M. R. Abdel-Aziz.
Safeguarded use of the implicit restarted Lanczos technique for
solving non-linear structural eigensystems.
Internat. J. Numer. Methods Engrg., 37:3117-3133, 1994.
A. Abramow and M. Neuhaus.
Bemerkungen über Eigenwertprobleme von Matrizen höherer
Ordnung.
In Les mathématiques de l'ingénieur, pages 176-179. Mém.
Publ. Soc. Sci. Arts Lett. Hainaut, Vol. hors Série, Maison Léon Losseau,
Mons, France, 1958.
P. R. Amestoy and I. S. Duff.
Memory management issues in sparse multifrontal methods on
multiprocessors.
Internat. J. Supercomputer Appl., 7:64-82, 1993.
P. R. Amestoy, I. S. Duff, J.-Y. L'Excellent, and J. Koster.
A fully asynchronous multifrontal solver using distributed dynamic
scheduling.
Technical Report RAL-TR-1999-059, Rutherford Appleton Laboratory,
Oxfordshire, UK, 1999.
Software available at
http://www.pallas.de/parasol.
G. S. Ammar, L. Reichel, and D. C. Sorensen.
Algorithm 730: An implementation of a divide and conquer method for
the unitary eigenproblem.
ACM Trans. Math. Software, 20:161-170, 1994.
E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra,
J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen.
LAPACK Users' Guide.
SIAM, Philadelphia, Third edition, 1999.
I. Andersson.
Experiments with the conjugate gradient algorithm for the
determination of eigenvalues of symmetric matrices.
Technical Report UMINF-4.71, University of Umeå, Sweden, 1971.
P. Arbenz and G. H. Golub.
On the spectral decomposition of Hermitian matrices modified by low
rank perturbations with applications.
SIAM J. Matrix Anal. Appl., 9:40-58, 1988.
T. Arias, A. Edelman, and S. Smith.
Curvature in conjugate gradient eigenvalue computation with
applications.
In J. G. Lewis, editor, Proceedings of the 1994 SIAM Applied
Linear Algebra Conference, pages 233-238. SIAM, Philadelphia, 1994.
M. Arioli, I. S. Duff, and D. Ruiz.
Stopping criteria for iterative solvers.
Report RAL-91-057, Central Computing Center, Rutherford Appleton
Laboratory, Oxfordshire, UK, 1992.
C. Ashcraft and R. Grimes.
SPOOLES: An object-oriented sparse matrix library.
In Proceedings of the Ninth SIAM Conference on Parallel
Processing. SIAM, Philadelphia, 1999.
Software available at
http://www.netlib.org/linalg/spooles.
J. Baglama, D. Calvetti, and L. Reichel.
Iterative methods for the computation of a few eigenvalues of a large
symmetric matrix.
BIT, 36(3):400-421, 1996.
J. Baglama, D. Calvetti, L. Reichel, and A. Ruttan.
Computation of a few close eigenvalues of a large matrix with
application to liquid crystal modeling.
J. Comput. Phys., 146:203-226, 1998.
Z. Bai.
The CSD, GSVD, their applications and computations.
Preprint Series 958, Institute for Mathematics and Its Applications,
University of Minnesota, Minneapolis, April 1992.
Available at http://www.cs.ucdavis.edu/\simbai.
Z. Bai.
A spectral transformation block Lanczos algorithm for solving
sparse non-Hermitian eigenproblems.
In J. G. Lewis, editor, Proceedings of the Fifth SIAM Conference
on Applied Linear Algebra, pages 307-311. SIAM, Philadelphia, 1994.
Z. Bai, D. Day, J. Demmel, and J. Dongarra.
A test matrix collection for non-Hermitian eigenvalue problems.
Technical Report CS-97-355, University of Tennessee, Knoxville, 1997.
LAPACK Working Note #123, Software and test data available at
http://math.nist.gov/MatrixMarket/.
Z. Bai, D. Day, and Q. Ye.
ABLE: An adaptive block lanczos method for non-hermitian eigenvalue
problems.
SIAM J. Matrix Anal. Appl., 20:1060-1082, 1999.
Z. Bai and J. Demmel.
Design of a parallel nonsymmetric eigenroutine toolbox, Part I.
In R. F. Sincovec et al., editors, Proceedings of the Sixth
SIAM Conference on Parallel Processing for Scientific Computing. SIAM,
Philadelphia, 1993.
Long version available as Computer Science Report CSD-92-718,
University of California, Berkeley, 1992.
Z. Bai, J. Demmel, and M. Gu.
An inverse free parallel spectral divide and conquer algorithm for
nonsymmetric eigenproblems.
Numer. Math., 76:279-308, 1997.
Z. Bai, P. Feldmann, and R. W. Freund.
How to make theoretically passive reduced-order models passive in
practice.
In Proceedings of the IEEE 1998 Custom Integrated Circuits
Conference, pages 207-210. IEEE Press, Piscataway, NJ, 1998.
Z. Bai and R. W. Freund.
A band symmetric Lanczos process based on coupled recurrences with
applications.
Technical Report Numerical Analysis Manuscript, Bell Laboratories,
Murray Hill, NJ, USA, 1998.
Z. Bai and G. Golub.
Some unusual matrix eigenvalue problems.
In J. Palma, J. Dongarra, and V. Hernandez, editors, Proceedings
of VECPAR'98 - Third International Conference for Vector and Parallel
Processing, Lecture Notes in Computer Science. Vol. 1573, pages
4-19. Springer-Verlag, New York, 1999.
Z. Bai and G. W. Stewart.
Algorithm 776. SRRIT -- A FORTRAN subroutine to calculate the
dominant invariant subspaces of a nonsymmetric matrix.
ACM Trans. Math. Software, 23:494-513, 1998.
S. Balay, W. Gropp, L. C. McInnes, and B. Smith.
PETSc 2.0 Users Manual.
Technical Report ANL-95/11 - Revision 2.0.28, Argonne National
Laboratory, Argonne, IL, 2000.
Software available at
http://www.mcs.anl.gov/petsc.
R. Barrett, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout,
R. Pozo, C. Romine, and H. van der Vorst.
Templates for the Solution of Linear Systems: Building Blocks
for Iterative Methods.
SIAM, Philadelphia, 1994.
