An Implementation of the Look-Ahead Lanczos Algorithm for Non-Hermitian Matrices
Abstract
The nonsymmetric Lanczos method can be used to compute eigenvalues of large sparse non-Hermitian matrices or to solve large sparse non-Hermitian linear systems. However, the original Lanczos algorithm is susceptible to possible breakdowns and potential instabilities. An implementation of a look-ahead version of the Lanczos algorithm is presented that, except for the very special situation of an incurable breakdown, overcomes these problems by skipping over those steps in which a breakdown or near-breakdown would occur in the standard process. The proposed algorithm can handle look-ahead steps of any length and requires the same number of matrix–vector products and inner products as the standard Lanczos process without look-ahead.
MSC codes
Keywords
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Daniel L. Boley, Sylvan Elhay, Gene H. Golub, Martin H. Gutknecht, Nonsymmetric Lanczos and finding orthogonal polynomials associated with indefinite weights, Numer. Algorithms, 1 (1991), 21–43
2.
D. L. Boley, G. H. Golub, The nonsymmetric Lanczos algorithm and controllability, Systems Control Lett., 16 (1991), 97–105
3.
Claude Brezinski, Padé-type approximation and general orthogonal polynomials, International Series of Numerical Mathematics, Vol. 50, Birkhäuser Verlag, Basel, 1980, 250–
4.
C. Brezinski, M. Redivo Zaglia, H. Sadok, A breakdown-free Lanczos type algorithm for solving linear systems, Tech. Rep., ANO-239, Université des Sciences et Techniques de Lille Flandres-Artois, France, 1991, Jan.
5.
C. Brezinski, M. Redivo Zaglia, H. Sadok, Avoiding breakdown and near-breakdown in Lanczos type algorithms, Numer. Algorithms, 1 (1991), 261–284
6.
Jane Cullum, Ralph A. Willoughby, J. Cullum, R. A. Willoughby, A practical procedure for computing eigenvalues of large sparse nonsymmetric matricesLarge scale eigenvalue problems (Oberlech, 1985), North-Holland Math. Stud., Vol. 127, North-Holland, Amsterdam, 1986, 193–240
7.
A. Draux, Polynômes orthogonaux formels, Lecture Notes in Mathematics, Vol. 974, Springer-Verlag, Berlin, 1983vi+625
8.
I. S. Duff, R. G. Grimes, J. G. Lewis, Sparse matrix test problems, ACM Trans. Math. Software, 15 (1989), 1–14
9.
Roland W. Freund, Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices, SIAM J. Sci. Statist. Comput., 13 (1992), 425–448
10.
R. W. Freund, G. H. Golub, M. Hochbruck, Krylov subspace methods for non-Hermitian p-cyclic matrices, Tech. Rep., RIACS, NASA Ames Research Center, Moffett Field, CA, in preparation
11.
R. W. Freund, M. H. Gutknecht, N. M. Nachtigal, An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices, Part I, Tech. Rep., 90.45, RIACS, NASA Ames Research Center, Moffett Field, CA, 1990, November
12.
R. W. Freund, N. M. Nachtigal, An implementation of the look-ahead Lanczos algorithm for non- Hermitian matrices, Part II, Tech. Rep., 90.46, RIACS, NASA Ames Research Center, Moffett Field, CA, 1990, November
13.
R. W. Freund, N. M. Nachtigal, QMR: a quasi-minimal residual method for non-Hermitian linear systems, Numer. Math., 60 (1991), 315–339
14.
William B. Gragg, Matrix interpretations and applications of the continued fraction algorithm, Rocky Mountain J. Math., 4 (1974), 213–225
15.
William B. Gragg, Anders Lindquist, On the partial realization problem, Linear Algebra Appl., 50 (1983), 277–319
16.
Martin H. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms. I, SIAM J. Matrix Anal. Appl., 13 (1992), 594–639
17.
M. H. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms, Part II, IPS Research Report, 90–16, Zürich, Switzerland, 1990, Septemper
18.
