Previous: Notation
Up: Contents
Next: Index
Previous Page: Notation
Next Page: Index

References

1

J. AARDEN AND K.-E. KARLSSON, Preconditioned CG-type methods for solving the coupled systems of fundamental semiconductor equations, BIT, 29 (1989), pp. 916-937.

2

L. ADAMS AND H. JORDAN, Is SOR color-blind?, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 490-506.

3

E. ANDERSON, ET. AL., LAPACK Users Guide, SIAM, Philadelphia, 1992.

4

J. APPLEYARD AND I. CHESHIRE, Nested factorization, in Reservoir Simulation Symposium of the SPE, 1983. Paper 12264.

5

M. ARIOLI, J. DEMMEL, AND I. DUFF, Solving sparse linear systems with sparse backward error, SIAM J. Matrix Anal. Appl., 10 (1989), pp. 165-190.

6

W. ARNOLDI, The principle of minimized iterations in the solution of the matrix eigenvalue problem, Quart. Appl. Math., 9 (1951), pp. 17-29.

7

S. ASHBY, CHEBYCODE: A Fortran implementation of Manteuffel's adaptive Chebyshev algorithm, Tech. Report UIUCDCS-R-85-1203, University of Illinois, 1985.

8

S. ASHBY, T. MANTEUFFEL, AND J. OTTO, A comparison of adaptive Chebyshev and least squares polynomial preconditioning for Hermitian positive definite linear systems, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 1-29.

9

S. ASHBY, T. MANTEUFFEL, AND P. SAYLOR, Adaptive polynomial preconditioning for Hermitian indefinite linear systems, BIT, 29 (1989), pp. 583-609.

10

S. F. ASHBY, T. A. MANTEUFFEL, AND P. E. SAYLOR, A taxonomy for conjugate gradient methods, SIAM J. Numer. Anal., 27 (1990), pp. 1542-1568.

11

C. ASHCRAFT AND R. GRIMES, On vectorizing incomplete factorizations and SSOR preconditioners, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 122-151.

12

O. AXELSSON, Incomplete block matrix factorization preconditioning methods. The ultimate answer?, J. Comput. Appl. Math., 12&13 (1985), pp. 3-18.

13

____, A general incomplete block-matrix factorization method, Linear Algebra Appl., 74 (1986), pp. 179-190.

14

O. AXELSSON AND A. BARKER, Finite element solution of boundary value problems. Theory and computation, Academic Press, Orlando, Fl., 1984.

15

O. AXELSSON AND V. EIJKHOUT, Vectorizable preconditioners for elliptic difference equations in three space dimensions, J. Comput. Appl. Math., 27 (1989), pp. 299-321.

16

____, The nested recursive two-level factorization method for nine-point difference matrices, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 1373-1400.

17

O. AXELSSON AND I. GUSTAFSSON, Iterative solution for the solution of the Navier equations of elasticity, Comput. Methods Appl. Mech. Engrg., 15 (1978), pp. 241-258.

18

O. AXELSSON AND G. LINDSKOG, On the eigenvalue distribution of a class of preconditioning matrices, Numer. Math., 48 (1986), pp. 479-498.

19

____, On the rate of convergence of the preconditioned conjugate gradient method, Numer. Math., 48 (1986), pp. 499-523.

20

O. AXELSSON AND B. POLMAN, On approximate factorization methods for block-matrices suitable for vector and parallel processors, Linear Algebra Appl., 77 (1986), pp. 3-26.

21

O. AXELSSON AND P. VASSILEVSKI, Algebraic multilevel preconditioning methods, I, Numer. Math., 56 (1989), pp. 157-177.

22

____, Algebraic multilevel preconditioning methods, II, SIAM J. Numer. Anal., 27 (1990), pp. 1569-1590.

23

O. AXELSSON AND P. S. VASSILEVSKI, A black box generalized conjugate gradient solver with inner iterations and variable-step preconditioning, SIAM J. Matrix Anal. Appl., 12 (1991), pp. 625-644.

24

R. BANK, Marching algorithms for elliptic boundary value problems; II: The variable coefficient case, SIAM J. Numer. Anal., 14 (1977), pp. 950-970.

25

R. BANK AND T. CHAN, An analysis of the composite step Bi-conjugate gradient method, Tech. Report CAM 92-, UCLA, Dept. of Math., Los Angeles, CA 90024-1555, 1992.

26

R. BANK, T. CHAN, W. COUGHRAN JR., AND R. SMITH, The Alternate-Block-Factorization procedure for systems of partial differential equations, BIT, 29 (1989), pp. 938-954.

27

R. BANK AND D. ROSE, Marching algorithms for elliptic boundary value problems. I: The constant coefficient case, SIAM J. Numer. Anal., 14 (1977), pp. 792-829.