P. Benner and H. Faßbender.
The symplectic eigenvalue problem, the butterfly form, the SR
algorithm, and the Lanczos method.
Linear Algebra Appl., 275/276:19-47, 1998.
P. Benner, V. Mehrmann, and H. Xu.
A new method for computing the stable invariant subspace of a real
hamiltonian matrix.
J. Comput. Appl. Math., 86:17-43, 1997.
P. Benner, V. Mehrmann, and H. Xu.
A numerical stable, structure preserving method for computing the
eigenvalues of real Hamiltonian or symplectic pencils.
Numer. Math., 78:329-358, 1998.
P. Benner, V. Mehrmann, and H. Xu.
A note on the numerical solution of complex Hamiltonian and
skew-Hamiltonian eigenvalue problem.
Electron. Trans. Numer. Anal., 8:115-126, 1999.
L. Bergamaschi, G. Gambolati, and G. Pini.
Asymptotic convergence of conjugate gradient methods for the partial
symmetric eigenproblem.
Numer. Linear Algebra Appl., 4(2):69-84, 1997.
L. S. Blackford, J. Choi, A. Cleary, E. D'Azevedo, J. Demmel, I. Dhillon,
J. Dongarra, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. Whaley.
ScaLAPACK Users' Guide.
SIAM, Philadelphia, 1997.
A. Bojanczyk and P. Van Dooren.
On propagating orthogonal transformations in a product of 2 \times 2 triangular matrices.
In Numerical Linear Algebra. de Gruyter, Berlin, 1993.
F. Bourquin.
Component mode synthesis and eigenvalues of second order operators:
discretization and algorithm.
RAIRO Modél. Math. Anal. Numér., 26(3):385-423, 1992.
F. Bourquin.
A domain decomposition method for the eigenvalue problem in elastic
multistructures.
In Asymptotic Methods for Elastic Structures (Lisbon, 1993),
pages 15-29. de Gruyter, Berlin, 1995.
F. Bourquin and P. G. Ciarlet.
Modelling and justification of eigenvalue problems for junctions
between elastic structures.
J. Funct. Anal., 87(2):392-427, 1989.
J. H. Bramble, J. E. Pasciak, and A. V. Knyazev.
A subspace preconditioning algorithm for eigenvector/eigenvalue
computation.
Adv. Comput. Math., 6(2):159-189, 1996.
A. Bunse-Gerstner, R. Byers, V. Mehrmann, and N. K. Nichols.
Numerical computation of an analytic singular value decomposition of
a matrix valued function.
Numer. Math., 60:1-39, 1991.
R. Byers, C. He, and Mehrmann.
The matrix sign function method and the computation of invariant
subspaces.
SIAM J. Matrix Anal. Appl., 18:615-632, 1997.
Z. Q. Cai, J. Mandel, and S. McCormick.
Multigrid methods for nearly singular linear equations and eigenvalue
problems.
SIAM J. Numer. Anal., 34:178-200, 1997.
C. Carey, G. H. Golub, and K. H. Law.
A Lanczos-based method for structural dynamics re-analysis
problems.
Manuscript na-93-03, Computer Science Department, Stanford
University, Stanford, CA, 1993.
J. Carrier, L. Greengard, and V. Rokhlin.
A fast adaptive multipole algorithm for particle simulations.
SIAM J. Sci. Statist. Comput., 9:669-686, 1988.
T. F. Chan, E. Gallopoulos, V. Simoncini, T. Szeto, and C. H. Tong.
A quasi-minimal residual variant of the Bi-CGSTAB algorithm for
nonsymmetric systems.
SIAM J. Sci. Comput., 15:338-347, 1994.
C. R. Crawford.
Algorithm 646 PDFIND: A routine to find a positive definite linear
combination of two real symmetric matrices.
ACM Trans. Math. Software, 12:278-282, 1986.
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.
In Proceedings of the 1994 IEEE Conference on Decision and
Control, pages 505-509. IEEE Press, Piscataway, NJ, 1974.
J. K. Cullum and R. A. Willoughby.
Computing eigenvalues of very large symmetric matrices--an
implementation of a Lanczos algorithm with no reorthogonalization.
J. Comput. Phys., 44:329-358, 1981.
J. K. Cullum and R. A. Willoughby.
A practical procedure for computing eigenvalues of large sparse
nonsymmetric matrices.
In J. K. Cullum and R. A. Willoughby, editors, Large Scale
Eigenvalue Problems, pages 193-240. Elsevier Science Publishers,
1986.
J. K. Cullum and R. A. Willoughby.
A QL procedure for computing the eigenvalues of complex symmetric
tridiagonal matrices.
SIAM J. Matrix Anal. Appl., 17:83-109, 1996.
J. W. Daniel, W. B. Gragg, L. Kaufman, and G. W. Stewart.
Reorthogonalization and stable algorithms for updating the
Gram-Schmidt QR factorization.
Math. Comp., 30:772-795, 1976.
D. F. Davidenko.
The method of variation of parameters as applied to the computation
of eigenvalues and eigenvectors of matrices.
Soviet Math. Dokl., 1:364-367, 1960.
E. R. Davidson.
The iterative calculation of a few of the lowest eigenvalues and
corresponding eigenvectors of large real symmetric matrices.
J. Comput. Phys., 17:87-94, 1975.
E. R. Davidson.
Matrix eigenvector methods.
In G. H. F. Direcksen and S. Wilson, editors, Methods in
Computational Molecular Physics, pages 95-113. Reidel, Boston, 1983.
T. A. Davis and I. S. Duff.
A combined unifrontal/multifrontal method for unsymmetric sparse
matrices.
Technical Report TR-95-020, Computer and Information Sciences
Department, University of Florida, Gainesville, 1995.
Software available at http://www.netlib.org/linalg/umfpack2.2.tgz.
J. de Leeuw and W. Heiser.
Theory of multidimensional scaling.
In P. R. Krishnaiah and L. N. Kanal, editors, Handbook of
Statistics, Vol. 2, pages 285-316. North-Holland, Amsterdam, 1982.
G. De Samblanx.
Filtering and restarting projection methods for eigenvalue
problems.
PhD Thesis, Katholieke Universiteit Leuven, Department of
Computer Science, 3001 Heverlee, Belgium, 1998.
