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  • Acta Numerica, Volume 1
  • 1992, pp. 57-100

Iterative solution of linear systems

  • Roland W. Freund (a1), Gene H. Golub (a2) and Noël M. Nachtigal (a3)
  • DOI:
  • Published online: 01 November 2008

Recent advances in the field of iterative methods for solving large linear systems are reviewed. The main focus is on developments in the area of conjugate gradient-type algorithms and Krylov subspace methods for nonHermitian matrices.

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A.W. Appel (1985), ‘An efficient program for many-body simulation’, SIAM J. Sci. Statist. Comput. 6, 85103.

S.F. Ashby , T.A. Manteuffel and P.E. Saylor (1990), ‘A taxonomy for conjugate gradient methods’, SIAM J. Numer. Anal. 27, 15421568.

O. Axelsson (1980), ‘Conjugate gradient type methods for unsymmetric and inconsistent systems of linear equations’, Lin. Alg. Appl. 29, 116.

O. Axelsson (1985), ‘A survey of preconditioned iterative methods for linear systems of algebraic equations’, BIT 25, 166187.

O. Axelsson (1987), ‘A generalized conjugate gradient, least square method’, Numer. Math. 51, 209227.

I.M. Barbour , N.-E. Behilil , P.E. Gibbs , M. Rafiq , K.J.M. Moriarty and G. Schierholz (1987), ‘Updating fermions with the Lanczos method’, J. Comput. Phys. 68, 227236.

A. Bayliss and C.I. Goldstein (1983), ‘An iterative method for the Helmholtz equation’, J. Comput. Phys. 49, 443457.

D.L. Boley and G.H. Golub (1991), ‘The nonsymmetric Lanczos algorithm and controllability’, Systems Control Lett. 16, 97105

D.L. Boley , S. Elhay , G.H. Golub and M.H. Gutknecht (1991), ‘Nonsymmetric Lanczos and finding orthogonal polynomials associated with indefinite weights’, Numer. Algorithms 1, 2143.

C.G. Broyden (1965), ‘A class of methods for solving nonlinear simultaneous equations’, Math. Comput. 19, 577593.

J. Carrier , L. Greengard and V. Rokhlin (1988), ‘A fast adaptive multipole algorithm for particle simulations’, SIAM J. Sci. Stat. Comput. 9, 669686.

R.H. Chan and G. Strang (1989), ‘Toeplitz equations by conjugate gradients with circulant preconditioner’, SIAM J. Sci. Stat. Comput. 10, 104119.

P. Concus and G.H. Golub (1976), ‘A generalized conjugate gradient method for nonsymmetric systems of linear equations’, in Computing Methods in Applied Sciences and Engineering (Lecture Notes in Economics and Mathematical Systems 134) (R. Glowinski and J.L. Lions , eds), Springer (Berlin) 5665.

E.J. Craig (1955), ‘The N-step iteration procedures’, J. Math. Phys. 34, 6473.

J. Cullum and R.A. Willoughby (1985), Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Volume 1, Theory, Birkhäuser (Basel).

P. Deuflhard , R.W. Freund and A. Walter (1990), ‘Fast secant methods for the iterative solution of large nonsymmetric linear systems’, IMPACT Comput. Sci. Eng. 2, 244276.

A. Draux (1983), Polynômes Orthogonaux Formels – Applications, (Lecture Notes in Mathematics 974) Springer (Berlin).

M. Eiermann , W. Niethammer and R.S. Varga (1985), ‘A study of semiiterative methods for nonsymmetric systems of linear equations’, Numer. Math. 47, 505533.

T. Eirola and O. Nevanlinna (1989), ‘Accelerating with rank-one updates’, Lin. Alg. Appl. 121, 511520.

S.C. Eisenstat (1983a), ‘A note on the generalized conjugate gradient method’, SIAM J. Numer. Anal. 20, 358361.

S.C. Eisenstat , H.C. Elman and M.H. Schultz (1983), ‘Variational iterative methods for nonsymmetric systems of linear equations’, SIAM J. Numer. Anal. 20, 345357.

V. Faber and T. Manteuffel (1984), ‘Necessary and sufficient conditions for the existence of a conjugate gradient method’, SIAM J. Numer. Anal. 21, 352362.

V. Faber and T. Manteuffel (1987), ‘Orthogonal error methods’, SIAM J. Numer. Anal. 24, 170187.

B. Fischer and R.W. Freund (1990), ‘On the constrained Chebyshev approximation problem on ellipses’, J. Approx. Theory 62, 297315.

B. Fischer and R.W. Freund (1991), ‘Chebyshev polynomials are not always optimal’, J. Approx. Theory 65, 261272.

R.W. Freund (1990), ‘On conjugate gradient type methods and polynomial preconditioners for a class of complex nonHermitian matrices’, Numer. Math. 57, 285312.

R.W. Freund (1992), ‘Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices’, SIAM J. Sci. Stat. Comput. 13, to appear.

R.W. Freund and N.M. Nachtigal (1991), ‘QMR: a quasi-minimal residual method for nonHermitian linear systems’, Numer. Math., to appear.

R.W. Freund and St. Ruscheweyh (1986), ‘On a class of Chebyshev approximation problems which arise in connection with a conjugate gradient type method’, Numer. Math. 48, 525542.

V.M. Fridman (1963), ‘The method of minimum iterations with minimum errors for a system of linear algebraic equations with a symmetrical matrix’, USSR Comput. Math, and Math. Phys. 2, 362363.