Wayne Joubert, Lanczos methods for the solution of nonsymmetric systems of linear equations, SIAM J. Matrix Anal. Appl., 13 (1992), 926–943
19.
S.-Y. Kung, Ph.D. Thesis, Multivariable and multidimensional systems: analysis and design, Dept. of Electrical Engineering, Stanford University, Stanford, 1977, June
20.
Cornelius Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Research Nat. Bur. Standards, 45 (1950), 255–282
21.
Cornelius Lanczos, Solution of systems of linear equations by minimized-iterations, J. Research Nat. Bur. Standards, 49 (1952), 33–53
22.
Thomas A. Manteuffel, The Tchebychev iteration for nonsymmetric linear systems, Numer. Math., 28 (1977), 307–327
23.
Beresford N. Parlett, Reduction to tridiagonal form and minimal realizations, SIAM J. Matrix Anal. Appl., 13 (1992), 567–593
24.
Beresford N. Parlett, Youcef Saad, Complex shift and invert strategies for real matrices, Linear Algebra Appl., 88/89 (1987), 575–595
25.
Beresford N. Parlett, Derek R. Taylor, Zhishun A. Liu, A look-ahead Lánczos algorithm for unsymmetric matrices, Math. Comp., 44 (1985), 105–124
26.
Eduard L. Stiefel, Kernel polynomials in linear algebra and their numerical applications, Nat. Bur. Standards Appl. Math. Ser., 1958 (1958), 1–22
27.
D. R. Taylor, Ph.D. Thesis, Analysis of the look ahead Lanczos algorithm, Dept. of Mathematics, University of California, Berkeley, 1982, November
28.
Homer F. Walker, Implementation of the GMRES method using Householder transformations, SIAM J. Sci. Statist. Comput., 9 (1988), 152–163
29.
J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965xviii+662
Information & Authors
Information
Published In
Copyright
Copyright © 1993 Society for Industrial and Applied Mathematics.
History
Submitted: 8 April 1991
Accepted: 8 January 1992
Published online: 13 July 2006
MSC codes
Keywords
Authors
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- A modified CMRH method for solving large nonsymmetric linear systemsApplied Mathematics Letters, Vol. 171 | 1 Dec 2025
- Stabilized Krylov Subspace Recurrences via Randomized SketchingNumerical Linear Algebra with Applications, Vol. 32, No. 3 | 29 May 2025
- A unified approach to Krylov subspace methods for solving linear systemsNumerical Algorithms, Vol. 96, No. 1 | 16 October 2023
- Numerical analytic continuationJapan Journal of Industrial and Applied Mathematics, Vol. 40, No. 3 | 26 June 2023
- Topological phase diagrams of exactly solvable non-Hermitian interacting Kitaev chainsPhysical Review Research, Vol. 5, No. 2 | 2 June 2023
- A Lanczos-type procedure for tensorsNumerical Algorithms, Vol. 92, No. 1 | 31 August 2022
- A Lanczos-like method for non-autonomous linear ordinary differential equationsBollettino dell'Unione Matematica Italiana, Vol. 85 | 25 June 2022
- Theoretical modeling of molecules in weakly interacting environments: trifluoride anions in argonPhysical Chemistry Chemical Physics, Vol. 24, No. 6 | 1 January 2022
- Computing low‐rank approximations of the Fréchet derivative of a matrix function using Krylov subspace methodsNumerical Linear Algebra with Applications, Vol. 28, No. 6 | 14 June 2021
- The Gauss quadrature for general linear functionals, Lanczos algorithm, and minimal partial realizationNumerical Algorithms, Vol. 88, No. 2 | 12 January 2021
- Computing the Eigenvectors of Nonsymmetric Tridiagonal MatricesComputational Mathematics and Mathematical Physics, Vol. 61, No. 5 | 1 July 2021