E.. De Sturler.
A parallel restructed version of GMRES(m).
Technical Report Tech. Report Preprint 91-085, Delft University of
Technology, Delft, The Netherlands, 1992.
E. De Sturler and H. A. van der Vorst.
Communication cost reduction for krylov methods on parallel
computers.
In W. Gentzsch and U. Harms, editors, High Performance and
Networking Tools, Vol. 2, Lecture Notes in Computer Science. Vol. 797, pages
190-195. Springer-Verlag, Berlin, 1994.
J. Demmel and A. Edelman.
The dimension of matrices (matrix pencils) with given Jordan
(Kronecker) canonical forms.
Linear Algebra Appl., 230:61-87, 1995.
J. Demmel and W. Gragg.
On computing accurate singular values and eigenvalues of matrices
with acyclic graphs.
Linear Algebra Appl., 185:203-217, 1993.
J. Demmel, M. Gu, S. Eisenstat, I. Slapnicar, K. Veselic, and
Z. Drmac.
Computing the singular value decomposition with high relative
accuracy.
Linear Algebra Appl., 299:21-80, 1999.
J. Demmel and B. Kågström.
The generalized Schur decomposition of an arbitrary pencil
{A} -\lambda {B}: Robust software with error bounds and applications. Part I:
Theory and algorithms.
ACM Trans. Math. Software, 19(2):160-174, 1993.
J. Demmel and B. Kågström.
The generalized Schur decomposition of an arbitrary pencil
{A} -\lambda {B}: Robust software with error bounds and applications. Part II:
Software and applications.
ACM Trans. Math. Software, 19(2):175-201, 1993.
J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, and J. W. H. Liu.
A supernodal approach to sparse partial pivoting.
SIAM J. Matrix Anal. Appl., 20(3):720-755, 1999.
Software available at http://www.nersc.gov/\simxiaoye/SuperLU.
J. W. Demmel, J. R. Gilbert, and X. S. Li.
An asynchronous parallel supernodal algorithm for sparse Gaussian
elimination.
SIAM J. Matrix Anal. Appl., 20(4):915-952, 1999.
Software available at http://www.nersc.gov/\simxiaoye/SuperLU.
I. Dhillon.
A New O(n^2) Algorithm for the Symmetric Tridiagonal
Eigenvalue/Eigenvector Problem.
Ph.D. thesis, University of California, Berkeley, 1997.
J. Dongarra, J. Gabriel, D. Kolling, and J. Wilkinson.
The eigenvalue problem for Hermitian matrices with time reversal
symmetry.
Linear Algebra Appl., 60:27-42, 1984.
J. J. Dongarra, J. Du Croz, S. Hammarling, and R. J. Hanson.
An extended set of FORTRAN basic linear algebra subprograms.
ACM Trans. Math. Software, 14:1-32, 1988.
J. J. Dongarra, I. S. Duff, D. C. Sorensen, and H. A. van der Vorst.
Numerical Linear Algebra for High-Performance Computers.
SIAM, Philadelphia, PA, 1998.
I. S. Duff and J. K. Reid.
The multifrontal solution of indefinite sparse symmetric linear
equations.
ACM Trans. Math. Software, 9(3):302-325, September 1983.
I. S. Duff and J. K. Reid.
MA47, a Fortran code for direct solution of indefinite sparse
symmetric linear systems.
Technical Report RAL-95-001, DRAL, Chilton Didcot, UK, 1995.
I. S. Duff and J. K. Reid.
The design of MA48, a code for the direct solution of sparse
unsymmetric linear systems of equations.
ACM Trans. Math. Software, 22:187-226, 1996.
I. S. Duff and J. A. Scott.
Computing selected eigenvalues of large sparse unsymmetric matrices
using subspace iteration.
ACM Trans. Math. Software, 19:137-159, 1993.
E. G. D'yakonov.
Optimization in solving elliptic problems.
CRC Press, Boca Raton, FL, 1996.
Translated from the 1989 Russian original; translated, edited, and
with a preface by Steve McCormick.
E. G. D'yakonov and A. V. Knyazev.
Group iterative method for finding lower-order eigenvalues.
Moscow University, Ser. 15, Computational Math. and
Cybernetics, 2:32-40, 1982.
E. G. D'yakonov and M. Yu. Orekhov.
Minimization of the computational labor in determining the first
eigenvalues of differential operators.
Math. Notes, 27(5-6):382-391, 1980.
A. Edelman, E. Elmroth, and B. Kågström.
A geometric approach to perturbation theory of matrices and matrix
pencils. Part I: Versal deformations.
SIAM J. Matrix Anal. Appl., 18(3):653-692, 1997.
A. Edelman, E. Elmroth, and B. Kågström.
A geometric approach to perturbation theory of matrices and matrix
pencils. Part II: A stratification-enhanced staircase algorithm.
SIAM J. Matrix Anal. Appl., 20(3):667-699, 1999.
A. Edelman and S. T. Smith.
On conjugate gradient-like methods for eigen-like problems.
BIT, 36:494-508, 1996.
See also Loyce Adams and J. L. Nazareth, editors,
Proc. Linear and Nonlinear Conjugate Gradient-Related Methods, SIAM, Philadelphia, 1996.
V. Eijkhout.
Distributed sparse data structures for linear algebra operations.
Technical Report CS 92-169, Computer Science Department, University
of Tennessee, Knoxville, TN, 1992.
LAPACK Working Note #50,
http://www.netlib.org/lapack/lawns/lawn50.ps.
E. Elmroth, P. Johansson, and B. Kågström.
Computation and presentation of graphs displaying closure hierarchies
of Jordan and Kronecker structures.
Technical Report UMINF-99.12, Department of Computing
Science, Umeå University, Umeå, Sweden, 1999.
E. Elmroth and B. Kågström.
The set of 2-by-3 matrix pencils--Kronecker structures and their
transitions under perturbations.
SIAM J. Matrix Anal. Appl., 17(1):1-34, 1996.
T. Ericsson.
A generalised eigenvalue problem and the Lanczos algorithm.
In J. K. Cullum and R. A. Willoughby, editors, Large Scale
Eigenvalue Problems, pages 95-119. Elsevier Science Publishers
(North-Holland), Amsterdam, 1986.