28

R. E. BANK AND T. F. CHAN, A composite step bi-conjugate gradient algorithm for nonsymmetric linear systems, Tech. Report CAM 92-, UCLA, Dept. of Math., Los Angeles, CA 90024-1555, 1992.

29

G. BAUDET, Asynchronous iterative methods for multiprocessors, J. Assoc. Comput. Mach., 25 (1978), pp. 226-244.

30

R. BEAUWENS, On Axelsson's perturbations, Linear Algebra Appl., 68 (1985), pp. 221-242.

31

____, Approximate factorizations with S/P consistently ordered -factors, BIT, 29 (1989), pp. 658-681.

32

R. BEAUWENS AND L. QUENON, Existence criteria for partial matrix factorizations in iterative methods, SIAM J. Numer. Anal., 13 (1976), pp. 615-643.

33

A. BJÖRCK AND T. ELFVING, Accelerated projection methods for computing pseudo-inverse solutions of systems of linear equations, BIT, 19 (1979), pp. 145-163.

34

D. BRAESS, The contraction number of a multigrid method for solving the Poisson equation, Numer. Math., 37 (1981), pp. 387-404.

35

J. H. BRAMBLE, J. E. PASCIAK, AND A. H. SCHATZ, The construction of preconditioners for elliptic problems by substructuring, I, Mathematics of Computation, 47 (1986), pp. 103- 134.

36

J. H. BRAMBLE, J. E. PASCIAK, J. WANG, AND J. XU, Convergence estimates for product iterative methods with applications to domain decompositions and multigrid, Math. Comp., to appear.

37

R. BRAMLEY AND A. SAMEH, Row projection methods for large nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 168-193.

38

C. BREZINSKI AND H. SADOK, Avoiding breakdown in the CGS algorithm, Numer. Alg., 1 (1991), pp. 199-206.

39

C. BREZINSKI, M. ZAGLIA, AND H. SADOK, Avoiding breakdown and near breakdown in Lanczos type algorithms, Numer. Alg., 1 (1991), pp. 261-284.

40

____, A breakdown free Lanczos type algorithm for solving linear systems, Numer. Math., 63 (1992), pp. 29-38.

41

W. BRIGGS, A Multigrid Tutorial, SIAM, Philadelphia, 1977.

42

X.-C. CAI AND O. WIDLUND, Multiplicative Schwarz algorithms for some nonsymmetric and indefinite problems, SIAM J. Numer. Anal., 30 (1993), pp. 936-952.

43

T. CHAN, Fourier analysis of relaxed incomplete factorization preconditioners, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 668-680.

44

T. CHAN, L. DE PILLIS, AND H. VAN DER VORST, A transpose-free squared Lanczos algorithm and application to solving nonsymmetric linear systems, Tech. Report CAM 91-17, UCLA, Dept. of Math., Los Angeles, CA 90024-1555, 1991.

45

T. CHAN, E. GALLOPOULOS, V. SIMONCINI, T. SZETO, AND C. TONG, A quasi-minimal residual variant of the Bi-CGSTAB algorithm for nonsymmetric systems, Tech. Report CAM 92-26, UCLA, Dept. of Math., Los Angeles, CA 90024-1555, 1992. SIAM J. Sci. Comput., to appear.

46

T. CHAN, R. GLOWINSKI, , J. PéRIAUX, AND O. WIDLUND, eds., Domain Decomposition Methods, Philadelphia, 1989, SIAM. Proceedings of the Second International Symposium on Domain Decomposition Methods, Los Angeles, CA, January 14 - 16, 1988.

47

____, eds., Domain Decomposition Methods, Philadelphia, 1990, SIAM. Proceedings of the Third International Symposium on Domain Decomposition Methods, Houston, TX, 1989.

48

____, eds., Domain Decomposition Methods, SIAM, Philadelphia, 1991. Proceedings of the Fourth International Symposium on Domain Decomposition Methods, Moscow, USSR, 1990.

49

T. CHAN AND C.-C. J. KUO, Two-color Fourier analysis of iterative algorithms for elliptic problems with red/black ordering, SIAM J. Sci. Statist. Comput., 11 (1990), pp. 767-793.

50

T. F. CHAN, T. P. MATHEW, AND J. P. SHAO, Efficient variants of the vertex space domain decomposition algorithm, Tech. Report CAM 92-07, UCLA, Dept. of Math., Los Angeles, CA 90024-1555, 1992. SIAM J. Sci. Comput., to appear.

51

T. F. CHAN AND J. SHAO, On the choice of coarse grid size in domain decomposition methods, tech. report, UCLA, Dept. of Math., Los Angeles, CA 90024-1555, 1993. to appear.