- A pair potential modeling study of F 3− in neon matricesPhysical Chemistry Chemical Physics, Vol. 23, No. 2 | 1 January 2021
- Generalized block anti-Gauss quadrature rulesNumerische Mathematik, Vol. 143, No. 3 | 30 July 2019
- Balanced Iterative Solvers for Linear Nonsymmetric Systems and Nonlinear Systems with PDE Origins: Efficient Black-Box Stopping CriteriaJournal of Scientific Computing, Vol. 81, No. 1 | 3 August 2019
- Compact Modeling Techniques for Magnetic Resonance DetectorsMicro and Nano Scale NMR | 18 May 2018
- An extended nonsymmetric block Lanczos method for model reduction in large scale dynamical systemsCalcolo, Vol. 55, No. 1 | 21 February 2018
- SPMR: A Family of Saddle-Point Minimum Residual SolversSIAM Journal on Scientific Computing, Vol. 40, No. 3 | 26 June 2018
- Modularity in complex multilayer networks with multiple aspects: a static perspectiveApplied Informatics, Vol. 4, No. 1 | 8 May 2017
- A two-sided short-recurrence extended Krylov subspace method for nonsymmetric matrices and its relation to rational moment matchingNumerical Algorithms, Vol. 76, No. 1 | 28 November 2016
- A new iterative method for solving complex symmetric linear systemsApplied Mathematics and Computation, Vol. 302 | 1 Jun 2017
- The Infinite Bi-Lanczos Method for Nonlinear Eigenvalue ProblemsSIAM Journal on Scientific Computing, Vol. 39, No. 5 | 26 October 2017
- BiCGCR2: A new extension of conjugate residual method for solving non-Hermitian linear systemsJournal of Computational and Applied Mathematics, Vol. 305 | 1 Oct 2016
- An Arnoldi-Frontal Approach for the Stability Analysis of Flows in a Collapsible ChannelInternational Journal of Applied Mechanics, Vol. 08, No. 06 | 19 December 2016
- vibro ‐Lanczos, a symmetric Lanczos solver for vibro‐acoustic simulationsInternational Journal for Numerical Methods in Engineering, Vol. 107, No. 4 | 11 January 2016
- BibliographyPassive Macromodeling | 6 November 2015
- A new quasi-minimal residual method based on a biconjugate A-orthonormalization procedure and coupled two-term recurrencesNumerical Algorithms, Vol. 70, No. 4 | 26 February 2015
- A hybridized iterative algorithm of the BiCORSTAB and GPBiCOR methods for solving non-Hermitian linear systemsComputers & Mathematics with Applications, Vol. 70, No. 12 | 1 Dec 2015
- Multi-GPU Accelerated Admittance Method for High-Resolution Human Exposure EvaluationIEEE Transactions on Biomedical Engineering, Vol. 62, No. 12 | 1 Dec 2015
- A transpose-free quasi-minimal residual variant of the CORS method for solving non-Hermitian linear systemsJournal of Computational Physics, Vol. 291 | 1 Jun 2015
- Recycling BiCGSTAB with an Application to Parametric Model Order ReductionSIAM Journal on Scientific Computing, Vol. 37, No. 5 | 29 October 2015
- Tuned preconditioners for inexact two-sided inverse and Rayleigh quotient iterationNumerical Linear Algebra with Applications, Vol. 22, No. 1 | 14 August 2014
- Iterative MethodsNumerical Methods in Matrix Computations | 7 October 2014
- ML( n )BiCGStabt: A ML( n )BiCGStab variant with A -transposeJournal of Computational and Applied Mathematics, Vol. 272 | 1 Dec 2014
- Power system network partitioning for SVD based information retrieval using PMU data2014 IEEE PES General Meeting | Conference & Exposition | 1 Jul 2014
- Simultaneous Similarity Reductions for a Pair of Matrices to Condensed FormsCommunications in Mathematics and Statistics, Vol. 2, No. 2 | 29 August 2014
- Auxiliary master equation approach to nonequilibrium correlated impuritiesPhysical Review B, Vol. 89, No. 16 | 7 April 2014