T. Ericsson and A. Ruhe.
The spectral transformation Lanczos method for the numerical
solution of large sparse generalized symmetric eigenvalue problems.
Math. Comp., 35:1251-1268, 1980.
V. Faber and T. A. Manteuffel.
Necessary and sufficient conditions for the existence of a conjugate
gradient method.
SIAM J. Numer. Anal., 21(2):352-362, 1984.
C. Farhat and M. Geradin.
On a component mode synthesis method and its application to
incompatible substructures.
Comput. & Structures, 51(5):459-473, 1994.
H. Faßbender, D. S. Mackey, and N. Mackey.
Hamilton and Jacobi come full circle: Jacobi algorithms for
structured Hamiltonian eigenproblems.
To appear in Linear Algebra Appl., 2000.
P. Feldmann and R. W. Freund.
Reduced-order modeling of large linear subcircuits via a block
Lanczos algorithm.
In Proceedings of the 32nd Design Automation Conference, pages
474-479. ACM, New York, 1995.
Y. T. Feng and D. R. J. Owen.
Conjugate gradient methods for solving the smallest eigenpair of
large symmetric eigenvalue problems.
Internat. J. Numer. Methods Engrg., 39(13):2209-2229, 1996.
D. R. Fokkema, G. L. G. Sleijpen, and H. A. van der Vorst.
Jacobi-Davidson style QR and QZ algorithms for the partial
reduction of matrix pencils.
SIAM J. Sci. Comput., 20:94-125, 1998.
R. W. Freund.
Conjugate gradient-type methods for linear systems with complex
symmetric coefficient matrices.
SIAM J. Sci. Statist. Comput., 13:425-448, 1992.
R. W. Freund.
Computing minimal partial realizations via a Lanczos-type algorithm
for multiple starting vectors.
In Proceedings of the 36th IEEE Conference on Decision and
Control, pages 4394-4399. IEEE Press, Piscataway, NJ, 1997.
R. W. Freund.
Reduced-order modeling techniques based on Krylov subspaces and
their use in circuit simulation.
In Applied and Computational Control, Signals, and Circuits,
Vol. 1, pages 435-498. Birkhäuser, Boston, 1999.
R. W. Freund and P. Feldmann.
Reduced-order modeling of large linear passive multi-terminal
circuits using matrix-Padé approximation.
In Proceedings of the Design, Automation and Test in Europe
Conference 1998, pages 530-537. IEEE Computer Society Press, Los Alamitos, CA, 1998.
R. W. Freund, M. H. Gutknecht, and N. M. Nachtigal.
An implementation of the look-ahead Lanczos algorithm for
non-Hermitian matrices.
SIAM J. Sci. Comput., 14:137-158, 1993.
S. Friedland, J. Nocedal, and M. L. Overton.
The formulation and analysis of numerical methods for inverse
eigenvalue problems.
SIAM J. Numer. Anal., 24:634-667, 1987.
C. Fu, X. Jiao, and T. Yang.
Efficient sparse LU factorization with partial pivoting on
distributed memory architectures.
IEEE Trans. Parallel and Distributed Systems, 9(2):109-125,
1998.
Software available at http://www.cs.ucsb.edu/research/S+.
Z. Fu and E. M. Dowling.
Conjugate gradient eigenstructure tracking for adaptive spectral
estimation.
IEEE Trans. Signal Processing, 43(5):1151-1160, 1995.
G. Gambolati, F. Sartoretto, and P. Florian.
An orthogonal accelerated deflation technique for large symmetric
eigenproblems.
Comput. Methods Appl. Mech. Engrg., 94(1):13-23, 1992.
M. Genseberger and G. L. G. Sleijpen.
Alternative correction equations in the Jacobi-Davidson method.
Preprint 1073, Department of Mathematics, Utrecht University,
Utrecht, the Netherlands, 1998.
I. Gohberg, T. Kailath, and V. Olshevsky.
Fast gaussian elimination with partial pivoting for matrices with
displacement structure.
Math. Comp., 64(212):1557-1576, 1995.
G. Golub and R. Underwood.
The block Lanczos method for computing eigenvalues.
In J. Rice, editor, Mathematical Software III, pages
364-377. Academic Press, New York, 1977.
G. H. Golub, Z. Zhang, and H. Zha.
Large sparse symmetric eigenvalue problems with homogeneous linear
constraints: The Lanczos process with inner-outer iterations.
Linear Algebra Appl., 309:289-306, 2000.
W. B. Gragg and T.-L. Wang.
Convergence of the shifted QR algorithm for unitary Hessenberg
matrices.
Technical Report NPS-53-90-007, Naval Postgraduate School, Monterey,
CA, 1990.
J. F. Grcar.
Analyses of the Lanczos Algorithm and the Approximation
Problem in Richardson's Method.
Ph.D. thesis, University of Illinois at Urbana-Champaign, 1981.
R. G. Grimes, J. G. Lewis, and H. D. Simon.
A shifted block Lanczos algorithm for solving sparse symmetric
generalized eigenproblems.
SIAM J. Matrix Anal. Appl., 15:228-272, 1994.
E. Grimme, D. Sorensen, and P. Van Dooren.
Model reduction of state space systems via an implicitly restarted
Lanczos method.
Numer. Algorithms, 12:1-32, 1996.
M. Gu, J. Demmel, and I. Dhillon.
Efficient computation of the singular value decomposition with
applications to least squares problems.
Computer Science Dept. Technical Report CS-94-257,
University of Tennessee, Knoxville, 1994.
LAPACK Working Note #88,
http://www.netlib.org/lapack/lawns/lawn88.ps.
A. Gupta, G. Karypis, and V. Kumar.
Highly scalable parallel algorithms for sparse matrix factorization.
IEEE Trans. Parallel and Distributed Systems, 8:502-520, 1997.
Software available at
http://www.cs.umn.edu/\simmjoshi/pspases.
A. Gupta, E. Rothberg, E. Ng, and B. W. Peyton.
Parallel sparse Cholesky factorization algorithms for shared-memory
multiprocessor systems.
In R. Vichnevetsky, D. Knight, and G. Richter, editors, Advances
in Computer Methods for Partial Differential Equations-VII, pages 622-628.
IMACS, New Brunswick, NJ, 1992.
