Bibliography

1

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.

2

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.

3

G. Adams, A. Bojanczyk, and F. T. Luk.
Computing the PSVD of two $2 \times 2$ triangular matrices.
SIAM J. Matrix Anal. Appl., 15(2):366-382, 1994.

4

L. Ahlfors.
Complex Analysis.
McGraw-Hill, New York, 1966.

5

J. I. Aliaga, D. L. Boley, R. W. Freund, and V. Hernández.
A Lanczos-type method for multiple starting vectors.
Math. Comp. 69:1577-1601, 2000.

6

P. R. Amestoy and I. S. Duff.
Vectorization of a multiprocessor multifrontal code.
Internat. J. Supercomputer Appl., 3:41-59, 1989.

7

P. R. Amestoy and I. S. Duff.
Memory management issues in sparse multifrontal methods on multiprocessors.
Internat. J. Supercomputer Appl., 7:64-82, 1993.

8

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.

9

G. S. Ammar, W. B. Gragg, and L. Reichel.
Downdating Szegö polynomials and data fitting applications.
Linear Algebra Appl., 172:315-336, 1992.

10

G. S. Ammar and C. He.
On an inverse eigenvalue problem for unitary Hessenberg matrices.
Linear Algebra Appl., 218:263-271, 1995.

11

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.

12

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.

13

P. J. Anderson and G. Loizou.
A Jacobi type method for complex symmetric matrices.
Numer. Math., 25:347-363, 1976.

14

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.

15

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.

16

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.

17

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.

18

V. I. Arnold.
On matrices depending on parameters.
Russian Math. Surveys, 26:29-43, 1971.

19

W. E. Arnoldi.
The principle of minimized iterations in the solution of the matrix eigenvalue problem.
Quart. Appl. Math., 9:17-29, 1951.

20

E. Artin.
Geometric Algebra.
Interscience, New York, 1957.

21

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.

22

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.

23

J. Baglama, D. Calvetti, and L. Reichel.
Fast Leja points.
Electron. Trans. Numer. Anal., 7:124-140, 1998.

24

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.

25

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/$\sim$bai.

26

Z. Bai.
Error analysis of the Lanczos algorithm for the nonsymmetric eigenvalue problem.
Math. Comp., 62:209-226, 1994.

27

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.

28

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/.

29

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.

30

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.

31

Z. Bai and J. Demmel.
Using the matrix sign function to compute invariant subspaces.
SIAM J. Matrix Anal. Appl., 19:205-225, 1998.

32

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.

33

Z. Bai and J. W. Demmel.
On swapping diagonal blocks in real Schur form.
Linear Algebra Appl., 186:73-95, 1993.

34

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.

35

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.

36

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.

37

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.

38

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.

39

R. E. Bank.
Analysis of a multilevel inverse iteration procedure for eigenvalue problems.
SIAM J. Numer. Anal., 19(5):886-898, 1982.

40

J. Barlow and J. Demmel.
Computing accurate eigensystems of scaled diagonally dominant matrices.
SIAM J. Numer. Anal., 27(3):762-791, 1990.

41

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.

42

K.-J. Bathe and E. L. Wilson.
Numerical Methods in Finite Element Analysis.
Prentice Hall, Englewood Cliffs, NJ, 1976.

43

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.

44

P. Benner, H. Fassbender, and D. Watkins.
SR and SZ algorithms for the symplectic (butterfly) eigenproblem.
Linear Algebra Appl., 287:41-76, 1999.

45

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.

46

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.

47

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.

48

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.

49

M. Berry.
Large scale singular value computations.
Internat. J. Supercomputer Appl., 6(1):13-49, 1992.

50

Å. Björck.
Numerical Solutions for Least Squares Problems.
SIAM, Philadelphia, 1996.

51

Å. Björck and V. Pereyra.
Solution of vandermonde systems of equations.
Math. Comp., 24:893-903, 1970.

52

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.

53

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.

54

A. Bojanczyk, P. Van Dooren, L. M. Ewerbring, and F. T. Luk.
An accurate product SVD algorithm.
J. Signal Processing, 25:189-201, 1991.

55

D. Boley.
The algebraic structure of pencils and block Toeplitz matrices.
Linear Algebra Appl., 279:255-279, April 1998.

56

D. Boley and G. H. Golub.
A survey of matrix inverse eigenvalue problems.
Inverse Problems, 3:595-622, 1987.

57

F. Bourquin.
Analysis and comparison of several component mode synthesis methods on one-dimensional domains.
Numer. Math., 58(1):11-33, 1990.

58

F. Bourquin.
Component mode synthesis and eigenvalues of second order operators: discretization and algorithm.
RAIRO Modél. Math. Anal. Numér., 26(3):385-423, 1992.

59

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.

60

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.

61

W. W. Bradbury and R. Fletcher.
New iterative methods for solution of the eigenproblem.
Numer. Math., 9:259-267, 1966.

62

J. H. Bramble.
Multigrid Methods.
Longman Scientific & Technical, Harlow, UK, 1993.

63

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.

64

A. Brandt, S. McCormick, and J. Ruge.
Multigrid methods for differential eigenproblems.
SIAM J. Sci. Statist. Comput., 4(2):244-260, 1983.

65

C. Brezinski, M. Redivo Zaglia, and H. Sadok.
Avoiding breakdown and near-breakdown in Lanczos type algorithms.
Numer. Algorithms, 1:261-284, 1991.

66

W. L. Briggs.
A Multigrid Tutorial.
SIAM, Philadelphia, 1987.

67

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.

68

A. Bunse-Gerstner and C. He.
On the Sturm sequence of polynomials for unitary Hessenberg matrices.
SIAM J. Matrix Anal. Appl., 16:1043-1055, 1995.