- Projected Krylov Methods for Saddle-Point SystemsSIAM Journal on Matrix Analysis and Applications, Vol. 35, No. 4 | 4 November 2014
- Deflated and Augmented Krylov Subspace Methods: A Framework for Deflated BiCG and Related SolversSIAM Journal on Matrix Analysis and Applications, Vol. 35, No. 4 | 4 December 2014
- Large-Scale Matrix ComputationsHandbook of Linear Algebra, Second Edition | 26 November 2013
- Model-free voltage stability assessments via singular value analysis of PMU data2013 IREP Symposium Bulk Power System Dynamics and Control - IX Optimization, Security and Control of the Emerging Power Grid | 1 Aug 2013
- Robust Padé Approximation via SVDSIAM Review, Vol. 55, No. 1 | 7 February 2013
- Exploiting the Composite Step Strategy to the Biconjugate A -Orthogonal Residual Method for Non-Hermitian Linear SystemsJournal of Applied Mathematics, Vol. 2013 | 1 Jan 2013
- Estimating power flow conditioning from phasor measurement data2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton) | 1 Oct 2012
- Recycling BiCG with an Application to Model ReductionSIAM Journal on Scientific Computing, Vol. 34, No. 4 | 12 July 2012
- Algorithm comparisons for Stark-profile calculationsHigh Energy Density Physics, Vol. 7, No. 4 | 1 Dec 2011
- Investigating the composite step biconjugate A-orthogonal residual method for non-hermitian linear systems in electromagneticsCEM'11 Computational Electromagnetics International Workshop | 1 Aug 2011
- A novel family of iterative solvers for method of moments discretizations of maxwell's equationsCEM'11 Computational Electromagnetics International Workshop | 1 Aug 2011
- Block Krylov subspace methods for the computation of structural response to turbulent windComputer Methods in Applied Mechanics and Engineering, Vol. 200, No. 23-24 | 1 Jun 2011
- The BiCOR and CORS Iterative Algorithms for Solving Nonsymmetric Linear SystemsSIAM Journal on Scientific Computing, Vol. 33, No. 5 | 27 October 2011
- A backward stability analysis of diagonal pivoting methods for solving unsymmetric tridiagonal systems without interchangesNumerical Linear Algebra with Applications, Vol. 18, No. 1 | 13 October 2009
- Nonlinear eigenvalue and frequency response problems in industrial practiceJournal of Mathematics in Industry, Vol. 1, No. 1 | 1 Jan 2011
- The Lanczos method applied to quasi-static stark broadening of spectral linesHigh Energy Density Physics, Vol. 6, No. 4 | 1 Dec 2010
- A comparative study of iterative solutions to linear systems arising in quantum mechanicsJournal of Computational Physics, Vol. 229, No. 22 | 1 Nov 2010
- A numerical method for diffusion–convection equation using high-order difference schemesComputer Physics Communications, Vol. 181, No. 7 | 1 Jul 2010
- Embedding Calderón Multiplicative Preconditioners in Multilevel Fast Multipole AlgorithmsIEEE Transactions on Antennas and Propagation, Vol. 58, No. 4 | 1 Apr 2010
- A stable primal–dual approach for linear programming under nondegeneracy assumptionsComputational Optimization and Applications, Vol. 44, No. 2 | 20 December 2007
- Lanczos-type variants of the COCR method for complex nonsymmetric linear systemsJournal of Computational Physics, Vol. 228, No. 17 | 1 Sep 2009
- Model reduction using the Vorobyev moment problemNumerical Algorithms, Vol. 51, No. 3 | 30 September 2008
- Non-splitting Tridiagonalization of Complex Symmetric MatricesComputational Science – ICCS 2009 | 1 Jan 2009