W. Hackbusch.
On the computation of approximate eigenvalues and eigenfunctions of
elliptic operators by means of a multi-grid method.
SIAM J. Numer. Anal., 16(2):201-215, 1979.
W. Hackbusch.
Multigrid solutions to linear and nonlinear eigenvalue problems for
integral and differential equations.
Rostock. Math. Kolloq., (25):79-98, 1984.
M. T. Heath and P. Raghavan.
Performance of a fully parallel sparse solver.
Internat. J. Supercomputer Appl., 11(1):49-64, 1997.
Software available at
http://www.netlib.org/scalapack.
P. Henon, P. Ramet, and J. Roman.
A mapping and scheduling algorithm for parallel sparse fan-in
numerical factorization.
In P. Amestoy, P. Berger, M. Daydé, I. Duff, V. Frayssé,
L. Giraud, and D. Ruiz, editors, EuroPar'99 Parallel Processing,
Lecture Notes in Computer Science, Vol. 1685, pages 1059-1067.
Springer-Verlag, New York, 1999.
S. A. Hutchinson, L. V. Prevost, J. N. Shadid, and R. S. Tuminaro.
Aztec user's guide, version 2.0 Beta.
Technical Report SAND95-1559, Sandia National Laboratories, Albuquerque, NM, 1998.
T. Hwang and I. D. Parsons.
A multigrid method for the generalized symmetric eigenvalue problem.
I. Algorithm and implementation.
Internat. J. Numer. Methods Engrg., 35(8):1663-1676, 1992.
T. Hwang and I. D. Parsons.
A multigrid method for the generalized symmetric eigenvalue problem.
II. Performance evaluation.
Internat. J. Numer. Methods Engrg., 35(8):1677-1696, 1992.
E.-J. Im and K. A. Yelick.
Optimizing sparse matrix vector multiplication on SMPs.
In Proceedings of the Ninth SIAM Conference on Parallel
Processing for Scientific Computing, SIAM, Philadelphia, 1999.
C. G. J. Jacobi.
Ueber ein leichtes Verfahren, die in der Theorie der
Säcularstörungen vorkommenden Gleichungen numerisch
aufzulösen.
J. Reine Angew. Math., 30:51-94,
1846.
Z. Jia.
Polynomial characterizations of the approximate eigenvectors by the
refined Arnoldi method and an implicitly restarted refined Arnoldi
algorithm.
Linear Algebra Appl., 287:191-214, 1998.
Z. Jia and G. W. Stewart.
An analysis of the Rayleigh-Ritz method for approximating
eigenspaces.
Technical Report TR-4015, Department of Computer Science, University
of Maryland, College Park, 1999.
Z. Jia and G. W. Stewart.
On the convergence of Ritz values, Ritz vectors and refined
Ritz vectors.
Technical Report TR-3986, Department of Computer Science, University
of Maryland, College Park, 1999.
P. Johansson.
Stratigraph users' guide. version 1.1.
Technical Report UMINF-99.11, Department of Computing
Science, Umeå University, Umeå, Sweden, 1999.
M. T. Jones and M. L. Patrick.
The Lanczos algorithm for the generalized symmetric eigenproblem on
shared-memory architectures.
Appl. Numer. Math., 12:377-389, 1993.
B. Kågström.
How to compute the Jordan normal form -- the choice between
similarity transformations and methods using the chain relations.
Technical Report UMINF-91.81, Department of Numerical Analysis,
Institute of Information Processing, University of Umeå, Umeå, Sweden, 1981.
B. Kågström.
RGSVD--an algorithm for computing the Kronecker canonical form
and reducing subspaces of singular A - \lambda B pencils.
SIAM J. Sci. Statist. Comput., 7(1):185-211, 1986.
B. Kågström and A. Ruhe.
ALGORITHM 560: An algorithm for the numerical computation of the
Jordan normal form of a complex matrix [F2].
ACM Trans. Math. Software, 6(3):437-443, 1980.
B. Kågström and A. Ruhe.
An algorithm for the numerical computation of the Jordan normal
form of a complex matrix.
ACM Trans. Math. Software, 6(3):389-419, 1980.
B. Kågström and P. Wiberg.
Extracting partial canonical structure for large scale eigenvalue
problems.
Technical Report UMINF-98.13, Department of Computing
Science, Umeå University, Umeå, Sweden, 1998.
Submitted to Numerical Algorithms.
W. Kahan.
Accurate eigenvalues of a symmetric tridiagonal matrix.
Technical Report CS41, Computer Science Department, Stanford
University, Stanford, CA, 1966
(revised June 1968).
H. M. Kim and R. R. Craig, Jr.
Structural dynamics analysis using an unsymmetric block Lanczos
algorithm.
Internat. J. Numer. Methods Engrg., 26:2305-2318, 1988.
H. M. Kim and R. R. Craig, Jr.
Computational enhancement of an unsymmetric block Lanczos
algorithm.
Internat. J. Numer. Methods Engrg., 30:1083-1089, 1990.
D. R. Kincaid, J. R. Respess, D. M. Young, and R. G. Grimes.
Algorithm 586 - ITPACK 2C: A Fortran package for solving
large sparse linear systems by adaptive accelerated iterative methods.
ACM Trans. Math. Software, 8(3):302-322, 1982.
A. V. Knyazev.
Computation of eigenvalues and eigenvectors for mesh problems:
Algorithms and error estimates.
Dept. of Numerical Math., USSR Academy of Sciences, Moscow, 1986.
(In Russian.)
A. V. Knyazev.
Convergence rate estimates for iterative methods for mesh symmetric
eigenvalue problem.
Soviet J. Numer. Anal. Math. Modelling,
2(5):371-396, 1987.
A. V. Knyazev.
A preconditioned conjugate gradient method for eigenvalue problems
and its implementation in a subspace.
In International Ser. Numerical Mathematics, v. 96,
Eigenwertaufgaben in Natur- und Ingenieurwissenschaften und ihre numerische
Behandlung, Oberwolfach, 1990, pages 143-154, Birkhauser, Basel, 1991.
A. V. Knyazev.
Toward the optimal preconditioned eigensolver: Locally optimal block
preconditioned conjugate gradient method.
Technical Report UCD-CCM 149, Center for Computational Mathematics,
University of Colorado, Denver, 2000.