69

A. Bunse-Gerstner and V. Mehrmann.
The quaternion QR algorithm.
Numer. Math., 55:83-95, 1989.

70

J. V. Burke, A. S. Lewis, and M. L. Overton.
Optimizing matrix stability.
Proc. Amer. Math. Soc., 1999, to appear.

71

R. Byers.
A Hamiltonian QR-algorithm.
SIAM J. Sci. Statist. Comput., 7:212-229, 1986.

72

R. Byers.
Solving the algebraic Riccati equation with the matrix sign function.
Linear Algebra Appl., 85:267-279, 1987.

73

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.

74

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.

75

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.

76

J. Carrier, L. Greengard, and V. Rokhlin.
A fast adaptive multipole algorithm for particle simulations.
SIAM J. Sci. Statist. Comput., 9:669-686, 1988.

77

F. Chaitin-Chatelin and V. Frayssé.
Lectures on Finite Precision Computations.
SIAM, Philadelphia, 1996.

78

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.

79

F. Chatelin.
Eigenvalues of Matrices.
Wiley, New York, 1993.

80

I. Chavel.
Riemannian Geometry--A Modern Introduction.
The Cambridge University Press, Cambridge, UK, 1993.

81

T.-Y. Chen.
Balancing sparse matrices for computing eigenvalues.
Master's thesis, University of California, Berkeley, May 1998.

82

T.-Y. Chen and J. Demmel.
Balancing sparse matrices for computing eigenvalues.
Linear Algebra Appl., 309:261-287, 2000.

83

X. Chi and M. Gu.
Updating the SVD.
CAM technical report, Department of Mathematics, University of California, Los Angeles, 2000.

84

M. T. Chu.
Inverse eigenvalue problems.
SIAM Rev., 40:1-39, 1998.

85

B. D. Craven.
Complex symmetric matrices.
J. Austral. Math. Soc., 10:341-354, 1969.

86

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.

87

C. R. Crawford and Y. S. Moon.
Finding a positive definite linear combination of two Hermitian matrices.
Linear Algebra Appl., 51:37-48, 1983.

88

M. Crouzeix, B. Philippe, and M. Sadkane.
The Davidson method.
SIAM J. Sci. Comput., 15:62-76, 1994.

89

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.

90

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.

91

J. K. Cullum and R. A. Willoughby.
Lanczos Algorithms for Large Symmetric Eigenvalue Computations. Volume $1$, Theory.
Birkhäuser, Boston, 1985.

92

J. K. Cullum and R. A. Willoughby.
Lanczos Algorithms for Large Symmetric Eigenvalue Computations. Volume $2$, Programs.
Birkhäuser, Boston, 1985.

93

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.

94

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.

95

H. Dai and P. Lancaster.
Numerical methods for finding multiple eigenvalues of matrices depending on parameters.
Numer. Math., 76:189-208, 1997.

96

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.

97

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.

98

D. F. Davidenko.
On the computation of eigenvalues and eigenvectors of matrices.
Dokl. Akad. Nauk SSSR, 141:277-280, 1961.

99

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.

100

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.

101

C. Davis and W. Kahan.
The rotation of eigenvectors by a perturbation. III.
SIAM J. Numer. Anal., 7:1-46, 1970.

102

G. J. Davis.
Numerical solution of a quadratic matrix equation.
SIAM J. Sci. Comput., 2:164-175, 1981.

103

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.

104

D. Day.
Semi-Duality in the Two-Sided Lanczos Algorithm.
Ph.D. thesis, University of California, Berkeley, 1993.

105

D. Day.
An efficient implementation of the nonsymmetric Lanczos algorithm.
SIAM J. Matrix Anal. Appl., 18:566-589, 1997.

106

I. De Hoyos.
Points of continuity of the Kronecker canonical form.
SIAM J. Matrix Anal. Appl., 11(2):278-300, April 1990.

107

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.

108

B. De Moor.
On the structure of generalized singular value and QR decompositions.
SIAM J. Matrix Anal. Appl., 15(1):347-358, 1994.

109

G. De Samblanx.
Filtering and restarting projection methods for eigenvalue problems.
PhD Thesis, Katholieke Universiteit Leuven, Department of Computer Science, 3001 Heverlee, Belgium, 1998.

110

G. De Samblanx and A. Bultheel.
Nested Lanczos: implicitly restarting a Lanczos algorithm.
Numer. Algorithms, 18:31-50, 1998.

111

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.

112

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.

113

R. S. Dembo, S. C. Eisenstat, and T. Steihaug.
Inexact Newton methods.
SIAM J. Numer. Anal., 19:400-408, 1982.

114

J. Demmel.
Applied Numerical Linear Algebra.
SIAM, Philadelphia, 1997.

115

J. Demmel.
Accurate SVDs of structured matrices.
SIAM J. Matrix Anal. Appl., 21(3):562-580, 2000.

116

J. Demmel and A. Edelman.
The dimension of matrices (matrix pencils) with given Jordan (Kronecker) canonical forms.
Linear Algebra Appl., 230:61-87, 1995.

117

J. Demmel and W. Gragg.
On computing accurate singular values and eigenvalues of matrices with acyclic graphs.
Linear Algebra Appl., 185:203-217, 1993.

118

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.

119

J. Demmel and B. Kågström.
Computing stable eigendecompositions of matrix pencils.
Linear Algebra Appl., 88/89:139-186, 1987.

120

J. Demmel and B. Kågström.
Accurate solutions of ill-posed problems in control theory.
SIAM J. Matrix Anal. Appl., 9(1):126-145, 1988.

121

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.

122

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.

123

J. Demmel and W. Kahan.
Accurate singular values of bidiagonal matrices.
SIAM J. Sci. Statist. Comput., 11:873-912, 1990.

124

J. Demmel and K. Veselic.
Jacobi's method is more accurate than ${Q}{R}$.
SIAM J. Matrix Anal. Appl., 13(4):1204-1245, 1992.

125

J. W. Demmel, L. Dieci, and M. Friedman.
SIAM J. Sci. Comput., 22(1):81-94, 2000.

126

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/$\sim$xiaoye/SuperLU.

127

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/$\sim$xiaoye/SuperLU.