- GPU Accelerated Krylov Subspace Methods for Computational Electromagnetics2008 38th European Microwave Conference | 1 Oct 2008
- A new implementation of the CMRH method for solving dense linear systemsJournal of Computational and Applied Mathematics, Vol. 213, No. 2 | 1 Apr 2008
- Combination Preconditioning and the Bramble–Pasciak$^{+}$ PreconditionerSIAM Journal on Matrix Analysis and Applications, Vol. 30, No. 2 | 6 June 2008
- Recursive Solutions to Large Eigenproblems in Molecular Spectroscopy and Reaction DynamicsReviews in Computational Chemistry | 30 April 2007
- Recent computational developments in Krylov subspace methods for linear systemsNumerical Linear Algebra with Applications, Vol. 14, No. 1 | 9 October 2006
- A simple form for the fourth order difference method for 3-D elliptic equationsApplied Mathematics and Computation, Vol. 184, No. 2 | 1 Jan 2007
- A dynamic thick restarted semi-refined ABLE algorithm for computing a few selected eigentriplets of large nonsymmetric matricesLinear Algebra and its Applications, Vol. 416, No. 2-3 | 1 Jul 2006
- The second-order biorthogonalization procedure and its application to quadratic eigenvalue problemsApplied Mathematics and Computation, Vol. 172, No. 2 | 1 Jan 2006
- New implementation of QMR-type algorithmsComputers & Structures, Vol. 83, No. 28-30 | 1 Nov 2005
- Numerical implementation of the QMR algorithm by using discrete stochastic arithmeticJournal of Applied Mathematics and Computing, Vol. 17, No. 1-2 | 1 Mar 2005
- Algorithm 841: BHESS: Gaussian reduction to a similar banded Hessenberg formACM Transactions on Mathematical Software, Vol. 31, No. 1 | 1 March 2005
- QRT: A QR-Based Tridiagonalization Algorithm for Nonsymmetric MatricesSIAM Journal on Matrix Analysis and Applications, Vol. 26, No. 3 | 31 July 2006
- Solution of Linear SystemsNumerical Methods in Electromagnetics | 1 Jan 2005
- The block Lanczos method for linear systems with multiple right-hand sidesApplied Numerical Mathematics, Vol. 51, No. 2-3 | 1 Nov 2004
- Estimation of uT ƒ( A ) v for large‐scale unsymmetric matricesNumerical Linear Algebra with Applications, Vol. 11, No. 1 | 5 June 2003
- LARGE-SCALE MATRIX COMPUTATIONS IN CONTROL: KRYLOV SUBSPACE METHODSNumerical Methods for Linear Control Systems | 1 Jan 2004
- Characterization of electromagnetic systems using Pade approximation with expansion at infinityIEEE Transactions on Advanced Packaging, Vol. 26, No. 4 | 1 Nov 2003
- Application of subspace projection approaches for reduced-order modeling of electromagnetic systemsIEEE Transactions on Advanced Packaging, Vol. 26, No. 4 | 1 Nov 2003
- A Krylov subspace projection method for simultaneous solution of Helmholtz problems at multiple frequenciesComputer Methods in Applied Mechanics and Engineering, Vol. 192, No. 41-42 | 1 Oct 2003
- VPAStab: Stabilised Vector-Padé Approximation with Application to Linear SystemsNumerical Algorithms, Vol. 33, No. 1-4 | 1 Aug 2003
- Modelling of diffractive structures in photorefractive Er/Yb–co-doped glass waveguidesOptics and Lasers in Engineering, Vol. 39, No. 3 | 1 Mar 2003
- A Quasi-Minimal Residual Method for Simultaneous Primal-Dual Solutions and Superconvergent Functional EstimatesSIAM Journal on Scientific Computing, Vol. 24, No. 5 | 25 July 2006
- Krylov subspace techniques for reduced-order modeling of large-scale dynamical systemsApplied Numerical Mathematics, Vol. 43, No. 1-2 | 1 Oct 2002