Available at
http://www-math.cudenver.edu/ccmreports/rep149.ps.gz.
A. V. Knyazev and A. L. Skorokhodov.
Preconditioned iterative methods in subspace for solving linear
systems with indefinite coefficient matrices and eigenvalue problems.
Soviet J. Numer. Anal. Math. Modelling,
4(4):283-310, 1989.
A. V. Knyazev and A. L. Skorokhodov.
The preconditioned gradient-type iterative methods in a subspace for
partial generalized symmetric eigenvalue problem.
Soviet Math. Dokl., 45(2):474-478, 1993.
A. V. Knyazev and A. L. Skorokhodov.
The preconditioned gradient-type iterative methods in a subspace for
partial generalized symmetric eigenvalue problem.
SIAM J. Numer. Anal., 31(4):1226-1239, 1994.
N. Kosugi.
Modification of the Liu-Davidson method for obtaning one or
simultaneously several eigensolutions of a large real symmetric matrix.
J. Comput. Phys., 55(3):426-436, 1984.
V. N. Kublanovskaja.
On an application to the solution of the generalized latent value
problem for \lambda-matrices.
SIAM J. Numer. Anal., 7:532-537, 1970.
V. N. Kublanovskaya.
On a method of solving the complete eigenvalue problem for a
degenerate matrix (in Russian).
Zh. Vychisl. Mat. Mat. Fiz., 6:611-620, 1966.
USSR Comput. Math. Phys., 6(4):1-16, 1968.
V. N. Kublanovskaya.
An approach to solving the spectral problem of A - \lambda B.
In B. Kågström and A. Ruhe, editors, Matrix Pencils,
Lecture Notes in Mathematics, Vol. 973, pages 17-29.
Springer-Verlag, Berlin, 1983.
K. Kundert.
Sparse matrix techniques.
In Albert Ruehli, editor, Circuit Analysis, Simulation and
Design. North-Holland, Amsterdam, 1986.
Software available at
http://www.netlib.org/sparse.
Yu. A. Kuznetsov.
Iterative methods in subspaces for eigenvalue problems.
In A. V. Balakrishinan, A. A. Dorodnitsyn, and J. L. Lions, editors,
Vistas in Applied Math., Numerical Analysis, Atmospheric Sciences,
Immunology, pages 96-113. Optimization Software, New York, 1986.
C. Lanczos.
An iteration method for the solution of the eigenvalue problem of
linear differential and integral operators.
J. Res. Nat. Bur. Standards, 45:255-282, 1950.
A. Laub.
Invariant subspace methods for the numerical solution of Riccati
equations.
In S. Bittanti A. Laub, and J. C. Willems, editors, Riccati
Equations. Springer-Verlag, New York, 1990.
R. B. Lehoucq and K. J. Maschhoff.
Implementation of an implicitly restarted block Arnoldi method.
Preprint MCS-P649-0297, Argonne National Laboratory, Argonne, IL,
1997.
R. B. Lehoucq and K. Meerbergen.
Using generalized Cayley transformations within an inexact rational
Krylov sequence method.
SIAM J. Matrix Anal. Appl., 20(1):131-148, 1998.
R. B. Lehoucq and J. A. Scott.
An evaluation of software for computing eigenvalues of sparse
nonsymmetric matrices.
Technical Report MCS-P547-1195, Argonne National Laboratory, Argonne,
IL, 1995.
R. B. Lehoucq and J. A. Scott.
Implicitly restarted Arnoldi methods and eigenvalues of the
discretized Navier-Stokes equations.
Technical Report SAND97-2712J, Sandia National Laboratory,
Albuquerque, NM, 1997.
R. B. Lehoucq, D. C. Sorensen, and C. Yang.
ARPACK Users' Guide: Solution of Large-Scale Eigenvalue
Problems with Implicitly Restarted Arnoldi Methods.
SIAM, Phildelphia, 1998.
A. S. Lewis and M. L. Overton.
Eigenvalue optimization.
In A. Iserles, editor, Acta Numerica, Volume 5, pages 149-190.
Cambridge University Press, Cambridge, UK, 1996.
H. Li, P. Aitchison, and A. Woodbury.
Methods for overcoming breakdown problems in the unsymmetric
Lanczos reduction method.
Internat. J. Numer. Methods Engrg., 42:389-408, 1998.
T.-Y. Li and Z. Zeng.
Homotopy continuation algorithm for the real nonsymmetric
eigenproblem: Further development and implementation.
SIAM J. Sci. Comput., 20:1627-1651, 1999.
X. S. Li and J. W. Demmel.
A scalable sparse direct solver using static pivoting.
In Proceedings of the Ninth SIAM Conference on Parallel
Processing for Scientific Computing, SIAM, Philadelphia, 1999.
Software available at
http://www.nersc.gov/\simxiaoye/SuperLU.
S. H. Lui and G. H. Golub.
Homotopy method for the numerical solution of the eigenvalue problem
of self-adjoint partial differential operators.
Numer. Algorithms, 10(3-4):363-378, 1995.
S. H. Lui, H. B. Keller, and T. W. C. Kwok.
Homotopy method for the large, sparse, real nonsymmetric eigenvalue
problem.
SIAM J. Matrix Anal. Appl., 18(2):312-333, 1997.
J.-C. Luo.
A domain decomposition method for eigenvalue problems.
In Fifth International Symposium on Domain Decomposition Methods
for Partial Differential Equations (Norfolk, VA, 1991), pages 306-321.
SIAM, Philadelphia, 1992.
J. Mandel and S. McCormick.
A multilevel variational method for
{A}{\bf u}=\lambda {B}{\bf u}
on composite grids.
J. Comput. Phys., 80(2):442-452, 1989.
K. Meerbergen.
The rational Lanczos method for Hermitian eigenvalue problems.
Technical Report RAL-TR-1999-025, Rutherford Appleton Laboratory,
Chilton, UK, 1999.
Available at
http://www.numerical.rl.ac.uk/reports/reports.html.
K. Meerbergen and D. Roose.
The restarted Arnoldi method applied to iterative linear system
solvers for the computation of rightmost eigenvalues.
SIAM J. Matrix Anal. Appl., 18:1-20, 1997.