128

I. Dhillon.
A New $O(n^2)$ Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem.
Ph.D. thesis, University of California, Berkeley, 1997.

129

I. S. Dhillon.
Current inverse iteration software can fail.
BIT, 38(4):685-704, 1998.

130

D. C. Dobson.
An efficient method for band structure calculations in 2D photonic crystals.
J. Comput. Phys., 149(2):363-376, 1999.

131

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.

132

J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W. Stewart.
LINPACK Users' Guide.
SIAM, Philadelphia, 1979.

133

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.

134

J. J. Dongarra, J. DuCroz, I. S. Duff, and S. Hammarling.
A set of level 3 basic linear algebra subprograms.
ACM Trans. Math. Software, 16:1-17, 1990.

135

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.

136

Z. Drmac.
A posteriori computation of the singular vectors in a preconditioned Jacobi SVD algorithm.
IMA J. Numer. Anal., 19:191-213, 1999.

137

I. S. Duff.
Direct methods.
Technical Report RAL-98-056, Rutherford Appleton Laboratory, Oxfordshire, UK, 1998.

138

I. S. Duff, A. M. Erisman, and J. K. Reid.
Direct Methods for Sparse Matrices.
Clarendon Press, Oxford, UK, 1986.

139

I. S. Duff, R. G. Grimes, and J. G. Lewis.
Sparse matrix test problems.
ACM Trans. Math. Software, 15:1-14, 1989.

140

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.

141

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.

142

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.

143

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.

144

I. S. Duff and J. A. Scott.
The design of a new frontal code for solving sparse unsymmetric systems.
ACM Trans. Math. Software, 22(1):30-45, 1996.

145

R. J. Duffin.
A minimax theory for overdamped networks.
J. Rational Mech. Anal., 4:221-233, 1955.

146

E. G. D'yakonov.
Iteration methods in eigenvalue problems.
Math. Notes, 34:945-953, 1983.

147

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.

148

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.

149

E. G. D'yakonov and A. V. Knyazev.
On an iterative method for finding lower eigenvalues.
Russian J. Numer. Anal. Math. Modelling, 7(6):473-486, 1992.

150

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.

151

A. Edelman, T. A. Arias, and S. T. Smith.
The geometry of algorithms with orthogonality constraints.
SIAM J. Matrix Anal. Appl., 20:303-353, 1999.

152

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.

153

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.

154

A. Edelman and Y. Ma.
Staircase failures explained by orthogonal versal forms.
SIAM J. Matrix Anal. Appl., 21(3):1004-1025, 2000.

155

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.

156

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.

157

S. C. Eisenstat and I. C. F. Ipsen.
Relative perturbation techniques for singular value problems.
SIAM J. Numer. Anal., 32:1972-1988, 1995.

158

L. Eldén.
Algorithms for the regularization of ill-conditioned least-squares problems.
BIT, 17:134-145, 1977.

159

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.

160

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.

161

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.

162

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.

163

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.

164

C. Farhat and M. Geradin.
On a component mode synthesis method and its application to incompatible substructures.
Comput. & Structures, 51(5):459-473, 1994.

165

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.

166

P. Feldmann and R. W. Freund.
Reduced-order modeling of large linear subcircuits via a block Lanczos algorithm.
In Proceedings of the $32$nd Design Automation Conference, pages 474-479. ACM, New York, 1995.

167

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.

168

K. V. Fernando and B. N. Parlett.
Accurate singular values and differential qd algorithms.
Numer. Math., 67:191-229, 1994.

169

R. Fletcher.
Conjugate gradient methods for indefinite systems.
Lecture Notes in Mathematics, Vol. 506, pages 73-89. Springer-Verlag, Berlin, 1976.

170

R. Fletcher.
Practical Methods of Optimization.
Wiley, New York, second edition, 1987.

171

D. R. Fokkema.
Subspace Methods for Linear, Nonlinear, and Eigen Problems.
Ph.D. thesis, Utrecht University, Utrecht, the Netherlands, 1996.

172

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.

173

R. W. Freund.
Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices.
SIAM J. Sci. Statist. Comput., 13:425-448, 1992.

174

R. W. Freund.
A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems.
SIAM J. Sci. Comput., 14:470-482, 1993.

175

R. W. Freund.
Computing minimal partial realizations via a Lanczos-type algorithm for multiple starting vectors.
In Proceedings of the $36$th IEEE Conference on Decision and Control, pages 4394-4399. IEEE Press, Piscataway, NJ, 1997.

176

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.

177

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.

178

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.

179

R. W. Freund and N. M. Nachtigal.
QMR: A quasi-minimal residual method for non-Hermitian linear systems.
Numer. Math., 60:315-339, 1991.

180

R. W. Freund and N. M. Nachtigal.
QMRPACK: A package of QMR algorithms.
ACM Trans. Math. Software, 22:46-77, 1996.

181

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.

182

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+.

183

Z. Fu and E. M. Dowling.
Conjugate gradient eigenstructure tracking for adaptive spectral estimation.
IEEE Trans. Signal Processing, 43(5):1151-1160, 1995.

184

K. Gallivan, E. Grimme, and P. Van Dooren.
A rational Lanczos algorithm for model reduction.
Numer. Algorithms, 12:33-64, 1996.

185

G. Gambolati, G. Pini, and M. Putti.
Nested iterations for symmetric eigenproblems.
SIAM J. Sci. Comput., 16(1):173-191, 1995.

186

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.

187

F. Gantmacher.
The Theory of Matrices, Vols. I and II (transl.).
Chelsea, New York, 1959.

188

I. Garcia-Planas.
Kronecker stratification of the space of quadruples of matrices.
SIAM J. Matrix Anal. Appl., 19(4):872-885, 1998.

189

G. Geist, G. Howell, and D. Watkins.
The BR eigenvalue algorithm.
SIAM J. Matrix Anal. Appl., 20(4):1083-1098, 1999.

190

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.