- A review of formal orthogonality in Lanczos-based methodsJournal of Computational and Applied Mathematics, Vol. 140, No. 1-2 | 1 Mar 2002
- Quantitative performance analysis of the improved quasi-minimal residual method on massively distributed memory computersAdvances in Engineering Software, Vol. 33, No. 3 | 1 Mar 2002
- The Breakdowns of BiCGStabNumerical Algorithms, Vol. 29, No. 1-3 | 1 Mar 2002
- A Case for a Biorthogonal Jacobi--Davidson Method: Restarting and Correction EquationSIAM Journal on Matrix Analysis and Applications, Vol. 24, No. 1 | 31 July 2006
- Two-Sided Arnoldi and Nonsymmetric Lanczos AlgorithmsSIAM Journal on Matrix Analysis and Applications, Vol. 24, No. 2 | 31 July 2006
- On the Numerical Solution of $(\lambda^2 A + \lambda B + C), x = b$ and Application to Structural DynamicsSIAM Journal on Scientific Computing, Vol. 23, No. 6 | 25 July 2006
- Computation of large-scale quadratic forms and transfer functions using the theory of moments, quadrature and Padé approximationModern Methods in Scientific Computing and Applications | 1 Jan 2002
- A partial Padé-via-Lanczos method for reduced-order modelingLinear Algebra and its Applications, Vol. 332-334 | 1 Aug 2001
- By How Much Can Residual Minimization Accelerate the Convergence of Orthogonal Residual Methods?Numerical Algorithms, Vol. 27, No. 2 | 1 Jun 2001
- Improved scalar analysis of integrated optical structures by the mapped Galerkin method and Arnoldi iterationJournal of the Optical Society of America A, Vol. 18, No. 4 | 1 April 2001
- Why Gaussian quadrature in the complex plane?Numerical Algorithms, Vol. 26, No. 3 | 1 Mar 2001
- The Quadratic Eigenvalue ProblemSIAM Review, Vol. 43, No. 2 | 4 August 2006
- Structure-Preserving Methods for Computing Eigenpairs of Large Sparse Skew-Hamiltonian/Hamiltonian PencilsSIAM Journal on Scientific Computing, Vol. 22, No. 6 | 25 July 2006
- Computation of matrix-valued formally orthogonal polynomials and applicationsJournal of Computational and Applied Mathematics, Vol. 127, No. 1-2 | 1 Jan 2001
- Residual Smoothing Techniques: Do They Improve the Limiting Accuracy of Iterative Solvers?BIT Numerical Mathematics, Vol. 41, No. 1 | 1 Jan 2001
- Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulationsNumerical Linear Algebra with Applications, Vol. 7, No. 7-8 | 1 January 2000
- Dual Lanczos simulation of dynamic nuclear magnetic resonance spectra for systems with many spins or exchange sitesThe Journal of Chemical Physics, Vol. 113, No. 8 | 22 Aug 2000
- CTLSS-an advanced electromagnetic simulation tool for designing high-power microwave sourcesIEEE Transactions on Plasma Science, Vol. 28, No. 3 | 1 Jun 2000
- Convergence properties of Krylov subspace methods for singular linear systems with arbitrary indexJournal of Computational and Applied Mathematics, Vol. 114, No. 2 | 1 Feb 2000
- Look-Ahead Procedures for Lanczos-Type Product Methods Based on Three-Term Lanczos RecurrencesSIAM Journal on Matrix Analysis and Applications, Vol. 21, No. 4 | 31 July 2006
- Implicitly Restarted GMRES and Arnoldi Methods for Nonsymmetric Systems of EquationsSIAM Journal on Matrix Analysis and Applications, Vol. 21, No. 4 | 31 July 2006
- Multiple Recursion Conjugate Gradient Algorithms Part I: Sufficient ConditionsSIAM Journal on Matrix Analysis and Applications, Vol. 21, No. 3 | 31 July 2006
- On Lanczos-Type Methods for Wilson FermionsNumerical Challenges in Lattice Quantum Chromodynamics | 1 Jan 2000