52

D. CHAZAN AND W. MIRANKER, Chaotic relaxation, Linear Algebra Appl., 2 (1969), pp. 199-222.

53

A. CHRONOPOULOS AND C. GEAR, -step iterative methods for symmetric linear systems, J. Comput. Appl. Math., 25 (1989), pp. 153-168.

54

P. CONCUS AND G. GOLUB, A generalized conjugate gradient method for nonsymmetric systems of linear equations, in Computer methods in Applied Sciences and Engineering, Second International Symposium, Dec 15-19, 1975; Lecture Notes in Economics and Mathematical Systems, Vol. 134, Berlin, New York, 1976, Springer-Verlag.

55

P. CONCUS, G. GOLUB, AND G. MEURANT, Block preconditioning for the conjugate gradient method, SIAM J. Sci. Statist. Comput., 6 (1985), pp. 220-252.

56

P. CONCUS, G. GOLUB, AND D. O'LEARY, A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations, in Sparse Matrix Computations, J. Bunch and D. Rose, eds., Academic Press, New York, 1976, pp. 309-332.

57

E. CUTHILL AND J. MCKEE, Reducing the bandwidth of sparse symmetric matrices, in ACM Proceedings of the 24th National Conference, 1969.

58

E. D'AZEVEDO, V. EIJKHOUT, AND C. ROMINE, LAPACK working note 56: Reducing communication costs in the conjugate gradient algorithm on distributed memory multiprocessor, tech. report, Computer Science Department, University of Tennessee, Knoxville, TN, 1993.

59

E. D'AZEVEDO AND C. ROMINE, Reducing communication costs in the conjugate gradient algorithm on distributed memory multiprocessors, Tech. Report ORNL/TM-12192, Oak Ridge National Lab, Oak Ridge, TN, 1992.

60

E. DE STURLER, A parallel restructured version of GMRES(m), Tech. Report 91-85, Delft University of Technology, Delft, The Netherlands, 1991.

61

E. DE STURLER AND D. R. FOKKEMA, Nested Krylov methods and preserving the orthogonality, Tech. Report Preprint 796, Utrecht University, Utrecht, The Netherlands, 1993.

62

S. DEMKO, W. MOSS, AND P. SMITH, Decay rates for inverses of band matrices, Mathematics of Computation, 43 (1984), pp. 491-499.

63

J. DEMMEL, The condition number of equivalence transformations that block diagonalize matrix pencils, SIAM J. Numer. Anal., 20 (1983), pp. 599-610.

64

J. DEMMEL, M. HEATH, AND H. VAN DER VORST, Parallel linear algebra, in Acta Numerica, Vol. 2, Cambridge Press, New York, 1993.

65

S. DOI, On parallelism and convergence of incomplete LU factorizations, Appl. Numer. Math., 7 (1991), pp. 417-436.

66

J. DONGARRA, J. DUCROZ, I. DUFF, AND S. HAMMARLING, A set of level 3 Basic Linear Algebra Subprograms, ACM Trans. Math. Soft., 16 (1990), pp. 1-17.

67

J. DONGARRA, J. DUCROZ, S. HAMMARLING, AND R. HANSON, An extended set of FORTRAN Basic Linear Algebra Subprograms, ACM Trans. Math. Soft., 14 (1988), pp. 1-32.

68

J. DONGARRA, I. DUFF, D. SORENSEN, AND H. VAN DER VORST, Solving Linear Systems on Vector and Shared Memory Computers, SIAM, Philadelphia, PA, 1991.

69

J. DONGARRA AND E. GROSSE, Distribution of mathematical software via electronic mail, Comm. ACM, 30 (1987), pp. 403-407.

70

J. DONGARRA, C. MOLER, J. BUNCH, AND G. STEWART, LINPACK Users' Guide, SIAM, Philadelphia, 1979.

71

J. DONGARRA AND H. VAN DER VORST, Performance of various computers using standard sparse linear equations solving techniques, in Computer Benchmarks, J. Dongarra and W. Gentzsch, eds., Elsevier Science Publishers B.V., New York, 1993, pp. 177-188.

72

F. DORR, The direct solution of the discrete Poisson equation on a rectangle, SIAM Rev., 12 (1970), pp. 248-263.

73

M. DRYJA AND O. B. WIDLUND, Towards a unified theory of domain decomposition algorithms for elliptic problems, Tech. Report 486, also Ultracomputer Note 167, Department of Computer Science, Courant Institute, 1989.

74

D. DUBOIS, A. GREENBAUM, AND G. RODRIGUE, Approximating the inverse of a matrix for use in iterative algorithms on vector processors, Computing, 22 (1979), pp. 257-268.

