathematics, University of Manchester, Manchester, M13 9PL, England ([email protected], http://www.ma.man.ac.uk/~ftisseur/). This work was supported by En- gineering and Physical Sciences Research Council grant GR/L76532 and Nu±eld Foundation grant NAL/00216/G. 1 resulting generalized eigenvalue problem solved by the QZ algorithm. This approach is numerically stable, but unfortunately the reduction to Hessenberg-triangular form destroys the symmetry. Moreover, in ¯nite precision arithmetic there is no guarantee that the set of left and right eigenvectors computed via the QZ algorithm will coincide, a property possessed by GEPs with real symmetric matrices. Also, by preserving symmetry, storage and computational costs can be reduced. The tridiagonal-diagonal reduction of a pair (A; B) is the most compact form we can obtain in a ¯nite number of steps. Such reductions have been proposed by Brebner and Grad [5] and by ZurmuhlÄ and Falk [26] for nonsingular B. They re- quire non-orthogonal transformations and can be unstable. Once (A; B) is reduced to tridiagonal-diagonal form the eigenvalues and eigenvectors can be obtained by ap- plying, for example, an HR iteration or associated iterations [5], [6], [16], [25], Uhlig's DQR algorithm [24], or, if one is interested in the eigenvalues only, Aberth's method can be used in an e±cient way [1]. A robust tridiagonal-diagonal reduction is therefore of prime importance before one can consider using any of the methods cited above. We note that Garvey et al. [8] have considered a less compact form that allows the second matrix to be in tridiagonal form. One feature of their approach is that no assumption is made on the nonsingularity of the two matrices. The simultaneous tridiagonalization is convenient if one needs to solve linear systems of the form (A !B)x = b ¡ for many values of !, as is required in frequency response computations [8], but it is less attractive than the tridiagonal-diagonal form for eigenvalue computations. Three di®erent tridiagonal-diagonal reductions for inde¯nite pairs (A; B) with B nonsingular are described in this paper. They all consist of two stages. The ¯rst, common to all, is the reduction of the symmetric inde¯nite pair (A; B) to symmetric- diagonal form (C; J) with the aid of a block LDLT factorization of B. During the second stage, C is tridiagonalized using a sequence of congruence transformations that preserve the diagonal form of the second matrix J. Each of the three reductions proposed in this paper uses di®erent types of transformations. These transformations are not necessarily orthogonal, so they may be unstable in ¯nite precision arithmetic. We descrive several techniques that can be used to make them more robust and to improve stability during the reduction process: in particular, pivoting and zeroing strategies in order to minimize the condition numbers of the transformations, and mixed application of hyperbolic rotations. The paper is organized as follows. Section 2 sets up notations and de¯nitions. It is shown that if the tridiagonal-diagonal reduction exists, it is determined up to signs by the ¯rst column of the transformation matrix. Section 3 describes the ¯rst stage of the reduction, that is, the reduction of (A; B) to symmetric-diagonal form (C; J). The description is accompanied by an error analysis. The second stage of the reduc- tion is described in section 4. Three algorithms are proposed. The ¯rst two are an improvement over Brebner and Grad's pseudosymmetric Givens and pseudosymmet- ric Householder methods [5]. The third algorithm is based on some transformations used to compute hyperbolic QR factorizations in inde¯nite least square problems [3]. Numerical comparisons of these algorithms and comparisons to Brebner and Grad's reductions are given in the last section. 2. Background material. Unless otherwise speci¯ed, denotes the 2-norm. We denote by diagn( 1) the set of all n n diagonal matrices withk ¢ k q diagonal elements q § £ 2 n equal to 1 and n q equal to 1. A matrix J diagq ( 1) for some q is called a signature matrix. ¡ ¡ 2 § n Let J, J diag ( 1). A matrix H Rn£n is said to be (J; J)-orthogonal 2 q § 2 if HT JH = J. Note that (J; J)-orthogonal matrices are sometimes called (J; J)- hyperexchangee or (J; J)-hypernormal matrices in the signal processing literaturee [17]. We recall ethat a tridiagonalematrix is unreduced if none of its next-to-diagonale elements (that is, theeelements on the ¯rst subdiagonal and the ¯rst superdiagonal) is zero. The following result, is related to the implicit Q theorem [11]. A more general form can be found in [18, Th.2.2]. Theorem 2.1. If C Rn£n admits a representation of the form 2 (2.1) QT CQ = T; where T is unreduced tridiagonal and Q is (J; J)-orthogonal then the columns of Q and the next-to-diagonal elements of T are determined up to signs by the ¯rst (or last) column of Q. e We give the proof since we need to refer to it later in the text. This is a construc- tive proof that describes a Lanczos process. Proof. Let J = diag(σ~ ), σ~ = 1, i = 1: n, and i i § e ®1 ¯2 ¯2 ®2 ¯3 2 .. .. 3 T = ¯3 . : 6 7 6 .. 7 6 . ®n¡1 ¯n 7 6 7 6 ¯n ®n 7 4 5 T We assume that q1 is given and normalized such that σ~1 = q1 Jq1. This yields T ®1 = q1 Cq1: Using the (J; J)-orthogonality of Q, equation (2.1) can be rewritten as (2.2) e JCQ = QJT: Equating the ¯rst column on each side of (2.2)egives p1 JCq1 ®1σ~1q1 = ¯2σ~2q2: ´ ¡ ¡2 T From the (J; J)-orthogonality of Q we get σ~2 = ¯2 p1 Jp1, which implies T T e σ~2 = sign(p Jp1); ¯2 = p Jp1 ; 1 § j 1 j q so that ¡1 q2 = σ~2¯2 p1 is determined up to the sign chosen for ¯2. The second diagonal element of T is uniquely determined by T ®2 = q2 Cq2: 3 Hence, the construction of q2, ®2, ¯2 and σ~2 requires just the knowledge of p1. Now suppose that the ¯rst j < n columns of Q and the leading j j principal submatrices of T and J are known. Then by equating the jth columns£on each side of (2.2) we obtain e p JCq σ~ ® q σ~ 1¯ q 1 = σ~ +1¯ +1q +1: j ´ j ¡ j j j ¡ j¡ j j¡ j j j Using once again the (J; J)-orthogonality of Q we have T T (2.3) σ~ +1e= sign(p Jp ); ¯ +1 = p Jp : j j j j § j j j j q Hence ¡1 T (2.4) qj+1 = σ~j+1¯j+1pj; ®j+1 = qj+1Cqj+1: Again, ¯j+1 and qj+1 are determined up to a sign. By induction on j all columns of Q and all next-to-diagonal elements of T are determined, up to a sign by q1.