191

A. George and J. Liu.
Computer Solution of Large Sparse Positive Definite Systems.
Prentice-Hall, Englewood Cliffs, NJ, 1981.

192

P. E. Gill, W. Murray, and M. H. Wright.
Practical Optimization.
Academic Press, New York, second edition, 1981.

193

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.

194

I. Gohberg, P. Lancaster, and L. Rodman.
Matrix Polynomials.
Academic Press, New York, 1982.

195

I. Gohberg and V. Olshevsky.
Complexity of multiplication with vectors for structured matrices.
Linear Algebra Appl., 202:163-192, 1994.

196

I. Gohberg and V. Olshevsky.
Fast algorithms with preprocessing for matrix-vector mupltiplication problems.
J. Complexity, 10(4):411-427, 1994.

197

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.

198

G. Golub and C. Van Loan.
Matrix Computations.
The Johns Hopkins University Press, Baltimore, third edition, 1996.

199

G. Golub and J. H. Wilkinson.
Ill-conditioned eigensystems and the computation of the Jordan canonical form.
SIAM Rev., 18(4):578-619, 1976.

200

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.

201

W. B. Gragg.
The QR algorithm for unitary Hessenberg matrices.
J. Comput. Appl. Math., 16:1-8, 1986.

202

W. B. Gragg and L. Reichel.
A divide and conquer method for unitary and orthogonal eigenproblems.
Numer. Math., 57:695-718, 1990.

203

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.

204

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.

205

L. Greengard and V. Rokhlin.
A fast algorithm for particle simulations.
J. Comput. Phys., 73:325-348, 1987.

206

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.

207

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.

208

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.

209

J.-S. Guo, W.-W. Lin, and C.-S. Wang.
Numerical solutions for large sparse quadratic eigenvalue problems.
Linear Alg. Appl., 225:57-89, 1995.

210

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/$\sim$mjoshi/pspases.

211

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.

212

I. Gustafsson.
A class of first order factorization methods.
BIT, 18:142-156, 1978.

213

M. H. Gutknecht.
A completed theory of the unsymmetric Lanczos process and related algorithms, Part I.
SIAM J. Matrix Anal. Appl., 13:594-639, 1992.

214

M. H. Gutknecht.
A completed theory of the unsymmetric Lanczos process and related algorithms, Part II.
SIAM J. Matrix Anal. Appl., 15:15-58, 1994.

215

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.

216

W. Hackbusch.
Multigrid solutions to linear and nonlinear eigenvalue problems for integral and differential equations.
Rostock. Math. Kolloq., (25):79-98, 1984.

217

W. Hackbusch.
Multigrid eigenvalue computation.
In Advances in Multigrid Methods $($Oberwolfach, 1984), pages 24-32. Vieweg, Braunschweig, Germany, 1985.

218

W. Hackbusch.
Multigrid Methods and Applications.
Springer-Verlag, Berlin, 1985.

219

M. Heath, E. Ng, and B. Peyton.
Parallel algorithms for sparse linear systems.
SIAM Rev., 33:420-460, 1991.

220

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.

221

R. Heeg.
Stability and Transistion of Attachment-Line Flow.
Ph.D. thesis, Universiteit Twente, Enschede, the Netherlands, 1998.

222

S. Helgason.
Differential Geometry, Lie Groups, and Symmetric Spaces.
Academic Press, New York, 1978.

223

B. Hendrickson and R. Leland.
The Chaco User's Guide: Version 2.0.
Technical Report SAND94-2692, Sandia National Laboratories, 1994.

224

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.

225

M. R. Hestenes and W. Karush.
Solutions of ${A}x=\lambda {B}x$.
J. Res. Nat. Bur. Standards, 47:471-478, 1951.

226

M. R. Hestenes and E. Stiefel.
Methods of conjugate gradients for solving linear systems.
J. Res. Nat. Bur. Standards, 49:409-436, 1954.

227

V. Heuveline and M. Sadkane.
Arnoldi-Faber method for large non-Hermitian eigenvalue problems.
Electron. Trans. Numer. Anal., 7:62-76, 1997.

228

N. J. Higham.
Accuracy and Stability of Numerical Algorithms.
SIAM, Philiadelphia, 1996.

229

N. J. Higham.
QR factorization with complete pivoting and accurate computation of the SVD.
Linear Algebra Appl., 309:153-174, 2000.

230

N. J. Higham.
Stability analysis of algorithms for solving confluent Vandermonde-like systems.
SIAM J. Matrix Anal. Appl., 11:23-41, 1990.

231

D. Hinrichsen and J. O'Halloran.
Orbit closures of singular pencils.
J. Pure and Applied Algebra, 81:117-137, 1992.

232

K. Hirao and H. Nakatsuji.
A generalization of the Davidson's method to large nonsymmetric eigenvalue problem.
J. Comput. Phys., 45(2):246-254, 1982.

233

R. A. Horn and C. R. Johnson.
Matrix Analysis.
Cambridge University Press, Cambridge, UK, 1985.

234

A. S. Householder.
The Theory of Matrices in Numerical Analysis.
Blaisdell, New York, 1964.
Dover edition, 1975.

235

L. J. Huang and T.-Y. Li.
Parallel homotopy algorithm for symmetric large sparse eigenproblems.
J. Comput. Appl. Math., 60(1-2):77-100, 1995.

236

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.

237

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.

238

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.

239

E.-J. Im.
Automatic Optimization of Sparse Matrix - Vector Multiplication.
Ph.D. thesis, University of California, Berkeley, May 2000.

240

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.

241

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.

242

A. Jennings.
Matrix Computation for Engineers and Scientists.
Wiley, New York, 1977.

243

Z. Jia.
A block incomplete orthogonalisation method for large nonsymmetric eigenproblems.
BIT, 35:516-539, 1995.

244

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.

245

Z. Jia.
A refined iterative algorithm based on the block Arnoldi process for large unsymmetric eigenproblems.
Linear Algebra Appl., 270:171-189, 1998.

246

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.

247

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.