75

I. DUFF, R. GRIMES, AND J. LEWIS, Sparse matrix test problems, ACM Trans. Math. Soft., 15 (1989), pp. 1-14.

76

I. DUFF AND G. MEURANT, The effect of ordering on preconditioned conjugate gradients, BIT, 29 (1989), pp. 635-657.

77

I. S. DUFF, A. M. ERISMAN, AND J.K.REID, Direct methods for sparse matrices, Oxford University Press, London, 1986.

78

T. DUPONT, R. KENDALL, AND H. RACHFORD, An approximate factorization procedure for solving self-adjoint elliptic difference equations, SIAM J. Numer. Anal., 5 (1968), pp. 559-573.

79

E. D'YAKONOV, The method of variable directions in solving systems of finite difference equations, Soviet Math. Dokl., 2 (1961), pp. 577-580. TOM 138, 271-274.

80

L. EHRLICH, An Ad-Hoc SOR method, J. Comput. Phys., 43 (1981), pp. 31-45.

81

M. EIERMANN AND R. VARGA, Is the optimal best for the SOR iteration method?, Linear Algebra Appl., 182 (1993), pp. 257-277.

82

V. EIJKHOUT, Analysis of parallel incomplete point factorizations, Linear Algebra Appl., 154-156 (1991), pp. 723-740.

83

____, Beware of unperturbed modified incomplete point factorizations, in Proceedings of the IMACS International Symposium on Iterative Methods in Linear Algebra, Brussels, Belgium, R. Beauwens and P. de Groen, eds., 1992.

84

____, LAPACK working note 50: Distributed sparse data structures for linear algebra operations, Tech. Report CS 92-169, Computer Science Department, University of Tennessee, Knoxville, TN, 1992.

85

____, LAPACK working note 51: Qualitative properties of the conjugate gradient and Lanczos methods in a matrix framework, Tech. Report CS 92-170, Computer Science Department, University of Tennessee, Knoxville, TN, 1992.

86

V. EIJKHOUT AND B. POLMAN, Decay rates of inverses of banded -matrices that are near to Toeplitz matrices, Linear Algebra Appl., 109 (1988), pp. 247-277.

87

S. EISENSTAT, Efficient implementation of a class of preconditioned conjugate gradient methods, SIAM J. Sci. Statist. Comput., 2 (1981), pp. 1-4.

88

R. ELKIN, Convergence theorems for Gauss-Seidel and other minimization algorithms, Tech. Report 68-59, Computer Science Center, University of Maryland, College Park, MD, Jan. 1968.

89

H. ELMAN, Approximate Schur complement preconditioners on serial and parallel computers, SIAM J. Sci. Statist. Comput., 10 (1989), pp. 581-605.

90

H. ELMAN AND M. SCHULTZ, Preconditioning by fast direct methods for non self-adjoint nonseparable elliptic equations, SIAM J. Numer. Anal., 23 (1986), pp. 44-57.

91

L. ELSNER, A note on optimal block-scaling of matrices, Numer. Math., 44 (1984), pp. 127-128.

92

V. FABER AND T. MANTEUFFEL, Necessary and sufficient conditions for the existence of a conjugate gradient method, SIAM J. Numer. Anal., 21 (1984), pp. 315-339.

93

G. FAIRWEATHER, A. GOURLAY, AND A. MITCHELL, Some high accuracy difference schemes with a splitting operator for equations of parabolic and elliptic type, Numer. Math., 10 (1967), pp. 56-66.

94

R. FLETCHER, Conjugate gradient methods for indefinite systems, in Numerical Analysis Dundee 1975, G. Watson, ed., Berlin, New York, 1976, Springer Verlag, pp. 73-89.

95

____, Conjugate gradient methods for indefinite systems, vol. 506 of Lecture Notes Math., Springer-Verlag, Berlin, New York, 1976, pp. 73-89.

96

G. FORSYTHE AND E. STRAUSS, On best conditioned matrices, Proc. Amer. Math. Soc., 6 (1955), pp. 340-345.

97

R. FREUND, Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 425-448.

98

R. FREUND, G. GOLUB, AND N. NACHTIGAL, Iterative solution of linear systems, Tech. Report NA-91-05, Stanford University, Stanford, CA, 1991.

99

R. FREUND, M. GUTKNECHT, AND N. NACHTIGAL, An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices, SIAM J. Sci. Comput., 14 (1993), pp. 137-158.

100

R. FREUND AND N. NACHTIGAL, QMR: A quasi-minimal residual method for non-Hermitian linear systems, Numer. Math., 60 (1991), pp. 315-339.