G.H. Golub and D.P. O'Leary (1989), ‘Some history of the conjugate gradient and Lanczos algorithms: 1948–1976’, SIAM Review 31, 50102.

G.H. Golub and C.F. Van Loan (1989), Matrix Computations, second edition, The Johns Hopkins University Press (Baltimore).

G.H. Golub and R.S. Varga (1961), ‘Chebyshev semi-iterative methods, successive overrelaxation iterative methods, and second order Richardson iterative methods’, Numer. Math. 3, 147168.

W.B. Gragg (1974), ‘Matrix interpretations and applications of the continued fraction algorithm’, Rocky Mountain J. Math. 4, 213225.

W.B. Gragg and A. Lindquist (1983), ‘On the partial realization problem’, Lin. Alg. Appl. 50, 277319.

P. Hanrahan , D. Salzman and L. Aupperle (1991), ‘A rapid hierarchical radiosity algorithm’, Computer Graphics (Proc. SIGGRAPH '91) 25, 197206.

M.T. Heath , E. Ng and B.W. Peyton (1991), ‘Parallel algorithms for sparse linear systems’, SIAM Review 33, 420460.

G. Heinig and K. Rost (1984), ‘Algebraic methods for Toeplitz-like matrices and operators’, Birkhauser (Basel).

M.R. Hestenes and E. Stiefel (1952), ‘Methods of conjugate gradients for solving linear systems’, J. Res. Natl Bur. Stand. 49, 409436.

W.D. Joubert and D.M. Young (1987), ‘Necessary and sufficient conditions for the simplification of generalized conjugate-gradient algorithms’, Lin. Alg. Appl. 88/89, 449485.

I.M. Khabaza (1963), ‘An iterative least-square method suitable for solving large sparse matrices’, Comput. J. 6, 202206.

C. Lanczos (1950), ‘An iteration method for the solution of the eigenvalue problem of linear differential and integral operators’, J. Res. Natl Bur. Stand. 45, 255282.

C. Lanczos (1952), ‘Solution of systems of linear equations by minimized iterations’, J. Res. Natl Bur. Stand. 49, 3353.

D.G. Luenberger (1969), ‘Hyperbolic pairs in the method of conjugate gradients’, SIAM J. Appl. Math. 17, 12631267.

T.A. Manteuffel (1977), ‘The Tchebychev iteration for nonsymmetric linear systems’, Numer. Math. 28, 307327.

T.A. Manteuffel (1978), ‘Adaptive procedure for estimating parameters for the nonsymmetric Tchebychev iteration’, Numer. Math. 31, 183208.

C.C. Paige and M.A. Saunders (1975), ‘Solution of sparse indefinite systems of linear equations’, SIAM J. Numer. Anal. 12, 617629.

C.C. Paige and M.A. Saunders (1982), ‘LSQR: an algorithm for sparse linear equations and sparse least squares’, ACM Trans. Math. Softw. 8, 4371.

B.N. Parlett , D.R. Taylor and Z.A. Liu (1985), ‘A look-ahead Lanczos algorithm for unsymmetric matrices’, Math. Comput. 44, 105124.

V. Rokhlin (1985), ‘Rapid solution of integral equations of classical potential theory’, J. Comput. Phys. 60, 187207.

A. Ruhe (1987), ‘Closest normal matrix finally found!’, BIT 27, 585598.

Y. Saad (1980), ‘Variations of Arnoldi's method for computing eigenelements of large unsymmetric matrices’, Lin. Alg. Appl. 34, 269295.

Y. Saad (1981), ‘Krylov subspace methods for solving large unsymmetric linear systems’, Math. Comput. 37, 105126.

Y. Saad (1982), ‘The Lanczos bi-orthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems’, SIAM J. Numer. Anal. 19, 485506.

Y. Saad (1984), ‘Practical use of some Krylov subspace methods for solving indefinite and nonsymmetric linear systems’, SIAM J. Sci. Stat Comput. 5, 203227.

Y. Saad (1989), ‘Krylov subspace methods on supercomputers’, SIAM J. Sci. Stat. Comput. 10, 12001232.

Y. Saad and M.H. Schultz (1985), ‘Conjugate gradient-like algorithms for solving nonsymmetric linear systems’, Math. Comput. 44, 417424.

Y. Saad and M.H. Schultz (1986), ‘GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems’, SIAM J. Sci. Stat. Comput. 7, 856869.

P. Sonneveld (1989), ‘CGS, a fast Lanczos-type solver for nonsymmetric linear systems’, SIAM J. Sci. Stat. Comput. 10, 3652.

E. Stiefel (1955), ‘Relaxationsmethoden bester Strategie zur Lösung linearer Gleichungssysteme’, Comm. Math. Helv. 29, 157179.

J. Stoer (1983), ‘Solution of large linear systems of equations by conjugate gradient type methods’, in Mathematical Programming – The State of the Art (A. Bachem , M. Grötschel and B. Korte , eds.), Springer (Berlin) 540565.

V.V. Voevodin (1983), ‘The problem of a nonselfadjoint generalization of the conjugate gradient method has been closed’, USSR Comput. Math. Math. Phys. 23, 143144.

O. Widlund (1978), ‘A Lanczos method for a class of nonsymmetric systems of linear equations’, SIAM J. Numer. Anal. 15, 801812.

D.M. Young and K.C. Jea (1980), ‘Generalized conjugate gradient acceleration of nonsymmetrizable iterative methods’, Lin. Alg. Appl. 34, 159194.

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  • EISSN: 1474-0508
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