248

P. Johansson.
Stratigraph users' guide. version 1.1.
Technical Report UMINF-99.11, Department of Computing Science, Umeå University, Umeå, Sweden, 1999.

249

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.

250

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.

251

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.

252

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.

253

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.

254

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.

255

W. Kahan.
Accurate eigenvalues of a symmetric tridiagonal matrix.
Technical Report CS41, Computer Science Department, Stanford University, Stanford, CA, 1966
(revised June 1968).

256

W. Kahan, B. N. Parlett, and E. Jiang.
Residual bounds on approximate eigensystems of nonnormal matrices.
SIAM J. Numer. Anal., 19:470-484, 1982.

257

T. Kailath and A.H. Sayed, editors.
Fast Reliable Algorithms for Matrices with Structure.
SIAM, Philadelphia, 1999.

258

W. Karush.
An iterative method for finding characteristics vectors of a symmetric matrix.
Pacific J. Math., 1:233-248, 1951.

259

G. Karypis and V. Kumar.
Metis, Version 4.0.
University of Minnesota/Army HPC Research Center, Mineeapolis, 1998.

260

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.

261

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.

262

S. Kim and A. Chronopoulos.
An efficient nonsymmetric Lanczos method on parallel vector computers.
J. Comput. Appl. Math., 42:357-374, 1992.

263

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.

264

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.)

265

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.

266

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.

267

A. V. Knyazev.
New estimates for Ritz vectors.
Math. Comp., 66(219):985-995, 1997.

268

A. V. Knyazev.
Preconditioned eigensolvers - an oxymoron?
Electron. Trans. Numer. Anal., 7:104-123, 1998.

269

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.

270

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.

271

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.

272

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.

273

S. Kobayashi and K. Nomizu.
Foundations of Differential Geometry.
Wiley, New York, 1969.

274

L. Komzsik.
MSC/NASTRAN Numerical Methods User's Guide, Version 70.5.
The MacNeal-Schwendler Corporation, Los Angeles, 1998.

275

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.

276

L. Kronecker.
Algebraische Reduction der Schaaren Bilinearer Formen.
S. B. Akad., Berlin, 1890.

277

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.

278

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.

279

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.

280

V. N. Kublanovskaya.
AB-algorithm and its modifications for the spectral problem of linear pencils of matrices.
Numer. Math., 43:329-342, 1984.

281

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.

282

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.

283

Y.-L. Lai, K.-Y. Lin, and W.-W. Lin.
An inexact inverse iteration for large sparse eigenvalue problems.
Numer. Linear Algebra Appl., 4:425-437, 1997.

284

P. Lancaster.
Lambda-Matrices and Vibrating Systems.
Pergamon Press, Oxford, UK, 1966.

285

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.

286

C. Lanczos.
Solution of systems of linear equations by minimized iterations.
J. Res. Nat. Bur. Standards, 49:33-53, 1952.

287

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.

288

C. Lawson, R. Hanson, D. Kincaid, and F. Krogh.
Basic linear algebra subprograms for FORTRAN usage.
ACM Trans. Math. Software, 5:308-325, 1979.

289

R. B. Lehoucq.
Analysis and Implementation of an Implicitly Restarted Arnoldi Iteration.
Ph.D. thesis, Rice University, Houston, TX, 1995.

290

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.

291

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.

292

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.

293

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.

294

R. B. Lehoucq and D. C. Sorensen.
Deflation techniques within an implicitly restarted Arnoldi iteration.
SIAM J. Matrix Anal. Appl., 17:789-821, 1996.

295

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.

296

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.

297

C.-K. Li and R. Mathias.
The Lidskii-Mirsky-Wielandt theorem - additive and multiplicative versions.
Numer. Math., 81:377-413, 1999.

298

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.

299

R.-C. Li.
On perturbations of matrix pencils with real spectra.
Math. Comp., 62:231-265, 1994.

300

R.-C. Li.
Relative perturbation theory III: More bounds on eigenvalue variation.
Linear Algebra Appl., 266:337-345, 1997.

301

R. C. Li.
Relative perturbation theory I: Eigenvalue and singular value variations.
SIAM J. Matrix Anal. Appl., 19:956-982, 1998.

302

R. C. Li.
Relative perturbation theory II: Eigenspace and singular subspace variations.
SIAM J. Matrix Anal. Appl., 20:471-492, 1999.

303

R. C. Li.
Relative perturbation theory IV: $\sin2\theta$ theorems.
Linear Algebra Appl., 311:45-60, 2000.

304

T.-Y. Li and Z. Zeng.
The Laguerre iteration in solving the symmetric tridiagonal eigenproblem, revisited.
SIAM J. Sci. Comput., 15:1145-1173, 1994.

305

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.

306

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/$\sim$xiaoye/SuperLU.

307

W.-W. Lin, V. Mehrmann, and H. Xu.
Canonical forms for Hamiltonian and symplectic matrices and pencils.
Linear Algebra Appl., 301/303:469-533, 1999.

308

R. Lucas.
Private communication, 2000.
Contact rflucas@lbl.gov.

309

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.

310

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.

311

J.-C. Luo.
Solving eigenvalue problems by implicit decomposition.
Numer. Methods Partial Differential Equations, 7(2):113-145, 1991.

312

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.

313

N. Mackey.
Hamilton and Jacobi meet again - quaternions and the eigenvalue problem.
SIAM J. Matrix Anal. Appl., 16:421-435, 1995.

314

S. Yu. Maliassov.
On the Schwarz alternating method for eigenvalue problems.
Russian J. Numer. Anal. Math. Modelling, 13(1):45-56, 1998.

315

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.

316

T. Manteuffel.
The Tchebyshev iteration for nonsymmetric linear systems.
Numer. Math., 28:307-327, 1977.

317

R. Mathias.
Accurate eigensystem computations by Jacobi methods.
SIAM J. Matrix Anal. Appl., 16:977-1003, 1995.

318

The MathWorks.
Partial Differential Equation Toolbox User's Guide, The MathWorks, Natick, MA, 1995.