101

____, An implementation of the QMR method based on coupled two-term recurrences, Tech. Report 92.15, RIACS, NASA Ames, Ames, CA, 1992.

102

R. FREUND AND T. SZETO, A quasi-minimal residual squared algorithm for non-Hermitian linear systems, tech. report, RIACS, NASA Ames, Ames, CA, 1991.

103

R. W. FREUND, A transpose-free quasi-minimum residual algorithm for non-Hermitian linear systems, SIAM J. Sci. Comput., 14 (1993), pp. 470-482.

104

R. W. FREUND, G. H. GOLUB, AND N. M. NACHTIGAL, Iterative solution of linear systems, Acta Numerica, (1992), pp. 57-100.

105

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 (1993), pp. 137-158.

106

R. GLOWINSKI, G. H. GOLUB, G. A. MEURANT, AND J. PéRIAUX, eds., Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1988. Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Differential Equations, Paris, France, January 1987.

107

G. GOLUB AND D. O'LEARY, Some history of the conjugate gradient and Lanczos methods, SIAM Rev., 31 (1989), pp. 50-102.

108

G. GOLUB AND C. VAN LOAN, Matrix Computations, second edition, The Johns Hopkins University Press, Baltimore, 1989.

109

A. GREENBAUM AND Z. STRAKOS, Predicting the behavior of finite precision Lanczos and conjugate gradient computations, SIAM J. Mat. Anal. Appl., 13 (1992), pp. 121-137.

110

W. D. GROPP AND D. E. KEYES, Domain decomposition with local mesh refinement, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 967-993.

111

I. GUSTAFSSON, A class of first-order factorization methods, BIT, 18 (1978), pp. 142-156.

112

M. H. GUTKNECHT, A completed theory of the unsymmetric Lanczos process and related algorithms, part II, Tech. Report 90-16, IPS Research Report, ETH Zürich, Switzerland, 1990.

113

____, The unsymmetric Lanczos algorithms and their relations to Páde approximation, continued fractions and the QD algorithm, in Proceedings of the Copper Mountain Conference on Iterative Methods, 1990.

114

____, Variants of Bi-CGSTAB for matrices with complex spectrum, Tech. Report 91-14, IPS ETH, Zürich, Switzerland, 1991.

115

____, A completed theory of the unsymmetric Lanczos process and related algorithms, part I, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 594-639.

116

W. HACKBUSCH, Multi-Grid Methods and Applications, Springer-Verlag, Berlin, New York, 1985.

117

____, Iterative Lösung großer schwachbestetzter Gleichungssysteme, Teubner, Stuttgart, 1991.

118

A. HADJIDIMOS, On some high accuracy difference schemes for solving elliptic equations, Numer. Math., 13 (1969), pp. 396-403.

119

L. HAGEMAN AND D. YOUNG, Applied Iterative Methods, Academic Press, New York, 1981.

120

W. HAGER, Condition estimators, SIAM J. Sci. Statist. Comput., 5 (1984), pp. 311-316.

121

M. HESTENES AND E. STIEFEL, Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Stand., 49 (1952), pp. 409-436.

122

M. R. HESTENES, Conjugacy and gradients, in A History of Scientific Computing, Addison-Wesley, Reading, MA, 1990, pp. 167-179.

123

N. HIGHAM, Experience with a matrix norm estimator, SIAM J. Sci. Statist. Comput., 11 (1990), pp. 804-809.

124

K. JEA AND D. YOUNG, Generalized conjugate-gradient acceleration of nonsym- metrizable iterative methods, Linear Algebra Appl., 34 (1980), pp. 159-194.

125

O. JOHNSON, C. MICCHELLI, AND G. PAUL, Polynomial preconditioning for conjugate gradient calculations, SIAM J. Numer. Anal., 20 (1983), pp. 363-376.

126

____, Polynomial preconditioning for conjugate gradient calculation, SIAM J. Numer. Anal., 20 (1983), pp. 362-376.

127

M. JONES AND P. PLASSMANN, Parallel solution of unstructed, sparse systems of linear equations, in Proceedings of the Sixth SIAM conference on Parallel Processing for Scientific Computing, R. Sincovec, D. Keyes, M. Leuze, L. Petzold, and D. Reed, eds., SIAM, Philadelphia, pp. 471-475.

128

____, A parallel graph coloring heuristic, SIAM J. Sci. Statist. Comput., 14 (1993), pp. 654-669.

129

W. JOUBERT, Lanczos methods for the solution of nonsymmetric systems of linear equations, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 926-943.

130

W. KAHAN, Gauss-Seidel methods of solving large systems of linear equations, PhD thesis, University of Toronto, 1958.

131

S. KANIEL, Estimates for some computational techniques in linear algebra, Mathematics of Computation, 20 (1966), pp. 369-378.

