A Lanczos-type method for multiple starting vectors

Aliaga, J.; Boley, D.; Freund, R.; Hernández, V.

doi:10.1090/S0025-5718-99-01163-1

A Lanczos-type method for multiple starting vectors
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by J. I. Aliaga, D. L. Boley, R. W. Freund and V. Hernández PDF

Math. Comp. 69 (2000), 1577-1601

Abstract:

Given a square matrix and single right and left starting vectors, the classical nonsymmetric Lanczos process generates two sequences of biorthogonal basis vectors for the right and left Krylov subspaces induced by the given matrix and vectors. In this paper, we propose a Lanczos-type algorithm that extends the classical Lanczos process for single starting vectors to multiple starting vectors. Given a square matrix and two blocks of right and left starting vectors, the algorithm generates two sequences of biorthogonal basis vectors for the right and left block Krylov subspaces induced by the given data. The algorithm can handle the most general case of right and left starting blocks of arbitrary sizes, while all previously proposed extensions of the Lanczos process are restricted to right and left starting blocks of identical sizes. Other features of our algorithm include a built-in deflation procedure to detect and delete linearly dependent vectors in the block Krylov sequences, and the option to employ look-ahead to remedy the potential breakdowns that may occur in nonsymmetric Lanczos-type methods.

References

J. I. Aliaga, Algoritmos paralelos basados en el método de Lanczos. Aplicaciones en problemas de control, Doctoral Thesis, Departamento de Sistemas Informáticos y Computación, Universidad Politécnica de Valencia, Valencia, Spain, 1995.
J. I. Aliaga, D. L. Boley, R. W. Freund, and V. Hernández, A Lanczos-type algorithm for multiple starting vectors, Numerical Analysis Manuscript No. 96–18, Bell Laboratories, Murray Hill, NJ, 1996. Also available online from http://cm.bell-labs.com/cs/doc/96.
J. I. Aliaga, D. L. Boley, and V. Hernández, A block clustered Lanczos algorithm, Presentation at the workshop on “Numerical Linear Algebra with Applications”, Oberwolfach, Germany, April 1994.
Z. Bai, D. Day, and Q. Ye, ABLE: an adaptive block Lanczos method for non-Hermitian eigenvalue problems, Research Report 95-04, Department of Mathematics, University of Kentucky, Lexington, KY, 1995.
Daniel L. Boley, Krylov space methods on state-space control models, Circuits Systems Signal Process. 13 (1994), no. 6, 733–758. MR 1286564, DOI 10.1007/BF02523124
Daniel Boley and Gene Golub, The nonsymmetric Lanczos algorithm and controllability, Systems Control Lett. 16 (1991), no. 2, 97–105. MR 1095250, DOI 10.1016/0167-6911(91)90003-W
Okko H. Bosgra and Antonius J. J. van der Weiden, Input-output invariants for linear multivariable systems, IEEE Trans. Automat. Control 25 (1980), no. 1, 20–36. MR 561336, DOI 10.1109/TAC.1980.1102260
Adhemar Bultheel, Recursive algorithms for the matrix Padé problem, Math. Comp. 35 (1980), no. 151, 875–892. MR 572862, DOI 10.1090/S0025-5718-1980-0572862-4
J. K. Cullum and W. E. Donath, A block Lanczos algorithm for computing the $q$ algebraically largest eigenvalues and a corresponding eigenspace for large, sparse symmetric matrices, Proc. 1974 IEEE Conference on Decision and Control, IEEE Press, New York, 1974, pp. 505–509.
Jane K. Cullum and Ralph A. Willoughby, Lánczos algorithms for large symmetric eigenvalue computations. Vol. I, Progress in Scientific Computing, vol. 3, Birkhäuser Boston, Inc., Boston, MA, 1985. Theory. MR 808962, DOI 10.1007/978-1-4684-9178-4_{6}
Jane Cullum and Ralph A. Willoughby, A practical procedure for computing eigenvalues of large sparse nonsymmetric matrices, Large scale eigenvalue problems (Oberlech, 1985) North-Holland Math. Stud., vol. 127, North-Holland, Amsterdam, 1986, pp. 193–240. MR 875438, DOI 10.1016/S0304-0208(08)72647-1
P. Feldmann and R. W. Freund, Efficient linear circuit analysis by Padé approximation via the Lanczos process, IEEE Trans. Computer-Aided Design 14 (1995), 639–649.
—, Reduced-order modeling of large linear subcircuits via a block Lanczos algorithm, Proc. 32nd Design Automation Conference, ACM, New York, 1995, pp. 474–479.
Roland W. Freund, The look-ahead Lanczos process for nonsymmetric matrices and its applications, Proceedings of the Cornelius Lanczos International Centenary Conference (Raleigh, NC, 1993) SIAM, Philadelphia, PA, 1994, pp. 33–47. MR 1298220
C. K. Chui and L. L. Schumaker (eds.), Approximation theory VIII. Vol. 1, Series in Approximations and Decompositions, vol. 6, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. Approximation and interpolation. MR 1471706
R. W. Freund and P. Feldmann, Efficient circuit analysis by Padé approximation via the Lanczos process, Presentation at the workshop on “Numerical Linear Algebra with Applications”, Oberwolfach, Germany, April 1994.