319

The MathWorks.
MATLAB User's Guide, The MathWorks, Natick, MA, 1996.

320

S. F. McCormick.
A mesh refinement method for ${A}x=\lambda {B}x$.
Math. Comp., 36(154):485-498, 1981.

321

S. F. McCormick.
Multilevel Projection Methods for Partial Differential Equations.
SIAM, Philadelphia, 1992.

322

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.

323

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.

324

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.

325

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.

326

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.

327

G. Meurant.
Computer solution of large linear systems.
North-Holland, Amsterdam, 1999.

328

C. B. Moler and G. W. Stewart.
An algorithm for generalized matrix eigenvalue problems.
SIAM J. Numer. Anal., 10:241-256, 1973.

329

R. B. Morgan.
Davidson's method and preconditioning for generalized eigenvalue problems.
J. Comput. Phys., 89:241-245, 1990.

330

R. B. Morgan.
Theory for preconditioning eigenvalue problems.
In Proceedings of the Copper Mountain Conference on Iterative Methods, 1990.

331

R. B. Morgan.
Computing interior eigenvalues of large matrices.
Linear Algebra Appl., 154/156:289-309, 1991.

332

R. B. Morgan.
Generalisations of Davidson's method for computing eigenvalues of large nonsymmetric matrices.
J. Comput. Phys., 101:287-291, 1992.

333

R. B. Morgan.
On restarting the Arnoldi method for large nonsymmetric eigenvalue problems.
Math. Comp., 65:1213-1230, 1996.

334

R. B. Morgan.
Implicitly restarted GMRES and Arnoldi methods for nonsymmetric systems of equations.
SIAM J. Matrix Anal. Appl., 21:1112-1135, 2000.

335

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.

336

R. B. Morgan and D. S. Scott.
Preconditioning the Lanczos algorithm for sparse symmetric eigenvalue problems.
SIAM J. Sci. Comput., 14:585-593, 1993.

337

R. B. Morgan and M. Zeng.
Harmonic projection methods for large non-symmetric eigenvalue problems.
Numer. Linear Algebra Appl., 5:33-55, 1998.

338

S. G. Nash and A. Sofer.
Linear and Nonlinear Programming.
McGraw-Hill, New York, 1995.

339

E. G. Ng and B. W. Peyton.
Block sparse Cholesky algorithms on advanced uniprocessor computers.
SIAM J. Sci. Comput., 14(5):1034-1056, 1993.

340

B. Nour-Omid, B. N. Parlett, T. Ericsson, and P. S. Jensen.
How to implement the spectral transformation.
Math. Comp., 48:663-673, 1987.

341

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.

342

S. Oliveira.
On the convergence rate of a preconditioned subspace eigensolver.
Computing, 63(3):219-231, 1999.

343

W. E. Olmstead, W. E. Davis, S. H. Rosenblat, and W. L. Kath.
Bifurcation with memory.
SIAM J. Appl. Math., 40:171-188, 1986.

344

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.

345

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.

346

E. E. Osborne.
On pre-conditioning of matrices.
J. Assoc. Comput. Mach., 7:338-345, 1960.

347

C. C. Paige.
The Computation of Eigenvalues and Eigenvectors of Very Large Sparse Matrices.
Ph.D. thesis, London University, London, England, 1971.

348

C. C. Paige.
Properties of numerical algorithms related to computing controllability.
IEEE Trans. Automat. Control, AC-26(1):130-138, 1981.

349

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.

350

C. C. Paige and M. A. Saunders.
Solution of sparse indefinite systems of linear equations.
SIAM J. Numer. Anal., 12:617-629, 1975.

351

C. C. Paige and M. A. Saunders.
LSQR: An algorithm for sparse linear equations and sparse least squares.
ACM Trans. Math. Software, 8:43-71, 1982.

352

C. C. Paige and C. F. Van Loan.
A Schur decomposition for Hamiltonian matrices.
Linear Algebra Appl., 14:11-32, 1981.

353

B. N. Parlett.
The Symmetric Eigenvalue Problem.
Prentice-Hall, Englewood Cliffs, NJ, 1980.
Reprinted as Classics in Applied Mathematics 20, SIAM, Philadelphia, 1997.

354

B. N. Parlett.
Reduction to tridiagonal form and minimal realizations.
SIAM J. Matrix Anal. Appl., 13(2):567-593, 1992.

355

B. N. Parlett.
The new qd algorithms.
In Acta Numerica, pages 459-491. Cambridge University Press, Cambridge, UK, 1995.

356

B. N. Parlett.
Invariant subspaces for tightly clustered eigenvalues of tridiagonals.
BIT, 36:542-562, 1996.

357

B. N. Parlett and H. C. Chen.
Use of indefinite pencils for computing damped natural modes.
Linear Algebra Appl., 140:53-88, 1990.

358

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.

359

B. N. Parlett and J. Le.
Forward instability of tridiagonal QR.
SIAM J. Matrix Anal. Appl., 14:279-316, 1993.
316.

360

B. N. Parlett and O. A. Marques.
An implementation of the dqds algorithm (positive case).
Linear Algebra Appl., 309:217-259, 2000.

361

B. N. Parlett and C. Reinsch.
Balancing a matrix for calculation of eigenvalues and eigenvectors.
Numer. Math., 13:293-304, 1969.

362

B. N. Parlett and Y. Saad.
Complex shift and invert strategies for real matrices.
Linear Algebra Appl., 88/89:575-595, 1987.

363

B. N. Parlett and D. Scott.
The Lanczos algorithm with selective orthogonalization.
Math. Comput., 33:217-238, 1979.

364

B. N. Parlett, D. R. Taylor, and Z. S. Liu.
A look-ahead Lanczos algorithm for nonsymmetric matrices.
Math. Comp., 44:105-124, 1985.

365

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.

366

B. G. Pfrommer, J. Demmel, and H. Simon.
Unconstrained energy functionals for electronic structure calculations.
J. Comput. Phys., 150(1):287-298, 1999.