132

D. KERSHAW, The incomplete Cholesky-conjugate gradient method for the iterative solution of systems of linear equations, J. Comput. Phys., 26 (1978), pp. 43-65.

133

R. KETTLER, Analysis and comparison of relaxation schemes in robust multigrid and preconditioned conjugate gradient methods, in Multigrid Methods, Lecture Notes in Mathematics 960, W. Hackbusch and U. Trottenberg, eds., Springer-Verlag, Berlin, New York, 1982, pp. 502-534.

134

____, Linear multigrid methods in numerical reservoir simulation, PhD thesis, Delft University of Technology, Delft, The Netherlands, 1987.

135

D. E. KEYES, T. F. CHAN, G. MEURANT, J. S. SCROGGS, AND R. G. VOIGT, eds., Domain Decomposition Methods For Partial Differential Equations, SIAM, Philadelphia, 1992. Proceedings of the Fifth International Symposium on Domain Decomposition Methods, Norfolk, VA, 1991.

136

D. E. KEYES AND W. D. GROPP, A comparison of domain decomposition techniques for elliptic partial differential equations and their parallel implementation, SIAM J. Sci. Statist. Comput., 8 (1987), pp. s166 -- s202.

137

S. K. KIM AND A. T. CHRONOPOULOS, A class of Lanczos-like algorithms implemented on parallel computers, Parallel Comput., 17 (1991), pp. 763-778.

138

D. R. KINCAID, J. R. RESPESS, D. M. YOUNG, AND R. G. GRIMES, ITPACK 2C: A Fortran package for solving large sparse linear systems by adaptive accelerated iterative methods, ACM Trans. Math. Soft., 8 (1982), pp. 302-322. Algorithm 586.

139

C. LANCZOS, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Nat. Bur. Stand., 45 (1950), pp. 255-282.

140

____, Solution of systems of linear equations by minimized iterations, J. Res. Nat. Bur. Stand., 49 (1952), pp. 33-53.

141

C. LAWSON, R. HANSON, D. KINCAID, AND F. KROGH, Basic Linear Algebra Subprograms for FORTRAN usage, ACM Trans. Math. Soft., 5 (1979), pp. 308-325.

142

J. MAITRE AND F. MUSY, The contraction number of a class of two-level methods; an exact evaluation for some finite element subspaces and model problems, in Multigrid methods, Proceedings, Köln-Porz, 1981, W. Hackbusch and U. Trottenberg, eds., vol. 960 of Lecture Notes in Mathematics, 1982, pp. 535-544.

143

T. MANTEUFFEL, The Tchebychev iteration for nonsymmetric linear systems, Numer. Math., 28 (1977), pp. 307-327.

144

____, An incomplete factorization technique for positive definite linear systems, Mathematics of Computation, 34 (1980), pp. 473-497.

145

S. MCCORMICK, Multilevel Adaptive Methods for Partial Differential Equations, SIAM, Philadelphia, 1989.

146

S. MCCORMICK AND J. THOMAS, The Fast Adaptive Composite grid (FAC) method for elliptic equations, Mathematics of Computation, 46 (1986), pp. 439-456.

147

U. MEIER AND A. SAMEH, The behavior of conjugate gradient algorithms on a multivector processor with a hierarchical memory, J. Comput. Appl. Math., 24 (1988), pp. 13-32.

148

U. MEIER-YANG, Preconditioned conjugate gradient-like methods for nonsymmetric linear systems, tech. report, CSRD, University of Illinois, Urbana, IL, April 1992.

149

J. MEIJERINK AND H. VAN DER VORST, An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix, Mathematics of Computation, 31 (1977), pp. 148-162.

150

____, Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems, J. Comput. Phys., 44 (1981), pp. 134-155.

151

R. MELHEM, Toward efficient implementation of preconditioned conjugate gradient methods on vector supercomputers, Internat. J. Sumpercomput. Appls., 1 (1987), pp. 77-98.

152

G. MEURANT, The block preconditioned conjugate gradient method on vector computers, BIT, 24 (1984), pp. 623-633.

153

____, Multitasking the conjugate gradient method on the CRAY X-MP/48, Parallel Comput., 5 (1987), pp. 267-280.

154

N. NACHTIGAL, S. REDDY, AND L. TREFETHEN, How fast are nonsymmetric matrix iterations?, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 778-795.

155

N. NACHTIGAL, L. REICHEL, AND L. TREFETHEN, A hybrid GMRES algorithm for nonsymmetric matrix iterations, Tech. Report 90-7, MIT, Cambridge, MA, 1990.