K. Meerbergen, A. Spence, and D. Roose.
Shift-invert and Cayley transforms for detection of rightmost
eigenvalues of nonsymmetric matrices.
BIT, 34:409-423, 1994.
V. Mehrmann and D. Watkins.
Structure-preserving methods for computing eigenpairs of large sparse
skew-Hamiltonian/Hamiltonian pencils.
Technical Report SFB393/00-02, Technische Universitaet Chemnitz,
Germany, 2000.
J. Meijerink and H. A. van der Vorst.
An iterative solution method for linear systems of which the
coefficient matrix is a symmetric M-matrix.
Math. Comp., 31:148-162, 1977.
R. B. Morgan and D. S. Scott.
Generalizations of Davidson's method for computing eigenvalues of
sparse symmetric matrices.
SIAM J. Sci. Statist. Comput., 7:817-825, 1986.
S. Oliveira.
A convergence proof of an iterative subspace method for eigenvalues
problems.
In Foundations of Computational Mathematics (Rio de Janeiro,
1997), pages 316-325. Springer-Verlag, Berlin, 1997.
J. Olsen, P. Jørgensen, and J. Simons.
Passing the one-billion limit in full configuration-interaction
(FCI) calculations.
Chem. Phys. Lett., 169:463-472, 1990.
T.C. Oppe, W. Joubert, and D. Kincaid.
NSPCG's user's guide: A package for solving large linear
systems by various iterative methods.
Technical report, University of Texas, Austin, TX, 1988.
C. C. Paige, B. N. Parlett, and H. A. van der Vorst.
Approximate solutions and eigenvalue bounds from Krylov subspaces.
Numer. Linear Algebra Appl., 2:115-133, 1995.
B. N. Parlett.
The Symmetric Eigenvalue Problem.
Prentice-Hall, Englewood Cliffs, NJ, 1980.
Reprinted as Classics in Applied Mathematics 20, SIAM, Philadelphia, 1997.
B. N. Parlett and I. S. Dhillon.
Fernando's solution to Wilkinson's problem: an application of
double factorization.
Linear Algebra Appl., 267:247-279, 1997.
W. V. Petryshyn.
On the eigenvalue problem
Tu-\lambda Su=0 with unbounded and
non-symmetric operators T and S.
Philos. Trans. Roy. Soc. Math. Phys. Sci., 262:413-458, 1968.
B. G. Pfrommer, J. Demmel, and H. Simon.
Unconstrained energy functionals for electronic structure
calculations.
J. Comput. Phys., 150(1):287-298, 1999.
C. Pommerell.
Solution of Large Unsymmetric Systems of Linear Equations.
Ph.D. thesis, Swiss Federal Institute of Technology, Zürich,
Switzerland, 1992.
E. Rothberg.
Exploiting the Memory Hierarchy in Sequential and Parallel
Sparse Cholesky Factorization.
Ph.D. thesis, Dept. of Computer Science, Stanford University,
Stanford, CA, 1992.
A. Ruhe.
Implementation aspects of band Lanczos algorithms for computation
of eigenvalues of large sparse symmetric matrices.
Math. Comp., 33:680-687, 1979.
A. Ruhe.
Eigenvalue algorithms with several factorizations - a unified theory
yet?
Technical Report 1998:11, Department of Mathematics, Chalmers University of
Technology, Göteborg, Sweden, 1998.
A. Ruhe and D. Skoogh.
Rational Krylov algorithms for eigenvalue computation and model
reduction.
In B. Kågström, J. Dongarra, E. Elmroth, and J. Wasniewski, editors,
Applied Parallel
Computing. Large Scale Scientific and Industrial Problems., volume 1541 of
Lecture Notes in Computer Science, pages 491-502, 1998.
Y. Saad.
Least squares polynomials in the complex plane and their use for
solving nonsymmetric linear systems.
SIAM J. Numer. Anal., 24(1):155-169, 1987.
Y. Saad and M. H. Schultz.
GMRES: A generalized minimal residual algorithm for solving
nonsymmetric linear systems.
SIAM J. Sci. Statist. Comput., 7:856-869, 1986.
M. Sadkane.
A block Arnoldi-Chebyshev method for computing the leading
eigenpairs of large sparse unsymmetric matrices.
Numer. Math., 64:181-193, 1993.
G. V. Savinov.
Investigation of the convergence of a generalized method of conjugate
gradients for determining the extremal eigenvalues of a matrix.
Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI),
111:145-150, 1981.
(In Russian.)
O. Schenk, K. Gärtner, and W. Fichtner.
Efficient sparse LU factorization with left-right looking strategy
on shared memory multiprocessors.
BIT, 40(1):158-176, 2000.
N. S. Sehmi.
Large order structural eigenanalysis techniques.
Ellis Horwood Series: Mathematics and Its Applications. Ellis Horwood,
Chichester, UK, 1989.
V. A. Shishov.
A method for partitioning a high order matrix into blocks in order to
find its eigenvalues.
USSR Comput. Math. and Math. Phys., 1(1):186-190,
1961.
J. P. Singh, W.-D. Webber, and A. Gupta.
Splash. Stanford parallel applications for shared-memory.
Computer Architecture News, 20(1):5-44, 1992.
Software available at
http://www-flash.stanford.edu/apps/SPLASH.
G. L. G. Sleijpen, G. L. Booten, D. R. Fokkema, and H. A. van der Vorst.
Jacobi Davidson type methods for generalized eigenproblems and
polynomial eigenproblems.
BIT, 36:595-633, 1996.
G. L. G. Sleijpen and D. R. Fokkema.
Bi-\mathrm{CGSTAB}(\ell) methods for linear equations involving matrices
with complex spectrum.
Electron. Trans. Numer. Anal., 1:11-32, 1993.
G. L. G. Sleijpen and H. A. van der Vorst.
A Jacobi-Davidson iteration method for linear eigenvalue
problems.
SIAM J. Matrix Anal. Appl., 17:401-425, 1996.
G. L. G. Sleijpen, H. A. van der Vorst, and E. Meijerink.
Efficient expansion of subspaces in the Jacobi-Davidson method
for standard and generalized eigenproblems.
Electron. Trans. Numer. Anal., 7:75-89, 1998.