367

A. Pokrzywa.
On perturbations and the equivalence orbit of a matrix pencil.
Linear Algebra Appl., 82:99-121, 1986.

368

E. Polak.
Computational Methods in Optimization.
Academic Press, New York, 1971.

369

C. Pommerell.
Solution of Large Unsymmetric Systems of Linear Equations.
Ph.D. thesis, Swiss Federal Institute of Technology, Zürich, Switzerland, 1992.

370

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.

371

A. Ruhe.
An algorithm for numerical determination of the structure of a general matrix.
BIT, 10:196-216, 1970.

372

A. Ruhe.
Algorithms for the nonlinear eigenvalue problem.
SIAM J. Numer. Anal., 10:674-689, 1973.

373

A. Ruhe.
SOR-methods for the eigenvalue problem with large sparse matrices.
Math. Comp., 28:695-710, 1974.

374

A. Ruhe.
Iterative eigenvalue algorithms based on convergent splittings.
J. Comput. Phys., 19:110-120, 1975.

375

A. Ruhe.
Implementation aspects of band Lanczos algorithms for computation of eigenvalues of large sparse symmetric matrices.
Math. Comp., 33:680-687, 1979.

376

A. Ruhe.
The rational Krylov algorithm for nonsymmetric eigenvalue problems, III: Complex shifts for real matrices.
BIT, 34:165-176, 1994.

377

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.

378

A. Ruhe.
Rational Krylov: A practical algorithm for large sparse nonsymmetric matrix pencils.
SIAM J. Sci. Comput., 19(5):1535-1551, 1998.

379

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.

380

A. Ruhe and T. Wiberg.
The method of conjugate gradients used in inverse iteration.
BIT, 12:543-554, 1972.

381

H. Rutishauser.
Computational aspects of F. L. Bauer's simultaneous iteration method.
Numer. Math., 13:4-13, 1969.

382

H. Rutishauser.
Simultaneous iteration method for symmetric matrices.
Numer. Math., 16:205-223, 1970.
Also in [458, pp. 284-301].

383

Y. Saad.
Chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems.
Math. Comp., 42(166):567-588, 1984.

384

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.

385

Y. Saad.
Numerical solution of large nonsymmetric eigenvalue problems.
Comp. Phys. Comm., 53:71-90, 1989.

386

Y. Saad.
SPARSKIT: A basic tool-kit for sparse matrix computation, version 2, 1994.
Software available at http://www.cs.umn.edu/$\sim$saad

387

Y. Saad.
Numerical Methods for Large Eigenvalue Problems.
Halsted Press, New York, 1992.

388

Y. Saad.
Iterative Methods for Linear Systems.
PWS Publishing, Boston, 1996.

389

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.

390

M. Sadkane.
Block-Arnoldi and Davidson methods for unsymmetric large eigenvalue problems.
Numer. Math., 64:195-211, 1993.

391

M. Sadkane.
A block Arnoldi-Chebyshev method for computing the leading eigenpairs of large sparse unsymmetric matrices.
Numer. Math., 64:181-193, 1993.

392

A. H. Sameh and J. A. Wisniewski.
A trace minimization algorithm for the generalized eigenvalue problem.
SIAM J. Numer. Anal., 19:1243-1259, 1982.

393

B. A. Samokish.
The steepest descent method for an eigenvalue problem with semi-bounded operators.
Izv. Vuzov Math., 5:105-114, 1958.
(In Russian.)

394

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.)

395

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.

396

W. H. A. Schilders.
Personal communication, September 1997.

397

D. S. Scott.
Solving sparse symmetric generalised eigenvalue problems without factorisation.
SIAM J. Numer. Anal., 18:102-110, 1981.

398

J. A. Scott.
An Arnoldi code for computing selected eigenvalues of sparse unsymmetric matrices.
ACM Trans. Math. Software, 21:432-475, 1995.

399

N. S. Sehmi.
A Newtonian procedure for the solution of the Kron characteristic value problem.
J. Sound Vibration, 100(3):409-421, 1985.

400

N. S. Sehmi.
Large order structural eigenanalysis techniques.
Ellis Horwood Series: Mathematics and Its Applications. Ellis Horwood, Chichester, UK, 1989.

401

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.

402

H. Simon.
Analysis of the symmetric Lanczos algorithm with reorthogonalization methods.
Linear Algebra Appl., 61:101-132, 1984.

403

H. Simon.
The Lanczos algorithm with partial reorthogonalization.
Math. Comp., 42:115-142, 1984.

404

A. Simpson and B. Tabarrok.
On Kron's eigenvalue procedure and related methods of frequency analysis.
Quart. J. Mech. Appl. Math., 21:1-39, 1968.

405

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.

406

I. Slapnicar.
Accurate Symmetric Reduction by a Jacobi Method.
Ph.D. thesis, Fernuniversität Gesamthochschule Hagen, Germany, 1992.

407

I. Slapnicar.
Accurate computation of singular values and eigenvalues of symmetric matrices.
Math. Commun., 1:153-168, 1996.

408

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.

409

G. L. G. Sleijpen and D. R. Fokkema.
Bi-CGSTAB($\ell$) methods for linear equations involving matrices with complex spectrum.
Electron. Trans. Numer. Anal., 1:11-32, 1993.

410

G. L. G. Sleijpen, D. R. Fokkema, and H. A van der Vorst.
BiCGSTAB($\ell$) and other hybrid Bi-CG methods.
Numer. Algorithms, 7:75-109, 1994.

411

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.

412

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.

413

G. L. G. Sleijpen, H. A. van der Vorst, and M. B. van Gijzen.
Quadratic eigenproblems are no problem.
SIAM News, 29:8-9, 1996.

414

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.

