Skip to main content
×
Home
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 104
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Selvitopi, Oguz and Aykanat, Cevdet 2016. Reducing latency cost in 2D sparse matrix partitioning models. Parallel Computing, Vol. 57, p. 1.


    Arioli, Mario and Duff, Iain S. 2015. Preconditioning Linear Least-Squares Problems by Identifying a Basis Matrix. SIAM Journal on Scientific Computing, Vol. 37, Issue. 5, p. S544.


    Xu, Wei and Coleman, Thomas F. 2014. Solving nonlinear equations with the Newton–Krylov method based on automatic differentiation. Optimization Methods and Software, Vol. 29, Issue. 1, p. 88.


    El Akkraoui, Amal Trémolet, Yannick and Todling, Ricardo 2013. Preconditioning of variational data assimilation and the use of a bi-conjugate gradient method. Quarterly Journal of the Royal Meteorological Society, Vol. 139, Issue. 672, p. 731.


    Freund, Roland 2013. Handbook of Linear Algebra, Second Edition.


    Jia, Zhongxiao and Zhang, Qian 2013. An Approach to Making SPAI and PSAI Preconditioning Effective for Large Irregular Sparse Linear Systems. SIAM Journal on Scientific Computing, Vol. 35, Issue. 4, p. A1903.


    Shin, Wonseok and Fan, Shanhui 2013. Accelerated solution of the frequency-domain Maxwell’s equations by engineering the eigenvalue distribution of the operator. Optics Express, Vol. 21, Issue. 19, p. 22578.


    Sadok, Hassane and Szyld, Daniel B. 2012. A new look at CMRH and its relation to GMRES. BIT Numerical Mathematics, Vol. 52, Issue. 2, p. 485.


    Du, Lei Sogabe, Tomohiro and Zhang, Shao-Liang 2011. A variant of the IDR(<mml:math altimg="si11.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>s</mml:mi></mml:math>) method with the quasi-minimal residual strategy. Journal of Computational and Applied Mathematics, Vol. 236, Issue. 5, p. 621.


    Mei, Ting Thornquist, Heidi Keiter, Eric and Hutchinson, Scott 2011. 2011 IEEE/ACM International Conference on Computer-Aided Design (ICCAD). p. 361.

    Mukherjee, Souvik and Everett, Mark E. 2011. 3D controlled-source electromagnetic edge-based finite element modeling of conductive and permeable heterogeneities. GEOPHYSICS, Vol. 76, Issue. 4, p. F215.


    Akkraoui, Amal El and Gauthier, Pierre 2010. Convergence properties of the primal and dual forms of variational data assimilation. Quarterly Journal of the Royal Meteorological Society, Vol. 136, Issue. 646, p. 107.


    Hickey, Mark S. Everett, Mark E. Helwig, Stefan L. and Mogilatov, Vladimir S. 2010. SEG Technical Program Expanded Abstracts 2010. p. 3914.

    Scott, Ian Vukovic, Ana and Sewell, Phillip 2010. Krylov Acceleration Techniques for Time-Reversal Design Applications. IEEE Transactions on Microwave Theory and Techniques, Vol. 58, Issue. 4, p. 917.


    Jia, Zhongxiao and Zhu, Baochen 2009. A power sparse approximate inverse preconditioning procedure for large sparse linear systems. Numerical Linear Algebra with Applications, Vol. 16, Issue. 4, p. 259.


    Brown, Peter N. Walker, Homer F. Wasyk, Rebecca and Woodward, Carol S. 2008. On Using Approximate Finite Differences in Matrix-Free Newton–Krylov Methods. SIAM Journal on Numerical Analysis, Vol. 46, Issue. 4, p. 1892.


    Crouzeix, Michel 2008. A functional calculus based on the numerical range: applications. Linear and Multilinear Algebra, Vol. 56, Issue. 1-2, p. 81.


    Pawlowski, Roger P. Simonis, Joseph P. Walker, Homer F. and Shadid, John N. 2008. Inexact Newton Dogleg Methods. SIAM Journal on Numerical Analysis, Vol. 46, Issue. 4, p. 2112.


    Wang, H. Liu, F. Xia, L. Li, B. K. and Crozier, S. 2008. 2008 30th Annual International Conference of the IEEE Engineering in Medicine and Biology Society. p. 5636.

    Hung, Kun-Chien and Lin, David W. 2007. Theory and design of near-optimal MIMO OFDM transmission system for correlated multipath Rayleigh fading channels. Journal of Communications and Networks, Vol. 9, Issue. 2, p. 150.


    ×
  • 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: http://dx.doi.org/10.1017/S0962492900002245
  • Published online: 01 November 2008
Abstract

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.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Acta Numerica
  • ISSN: 0962-4929
  • EISSN: 1474-0508
  • URL: /core/journals/acta-numerica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×