Roland W. Freund, Martin H. Gutknecht, and Noël M. Nachtigal, An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices, SIAM J. Sci. Comput. 14 (1993), no. 1, 137–158. MR 1201315, DOI 10.1137/0914009
Roland W. Freund and Manish Malhotra, A block QMR algorithm for non-Hermitian linear systems with multiple right-hand sides, Proceedings of the Fifth Conference of the International Linear Algebra Society (Atlanta, GA, 1995), 1997, pp. 119–157. MR 1436677, DOI 10.1016/S0024-3795(96)00529-0
Roland W. Freund and Noël M. Nachtigal, QMR: a quasi-minimal residual method for non-Hermitian linear systems, Numer. Math. 60 (1991), no. 3, 315–339. MR 1137197, DOI 10.1007/BF01385726
Roland W. Freund and Noël M. Nachtigal, An implementation of the QMR method based on coupled two-term recurrences, SIAM J. Sci. Comput. 15 (1994), no. 2, 313–337. Iterative methods in numerical linear algebra (Copper Mountain Resort, CO, 1992). MR 1261456, DOI 10.1137/0915022
G. H. Golub and R. Underwood, The block Lanczos method for computing eigenvalues, Mathematical software, III (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1977) Publ. Math. Res. Center, No. 39, Academic Press, New York, 1977, pp. 361–377. MR 0474742
William B. Gragg, Matrix interpretations and applications of the continued fraction algorithm, Rocky Mountain J. Math. 4 (1974), 213–225. MR 341830, DOI 10.1216/RMJ-1974-4-2-213
William B. Gragg and Anders Lindquist, On the partial realization problem, Linear Algebra Appl. 50 (1983), 277–319. MR 699565, DOI 10.1016/0024-3795(83)90059-9
Martin H. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms. I, SIAM J. Matrix Anal. Appl. 13 (1992), no. 2, 594–639. MR 1152770, DOI 10.1137/0613037
H. M. Kim and R. R. Craig, Jr., Structural dynamics analysis using an unsymmetric block Lanczos algorithm, Internat. J. Numer. Methods Engrg. 26 (1988), 2305–2318.
—, Computational enhancement of an unsymmetric block Lanczos algorithm, Internat. J. Numer. Methods Engrg. 30 (1990), 1083–1089.
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Manish Malhotra, Roland W. Freund, and Peter M. Pinsky, Iterative solution of multiple radiation and scattering problems in structural acoustics using a block quasi-minimal residual algorithm, Comput. Methods Appl. Mech. Engrg. 146 (1997), no. 1-2, 173–196. MR 1453438, DOI 10.1016/S0045-7825(96)01227-3
A. A. Nikishin and A. Yu. Yeremin, Variable block CG algorithms for solving large sparse symmetric positive definite linear systems on parallel computers. I. General iterative scheme, SIAM J. Matrix Anal. Appl. 16 (1995), no. 4, 1135–1153. MR 1351461, DOI 10.1137/S0895479893247679
Dianne P. O’Leary, The block conjugate gradient algorithm and related methods, Linear Algebra Appl. 29 (1980), 293–322. MR 562766, DOI 10.1016/0024-3795(80)90247-5
Beresford N. Parlett, Reduction to tridiagonal form and minimal realizations, SIAM J. Matrix Anal. Appl. 13 (1992), no. 2, 567–593. MR 1152769, DOI 10.1137/0613036
Beresford N. Parlett, Derek R. Taylor, and Zhishun A. Liu, A look-ahead Lánczos algorithm for unsymmetric matrices, Math. Comp. 44 (1985), no. 169, 105–124. MR 771034, DOI 10.1090/S0025-5718-1985-0771034-2
J. Rissanen, Algorithms for triangular decomposition of block Hankel and Toeplitz matrices with application to factoring positive matrix polynomials, Math. Comp. 27 (1973), 147–154. MR 329235, DOI 10.1090/S0025-5718-1973-0329235-5
Axel Ruhe, Implementation aspects of band Lanczos algorithms for computation of eigenvalues of large sparse symmetric matrices, Math. Comp. 33 (1979), no. 146, 680–687. MR 521282, DOI 10.1090/S0025-5718-1979-0521282-9
T.-J. Su, A decentralized linear quadratic control design method for flexible structures, Ph.D. Thesis, Department of Aerospace and Engineering Mechanics, The University of Texas at Austin, Austin, TX, 1989.
T.-J. Su and R. R. Craig, Jr., Model reduction and control of flexible structures using Krylov vectors, J. Guidance Control Dynamics 14 (1991), 260–267.
D. R. Taylor, Analysis of the look ahead Lanczos algorithm, Ph.D. Thesis, Department of Mathematics, University of California, Berkeley, CA, 1982.
R. Underwood, An iterative block Lanczos method for the solution of large sparse symmetric eigenproblems, Ph.D. Thesis, Computer Science Department, Stanford University, Stanford, CA, 1975.
A. J. J. van der Weiden and O. H. Bosgra, The determination of structural properties of a linear multivariable system by operations of system similarity. II. Nonproper systems in generalized state-space, Internat. J. Control 32 (1980), no. 3, 489–537. MR 587183, DOI 10.1080/00207178008922870
Guo Liang Xu and Adhemar Bultheel, Matrix Padé approximation: definitions and properties, Linear Algebra Appl. 137/138 (1990), 67–136. MR 1067675, DOI 10.1016/0024-3795(90)90127-X