415

B. F. Smith, P. E. Bjørstad, and W. D. Gropp.
Domain decomposition.
Cambridge University Press, Cambridge, UK, 1996.

416

S. T. Smith.
Optimization techniques on Riemannian manifolds.
Fields Inst. Commun., 3:113-146, 1994.

417

E. Snapper and R. Troyer.
Metric Affine Geometry.
Academic Press, New York, 1971.

418

P. Sonneveld.
CGS: A fast Lanczos-type solver for nonsymmetric linear systems.
SIAM J. Sci. Statist. Comput., 10:36-52, 1989.

419

D. C. Sorensen.
Implicit application of polynomial filters in a $k$-step Arnoldi method.
SIAM J. Matrix Anal. Appl., 13:357-385, 1992.

420

D. C. Sorensen.
Deflation for implicitly restarted Arnoldi methods.
Technical Report TR98-12, Department of Computational and Applied Mathematics, Rice University, Houston, TX, 1998.

421

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.

422

G. W. Stewart.
Simultaneous iteration for computing invariant subspaces of non-Hermitian matrices.
Numer. Math., 25:123-136, 1976.

423

G. W. Stewart.
Perturbation bounds for the definite generalized eigenvalue problem.
Linear Algebra Appl., 23:69-86, 1979.

424

G. W. Stewart.
Computing the CS decomposition of a partitioned orthogonal matrix.
Numer. Math., 40:297-306, 1982.

425

G. W. Stewart and J.-G. Sun.
Matrix Perturbation Theory.
Academic Press, New York, 1990.

426

W. J. Stewart and A. Jennings.
Algorithm 570: LOPSI a simultaneous iteration method for real matrices.
ACM Trans. Math. Software, 7:230-232, 1981.

427

W. J. Stewart and A. Jennings.
A simultaneous iteration algorithm for real matrices.
ACM Trans. Math. Software, 7:184-198, 1981.

428

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.

429

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.

430

J.-G. Sun.
Perturbation bounds for eigenspaces of a definite matrix pair.
Numer. Math., 41:321-343, 1983.

431

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.

432

J. G. Sun.
Perturbation analysis of quadratic eigenvalue problems.
BIT, 1999, submitted.

433

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.

434

D. R. Taylor.
Analysis of the Look-Ahead Lanczos Algorithm.
Ph.D. thesis, University of California, Berkeley, 1982.

435

F. Tisseur.
Backward error and condition of polynomial eigenvalue problems.
Linear Algebra Appl., 309:339-361, 2000.

436

F. Tisseur.
Stability of structured Hamiltonian eigensolvers.
Numerical Analysis Report No. 357, Manchester Centre for Computational Mathematics, Manchester, UK, February 2000.

437

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.

438

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.

439

S. Toledo.
Improving the memory-system performance of sparse-matrix vector multiplication.
IBM J. Res. Develop., 41(6):711-726, 1997.

440

L. N. Trefethen.
Computation of pseudospectra.
In A. Iserles, editor, Acta Numerica, Volume 8, pages 247-295. Cambridge University Press, Cambridge, MA, 1999.

441

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.

442

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.

443

H. van der Veen and C. Vuik.
Bi-Lanczos with partial orthogonalization.
Computers and Structures, 56:605-613, 1995.

444

H. A. van der Vorst.
A generalized Lanczos scheme.
Math. Comp., 39:559-561, 1982.

445

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.

446

P. Van Dooren.
The computation of Kronecker's canonical form of a singular pencil.
Linear Algebra Appl., 27:103-141, 1979.

447

P. Van Dooren.
The generalized eigenstructure problem in linear system theory.
IEEE Trans. Automat. Control, AC-26(1):111-129, 1981.

448

P. Van Dooren.
A generalized eigenvalue approach for solving Ricatti equations.
SIAM J. Sci. Comput., 2:121-135, 1981.

449

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.

450

P. Van Dooren.
Reducing subspaces: Computational aspects and applications in linear systems theory.
In Proceedings of the $5$th Int. Conf. on Analysis and Optimization of Systems, 1982, Lecture Notes on Control and Information Sciences. Volume 44. Springer-Verlag, New York, 1983.

451

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.

452

C. F. Van Loan.
A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix.
Linear Algebra Appl., 61:233-251, 1984.

453

C. F. Van Loan.
Computational Frameworks for the Fast Fourier Transform.
SIAM, Philadelphia, 1992.

454

E. L. Wachspress.
Iterative solution of elliptic systems, and applications to the neutron diffusion equations of reactor physics.
Prentice-Hall, Englewood Cliffs, NJ, 1966.

455

H. F. Walker.
Implementation of the GMRES method using Householder transformations.
SIAM J. Sci. Statist. Comput., 9:152-163, 1988.

456

W. Waterhouse.
The codimension of singular matrix pairs.
Linear Algebra Appl., 57:227-245, 1984.

457

J. H. Wilkinson.
The Algebraic Eigenvalue Problem.
Clarendon Press, Oxford, UK, 1965.

458

J. H. Wilkinson and C. Reinsch.
Handbook for Automatic Computation. Vol. II, Linear Algebra.
Springer-Verlag, New York, 1971.

459

Y.-C. Wong.
Differential geometry of Grassmann manifolds.
Proc. Nat. Acad. Sci. USA, 57:589-594, 1967.

460

M. Wonham.
Linear Multivariable Control Theory: A Geometric Approach.
Springer-Verlag, New York, second edition, 1979.

461

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.

462

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.

463

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.

464

H. Yang.
Conjugate gradient methods for the Rayleigh quotient minimization of generalized eigenvalue problems.
Computing, 51(1):79-94, 1993.

465

Q. Ye.
A breakdown-free variation of the nonsymmetric Lanczos algorithms.
Math. Comp., 62:179-207, 1994.

466

H. Zha and H. Simon.
On updating problems in latent semantic indexing.
SIAM J. Sci. Comput., 21:782-791, 1999.

467

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.

468

T. Zhang, G. H. Golub, and K. H. Law.
Subspace iterative methods for eigenvalue problems.
Linear Algebra Appl., 294(1-3):239-258, 1999.

469

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.

470

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.)

471

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.

472

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.

473

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.