Robust Parallel Lanczos Methods for Clustered Eigenvalues* M. Szularz 1, J. Weston 1, and M. Clint 2 1 School of Information & Software Engineering, University of Ulster, Coleraine BT52 1SA, Northern Ireland 2 Department of Computer Science, The Queen's University of Belfast, Belfast BT7 INN, Northern Ireland Abstract. In this paper two recently proposed single-vector Lanczos methods based on a simple restarting strategy are analysed and their suitability for the computation of closely clustered eigenvalues is eval- uated. Both algorithms adopt an approach which yields a fixed k-step restarting scheme in which one eigenpair at a time is computed using a deflation technique in which each Lanczos vector generated is orthogonal- ized against all previously converged eigenvectors. In the first algorithm each newly generated Lanczos vector is also orthogonalised with respect to all of its predecessors; in the second, a selective orthogonalisation strategy permits re-orthogonalization between the Lanczos vectors to be almost completely eliminated. 'Reverse communication' implementations of the algorithms on an MPP Connection Machine CM~200 with 8K pro- cessors are discussed. Advantages of the algorithms include the ease with which they cope with genuinely multiple eigenvalues, their guaranteed convergence and their fixed storage requirements. Key words : Lanczos, restart, deflation, orthogonalization, MPP. 1 The Lanczos Algorithm Essentially, the Lanczos algorithm [2], [5] generates a sequence of tridiagonal matrices Tj, each with values (al,..., aj) on its main main diagonal and val- ues (~1,...,~j-1) on its main sub- and super-diagonals, together with a se- quence of orthonormal matrices of Lanczos vectors Qj = [ql,...,qj], where j _< n, such that Q~AQ = Tj. It can be shown that Qj is an orthonormal basis for tC(A, ql,j), the Krylov subspace of order j generated by A and ql. Let XTAX = diag(A1,...,An) where A1 _> A2 _> -.- _> An, and where X = Ix1,..., xn] is orthonormal, be the spectral decomposition of A. Similarly, let SfTjSj .= diag(81(Tj),...,8j(Tj)) where 81 > 82 > ... > 8j, and Sj = (spq), be the spectral decomposition of Tj. Then yi E ~n in QjSj = Yj = [yl,-.., yj] is known as the i-th Ritz vector of A for the subspace range(Qj), and 8i(Tj) This work was supported by the Engineering and Physical Sciences Research Council under grants GR/J41857 and GR/J41864 and was carried out using the facilities of the University of Edinburgh Parallel Computing Centre 711 is known as the corresponding Ritz value. It can be shown that, for surpris- ingly small j, the Ritz pair (Oi, Yi) closely approximates the i-th eigenpair of A, (;~i, xi), provided that the ratio I )~i I/ll A 112 is reasonably close to unity. 1.1 Estimating the Largest Eigenvalue Let ql C ~n be such that II ql 112 = 1. Then the Lanczos algorithm for the computation of the largest eigenvalue only may be expressed as follows: Algorithm 1 function[(01, Yl)] -- Lanczos(A, ql, k) flo +- 1, q0 +-- 0 for j --1, 2, . , k aj +- qTAqj qj+l ~ (A - ~jI)qj - Zj-lqj-1 +-II qj+1112 if ~j = 0 then STOP (eigenvalues of Tj are the j largest eigenvalues of A) qj+l ~ qj+l/Z~ (A) orthonormalize qj+l against ql,.-.,qj end_for compute (01, Yl). It follows that the Ritz pair (01, Yl) returned by Algorithm 1 is an approximation to the eigenpair (A1, Xl) of A. The accuracy of this approximation may be estimated before Yl is computed on the basis of the error bounds I ~j II Sjl I [2], since I ~J II Sjl I----IIAyl - 01yl 112 (1) Observe that the orthogonality of the Lanczos vectors has been guaranteed throughout by the inclusion of a complete re-orthogonalization of the Lanczos vectors in each iteration (Step A). A major problem associated with the above algorithm is that k must be chosen to be sufficiently large to guarantee the required accuracy in the solution. Moreover, the value of k required is not known in advance. However, two explicit restarting schemes for the computation of the p << n largest eigenvalues of a symmetric matrix A have been developed [8], each of which incorporates a fixed k-step variant of Algorithm 1 where k is assumed to be small. Brief outlines of these schemes are now given. 2 Lanczos with Explicit Restart: EXPRES1 Let approximated quantities be decorated with the : symbol. Thus, at the j-th Lanczos step, the Ritz pair (Oi,yi) is synonymous with (ii,~i). Suppose that the approximated eigenpairs (il,Xl),-.., (ii,xi), where i < p, are given. Let 2~ = span{21,..., 2~} and let/~ be its orthogonal complement in ~n. Observe that, if the Lanczos vectors ql,.., qk are constrained to stay in the subspace A'~-, 712 the Lanczos algorithm will converge to the Ritz values of A in the subspace ?