156

N. M. NACHTIGAL, A Look-Ahead Variant of the Lanczos Algorithm and its Application to the Quasi-Minimal Residual Methods for Non-Hermitian Linear Systems, PhD thesis, MIT, Cambridge, MA, 1991.

157

Y. NOTAY, Solving positive (semi)definite linear systems by preconditioned iterative methods, in Preconditioned Conjugate Gradient Methods, O. Axelsson and L. Kolotilina, eds., vol. 1457 of Lecture Notes in Mathematics, Nijmegen, 1989, pp. 105-125.

158

____, On the robustness of modified incomplete factorization methods, Internat. J. Comput. Math., 40 (1992), pp. 121-141.

159

D. O'LEARY, The block conjugate gradient algorithm and related methods, Linear Algebra Appl., 29 (1980), pp. 293-322.

160

____, Ordering schemes for parallel processing of certain mesh problems, SIAM J. Sci. Statist. Comput., 5 (1984), pp. 620-632.

161

T. C. OPPE, W. D. JOUBERT, AND D. R. KINCAID, NSPCG user's guide, version 1.0: A package for solving large sparse linear systems by various iterative methods, Tech. Report CNA--216, Center for Numerical Analysis, University of Texas at Austin, Austin, TX, April 1988.

162

J. M. ORTEGA, Introduction to Parallel and Vector Solution of Linear Systems, Plenum Press, New York and London, 1988.

163

C. PAIGE, B. PARLETT, AND H. VAN DER VORST, Approximate solutions and eigenvalue bounds from Krylov subspaces, Linear Algebra Appl., to appear.

164

C. PAIGE AND M. SAUNDERS, Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal., 12 (1975), pp. 617-629.

165

C. C. PAIGE AND M. A. SAUNDERS, LSQR: An algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Soft., 8 (1982), pp. 43-71.

166

G. PAOLINI AND G. RADICATI DI BROZOLO, Data structures to vectorize CG algorithms for general sparsity patterns, BIT, 29 (1989), pp. 703-718.

167

B. PARLETT, The symmetric eigenvalue problem, Prentice-Hall, London, 1980.

168

B. N. PARLETT, D. R. TAYLOR, AND Z. A. LIU, A look-ahead Lanczos algorithm for unsymmetric matrices, Mathematics of Computation, 44 (1985), pp. 105-124.

169

D. PEACEMAN AND J. H.H. RACHFORD, The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math., 3 (1955), pp. 28-41.

170

C. POMMERELL, Solution of large unsymmetric systems of linear equations, PhD thesis, Swiss Federal Institute of Technology, Zürich, Switzerland, 1992.

171

E. POOLE AND J. ORTEGA, Multicolor ICCG methods for vector computers, Tech. Report RM 86-06, Department of Applied Mathematics, University of Virginia, Charlottesville, VA, 1986.

172

A. QUARTERONI, ed., Domain Decomposition Methods, Proceedings of the Sixth International Symposium on Domain Decomposition Methods, Como, Italy,, Providence, RI, 1993, AMS. to appear.

173

G. RADICATI DI BROZOLO AND Y. ROBERT, Vector and parallel CG-like algorithms for sparse non-symmetric systems, Tech. Report 681-M, IMAG/TIM3, Grenoble, France, 1987.

174

J. REID, On the method of conjugate gradients for the solution of large sparse systems of linear equations, in Large Sparse Sets of Linear Equations, J. Reid, ed., Academic Press, London, 1971, pp. 231-254.

175

G. RODRIGUE AND D. WOLITZER, Preconditioning by incomplete block cyclic reduction, Mathematics of Computation, 42 (1984), pp. 549-565.

176

Y. SAAD, The Lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems, SIAM J. Numer. Anal., 19 (1982), pp. 485-506.

177

____, Practical use of some Krylov subspace methods for solving indefinite and nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 5 (1984), pp. 203-228.

178

____, Practical use of polynomial preconditionings for the conjugate gradient method, SIAM J. Sci. Statist. Comput., 6 (1985), pp. 865-881.

179

____, Krylov subspace methods on supercomputers, tech. report, RIACS, Moffett Field, CA, September 1988.

180

____, Preconditioning techniques for indefinite and nonsymmetric linear systems, J. Comput. Appl. Math., 24 (1988), pp. 89-105.

181

____, Krylov subspace methods on supercomputers, SIAM J. Sci. Statist. Comput., 10 (1989), pp. 1200-1232.

182

____, SPARSKIT: A basic tool kit for sparse matrix computation, Tech. Report CSRD TR 1029, CSRD, University of Illinois, Urbana, IL, 1990.

183

____, A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14 (1993), pp. 461-469.

184

Y. SAAD AND M. SCHULTZ, Conjugate gradient-like algorithms for solving nonsymmetric linear systems, Mathematics of Computation, 44 (1985), pp. 417-424.