P. Smit and M. H. C. Paardekooper.
The effects of inexact linear solvers in algorithms for symmetric
eigenvalue problems.
Linear Algebra Appl., 287:337-357, 1998.
D. C. Sorensen.
Deflation for implicitly restarted Arnoldi methods.
Technical Report TR98-12, Department of Computational and Applied
Mathematics, Rice University, Houston, TX, 1998.
A. Stathopoulos, Y. Saad, and K. Wu.
Dynamic thick restarting of the Davidson, and the implicitly
restarted Arnoldi methods.
SIAM J. Sci. Comput., 19:227-245, 1998.
I. Štich, R. Car, M. Parrinello, and S. Baroni.
Conjugate gradient minimization of the energy functional: A new
method for electronic structure calculation.
Phys. Rev. B., 39:4997-5004, 1989.
E. Suetomi and H. Sekimoto.
Conjugate gradient like methods and their application to eigenvalue
problems for neutron diffusion equation.
Annals of Nuclear Energy, 18(4):205, 1991.
J. G. Sun.
Stability and accuracy: Perturbation analysis of algebraic
eigenproblems.
Technical Report UMINF 98.07, Department of Computing Science,
Umeå University, Umeå, Sweden, 1998.
D. B. Szyld and O. B. Widlund.
Applications of conjugate gradient type methods to eigenvalue
calculations.
In Advances in Computer Methods for Partial Differential
Equations, III (Proc. Third IMACS Internat. Sympos., Lehigh Univ., Bethlehem,
Pa., 1979), pages 167-173. IMACS, New Brunswick, NJ, 1979.
F. Tisseur.
Stability of structured Hamiltonian eigensolvers.
Numerical Analysis Report No. 357, Manchester Centre for
Computational Mathematics, Manchester, UK, February 2000.
F. Tisseur and N. J. Higham.
Structured pseudospectra for polynomial eigenvalue problems, with
applications.
Numerical Analysis Report No. 359, Manchester Centre for
Computational Mathematics, Manchester, UK, 2000.
S. Toledo.
Improving instruction-level parallelism in sparse matrix-vector
multiplication using reordering, blocking, and prefetching.
In Proceedings of the Eighth SIAM Conference on Parallel Processing
for Scientific Computing. SIAM, Philadelphia, 1997.
L. N. Trefethen.
Computation of pseudospectra.
In A. Iserles, editor, Acta Numerica, Volume 8, pages 247-295.
Cambridge University Press, Cambridge, MA, 1999.
L. N. Trefethen.
Spectra and pseudospectra: The behavior of non-normal matrices and
operators.
In J. Levesley, M. Ainsworth, and M. Marletta, editors, The
Graduate Student's Guide to Numerical Analysis, Volume 26. Springer-Verlag, Berlin,
2000.
C. Trefftz, C. C. Huang, P. K. Mckinley, T.-Y. Li, and Z. Zeng.
A scalable eigenvalue solver for symmetric tridiagonal matrices.
Parallel Computing, 21:1213-1240, 1995.
H. A. van der Vorst.
Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for
the solution of non-symmetric linear systems.
SIAM J. Sci. Statist. Comput., 13:631-644, 1992.
P. Van Dooren.
Algorithm 590, DUSBSP and EXCHQZ: FORTRAN subroutines for
computing deflating subspaces with specified spectrum.
ACM Trans. Math. Software, 8:376-382, 1982.
P. Van Dooren.
Reducing subspaces: Computational aspects and applications in linear
systems theory.
In Proceedings of the 5th Int. Conf. on Analysis and
Optimization of Systems, 1982, Lecture Notes on Control
and Information Sciences. Volume 44. Springer-Verlag, New York, 1983.
P. Van Dooren.
Reducing subspaces: Definitions, properties and algorithms.
In B. Kågström and A. Ruhe, editors, Matrix Pencils,
Lecture Notes in Mathematics, Volume 973, pages 58-73.
Springer-Verlag, Berlin, 1983.
E. L. Wachspress.
Iterative solution of elliptic systems, and applications to the
neutron diffusion equations of reactor physics.
Prentice-Hall, Englewood Cliffs, NJ, 1966.
K. Wu, Y. Saad, and A. Stathopoulos.
Inexact Newton preconditioning techniques for eigenvalue problems.
Technical Report Technical Report LBNL-41382, Lawrence Berkeley
National Laboratory, Berkeley, CA, 1998.
Also published as Minnesota Super Computer Centre report number UMSI 98-10,
Minneapolis.
K. Wu and H. D. Simon.
A parallel Lanczos method for symmetric generalized eigenvalue
problems.
Technical Report LBNL-41284, National Energy Research Scientific
Computing Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 1997.
Software available at
http://www.nersc.gov/research/SIMON/planso.html.
K. Wu and H. D. Simon.
Dynamic restarting schemes for eigenvalue problems.
Technical Report LBNL-42982, National Energy Research Scientific
Computing Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 1999.
H. Zha and Z. Zhang.
On matrices with low-rank-plus-shift structures: partial SVD and
latent semantic indexing.
SIAM J. Matrix Anal. Appl., 21:522-280, 1999.
T. Zhang, K. H. Law, and G. H. Golub.
On the homotopy method for perturbed symmetric generalized eigenvalue
problems.
SIAM J. Sci. Comput., 19(5):1625-1645, 1998.
S. Zhou and H. Dai.
Dai Shu Te Zheng Zhi Fan Wen Ti (The Algebraic Inverse
Eigenvalue Problems).
Henan Science and Technology Press, Zhengzhou, China, 1991.
(In Chinese.)
Z. Zlatev, J. Wasniewski, P. C. Hansen, and Tz. Ostromsky.
PARASPAR: A package for the solution of large linear algebraic
equations on parallel computers with shared memory.
Technical Report 95-10, Technical University of Denmark, Lyngby, September
1995.
P. I. Davies, N. J. Higham, and F. Tisseur.
Analysis of the Cholesky Method with Iterative Refinement for Solving
the Symmetric Definite Generalized Eigenproblem.
Manchester Centre for Computational Mathematics,
Manchester, England, Numerical Analysis Report 360, 2000.
D. J. Higham and N. J. Higham.
Structured backward error and condition of generalized eigenvalue problems.
SIAM J. Matrix Anal. Appl., 20:493-512, 1998.