~/_L, viz, the desired approximations Ai+l, A/+2,... This observation provides the basis for the two fixed k-step restarting schemes described below. The first may be expressed as the following algorithm: Algorithm 2 function[(A1, :~1),..., (,~p, Xp)] : EXPRESI(A,p, k, tol) 20=0 (1) fori=l:p (1) choose ql ¢ 2i--1 (2) Yl +'- ql ; O1 +- y~Ayl (3) while ~,(IIAya-OlYllI21011) tol) (1) (Ol,yl) < Lanczos(A, ql,k) (2) ql +- Yl" end_while (4) (£i,~ci) +- (01,Yl) (5) if/<p then Xi +- Xi-1 @ span{Zi} end_for tol is the user supplied tolerance (normally set to u, the relative machine ac- curacy). It is assumed that Algorithm 1, as used in Step (1.3.1) above, is modified to include a mechanism for projecting each newly generated Lanczos vector (including ql) into X~ ¢ 0. This can obviously be achieved by the ex- plicit orthogonalization of each qj against 51,. •., ~i. Algorithm 2 as described above is henceforth referred to as EXPRES1. Observe that, since k is completely independent of p, it can always be chosen to be small. Clearly, since the i-th eigenvalue of A is sought in the subspace 2(~], Al- gorithm 2 is theoretically ideal for coping with closely clustered and genuinely multiple eigenvalues of A. This paper provides numerical evidence that this is indeed the case in practice. Further, it is demonstrated that, in many cases where closely clustered and genuinely multiple eigenvalues occur, Algorithm 2 is signif- icantly more efficient than Sorensen's state-of-the-art routine when implemented in a massively parallel SIMD environment. 3 Lanczos with Explicit Restart: EXPRES2 A finely tuned version of Algorithm 1 has also been developed for use with Algorithm 2 in which the complete re-orthogonalization of the Lanczos vectors (Step A) is replaced by a selective re-orthogonalization strategy [5], [6]. Thus, as soon as the error bound associated with a Ritz vector satisfies I /3j I[ sii [<- ~ II A [Iz (2) the Lanczos process is immediately restarted with the current value of Yl, even when j < k. Further, if Yl triggered the restart, all subsequent qj+l computed are not only projected into X~ 7~ 0 but are also orthogonalized against the recently computed, 'converging' Ritz vector yl. This orthogonalization strategy purges all 713 unwanted, 'converging' Ritz vectors, yi : i # 1, from the system and consider- ably reduces the computational overhead associated with the orthogonalisation process. An outline of this version of Algorithm 1 is given below: Algorithm 3 function[(01, Yl)] = Lanczos(A, ql, k, converging) /3o +-- 1, q0 +- 0 (1) for j = 1,2,...,k (1) if i ¢ 0 then orthogonalize qj against 21,... 2i (2) aj +- qTdq3 (3) qj+l +- (A - (~jX)qj - ~j-lqj-1 (4) 9j -I1 qj+1112 (5) if ~j = 0 then STOP (eigenvalues of Tj are the j largest eigenvalues of A) (6) qj+l qj+i/Z¢ (7) if converging = false then (1) compute sjl,...,sjj (2) if min{I /3j {] Sjl I,...,] flJ II Sjj I} = I /3J II Sjr I~- V ~ II A 112 then (1) compute (01,yl) (2) if r = 1 then converging +- true (3) exit else (3) orthogonalize qj+l against ql end_if end_for (2) compute (01,yl). The variant of Algorithm 2 which incorporates the finely tuned version of Algo- rithm 1 above is referred to as EXPRES2. Note also that this variant requires converging +-- false to be added to Step (1.2) in Algorithm 2 and Step (1.3.1) to be replaced by (01, Yl) +--- Lanczos(A, ql, k, converging). Further, it can be established formally that each of the restart schemes, EXPRES1 and EXPRES2, guarantees convergence to the p required eigenvalues of A. 4 Implementation on the Connection Machine CM-200 The algorithms have been implemented using a Reverse Communication strategy [7], [3] in which the user is responsible for supplying the code for all matrix-vector products of the form Aqj. Such an approach enables the user to take advantage of the architecture of the target machine and of any available highly optimised code when implementing these computationally expensive operations. Thus, all vectors of length n are declared and stored as distributed CM arrays, whereas all other quantities are confined to the 'front-end' machine. It follows that all massive saxpy type operations involving the n-vectors are fine-grain parallel op- erations and, consequently, they are performed on the CM itself using the highly optimised matrix-vector product routines which are available in the Connection 714 Machine Scientific Support Library. These operations include, for example, the dense matrix-vector products in the reorthogonalization steps (1.1) and (1.7.3) of Algorithm 3 and they occur also in the computation of the Ritz vectors in steps (2) and (1.7.2.1) of this algorithm.