185

____, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856-869.

186

G. L. G. SLEIJPEN AND D. R. FOKKEMA, Bi-CGSTAB() for linear equations involving unsymmetric matrices with complex spectrum, Tech. Report 772, University of Utrecht, Deptartment of Mathematics, Utrecht, The Netherlands, 1993.

187

B. F. SMITH, Domain decomposition algorithms for partial differential equations of linear elasticity, Tech. Report 517, Department of Computer Science, Courant Institute, 1990.

188

P. SONNEVELD, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 10 (1989), pp. 36-52.

189

R. SOUTHWELL, Relaxation Methods in Theoretical Physics, Clarendon Press, Oxford, 1946.

190

H. STONE, Iterative solution of implicit approximations of multidimensional partial differetntial equations, SIAM J. Numer. Anal., 5 (1968), pp. 530-558.

191

P. SWARZTRAUBER, The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle, SIAM Rev., 19 (1977), pp. 490-501.

192

C. TONG, A comparative study of preconditioned Lanczos methods for nonsymmetric linear systems, Tech. Report SAND91-8240, Sandia Nat. Lab., Livermore, CA, 1992.

193

A. VAN DER SLUIS, Condition numbers and equilibration of matrices, Numer. Math., 14 (1969), pp. 14-23.

194

A. VAN DER SLUIS AND H. VAN DER VORST, The rate of convergence of conjugate gradients, Numer. Math., 48 (1986), pp. 543-560.

195

H. VAN DER VORST, Iterative solution methods for certain sparse linear systems with a non-symmetric matrix arising from PDE-problems, J. Comput. Phys., 44 (1981), pp. 1-19.

196

____, A vectorizable variant of some ICCG methods, SIAM J. Sci. Statist. Comput., 3 (1982), pp. 350-356.

197

____, Large tridiagonal and block tridiagonal linear systems on vector and parallel computers, Parallel Comput., 5 (1987), pp. 45-54.

198

____, (M)ICCG for 2D problems on vector computers, in Supercomputing, A.Lichnewsky and C.Saguez, eds., North-Holland, 1988. Also as Report No.A-17, Data Processing Center, Kyoto University, Kyoto, Japan, December 17, 1986.

199

____, High performance preconditioning, SIAM J. Sci. Statist. Comput., 10 (1989), pp. 1174-1185.

200

____, ICCG and related methods for 3D problems on vector computers, Computer Physics Communications, (1989), pp. 223-235. Also as Report No.A-18, Data Processing Center, Kyoto University, Kyoto, Japan, May 30, 1987.

201

____, The convergence behavior of preconditioned CG and CG-S in the presence of rounding errors, in Preconditioned Conjugate Gradient Methods, O. Axelsson and L. Y. Kolotilina, eds., vol. 1457 of Lecture Notes in Mathematics, Berlin, New York, 1990, Springer-Verlag.

202

____, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 631-644.

203

H. VAN DER VORST AND J. MELISSEN, A Petrov-Galerkin type method for solving where is symmetric complex, IEEE Trans. Magnetics, 26 (1990), pp. 706-708.

204

H. VAN DER VORST AND C. VUIK, GMRESR: A family of nested GMRES methods, Tech. Report 91-80, Delft University of Technology, Faculty of Tech. Math., Delft, The Netherlands, 1991.

205

J. VAN ROSENDALE, Minimizing inner product data dependencies in conjugate gradient iteration, Tech. Report 172178, ICASE, NASA Langley Research Center, 1983.

206

R. VARGA, Matrix Iterative Analysis, Prentice-Hall Inc., Englewood Cliffs, NJ, 1962.

207

P. VASSILEVSKI, Preconditioning nonsymmetric and indefinite finite element matrices, J. Numer. Alg. Appl., 1 (1992), pp. 59-76.

208

V. VOEVODIN, The problem of non-self-adjoint generalization of the conjugate gradient method is closed, U.S.S.R. Comput. Maths. and Math. Phys., 23 (1983), pp. 143-144.

209

H. F. WALKER, Implementation of the GMRES method using Householder transformations, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 152-163.

210

P. WESSELING, An Introduction to Multigrid Methods, Wiley, Chichester, 1991.

211

O. WIDLUND, A Lanczos method for a class of non-symmetric systems of linear equations, SIAM J. Numer. Anal., 15 (1978), pp. 801-812.

212

D. YOUNG, Iterative solution of large linear systems, Academic Press, New York, 1971.

213

H. YSERENTANT, On the multilevel splitting of finite element spaces, Numer. Math., 49 (1986), pp. 379-412.