- Application of the Pade approximation via Lanczos (PVL) algorithm to electromagnetic systems with expansion at infinity2000 Proceedings. 50th Electronic Components and Technology Conference (Cat. No.00CH37070) | 1 Jan 2000
- Data distribution and communication schemes for IQMR method on massively distributed memory computersProceedings 2000. International Workshop on Parallel Processing | 1 Jan 2000
- The parallel waveform IQMR algorithm for transient simulation of semiconductor devicesProceedings 2000. International Workshop on Parallel Processing | 1 Jan 2000
- Estimating the parallel performance of IQMR method for unsymmetric large and sparse linear systemsProceedings Seventh International Conference on Parallel and Distributed Systems: Workshops | 1 Jan 2000
- Comparison of quasi minimal residual and bi-conjugate gradient iterative methods to solve complex symmetric systems arising from time-harmonic magnetic simulationsCOMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 18, No. 3 | 1 Sep 1999
- CMRH: A new method for solving nonsymmetric linear systems based on the Hessenberg reduction algorithmNumerical Algorithms, Vol. 20, No. 4 | 1 Aug 1999
- SUBSPACE ITERATION METHODS WITH PRECONDITIONING FOR EIGENVALUES OF VERY LARGE MATRICESRecent Advances in Numerical Methods and Applications II | 29 May 2013
- Parallel Performance Analysis of the Improved Quasi-Minimal Residual Method on Bulk Synchronous Parallel ArchitecturesThe Journal of Supercomputing, Vol. 13, No. 2 | 1 Mar 1999
- The BR Eigenvalue AlgorithmSIAM Journal on Matrix Analysis and Applications, Vol. 20, No. 4 | 1 August 2006
- ABLE: An Adaptive Block Lanczos Method for Non-Hermitian Eigenvalue ProblemsSIAM Journal on Matrix Analysis and Applications, Vol. 20, No. 4 | 1 August 2006
- Reduced-Order Modeling Techniques Based on Krylov Subspaces and Their Use in Circuit SimulationApplied and Computational Control, Signals, and Circuits | 1 Jan 1999
- Reducing Global Synchronization in the Biconjugate Gradient MethodParallel Numerical Computation with Applications | 1 Jan 1999
- Theoretical Performance Analysis of the IQMR Method on Distributed Memory ComputersParallel Numerical Computation with Applications | 1 Jan 1999
- ReferencesComputer Solution of Large Linear Systems | 1 Jan 1999
- Application of subspace projection approaches for reduced-order modeling of electromagnetic systemsIEEE 8th Topical Meeting on Electrical Performance of Electronic Packaging (Cat. No.99TH8412) | 1 Jan 1999
- Nested Lanczos: implicitly restarting an unsymmetric Lanczos algorithmNumerical Algorithms, Vol. 18, No. 1 | 1 Sep 1998
- The symplectic eigenvalue problem, the butterfly form, the SR algorithm, and the Lanczos methodLinear Algebra and its Applications, Vol. 275-276 | 1 May 1998
- Error Analysis of Krylov Methods In a NutshellSIAM Journal on Scientific Computing, Vol. 19, No. 2 | 25 July 2006
- An Efficient Solver for Multi--Right-Hand-Side Linear Systems Based on the CCCG($\eta$) Method with Applications to Implicit Time-Dependent Partial Differential EquationsSIAM Journal on Scientific Computing, Vol. 19, No. 1 | 25 July 2006
- Restarted GMRES for Shifted Linear SystemsSIAM Journal on Scientific Computing, Vol. 19, No. 1 | 25 July 2006
- QMR Smoothing for Lanczos-Type Product Methods Based on Three-Term RrecurrencesSIAM Journal on Scientific Computing, Vol. 19, No. 1 | 25 July 2006
- Further optimized look-ahead recurrences for adjacent rows in the Padé table and Toeplitz matrix factorizationsJournal of Computational and Applied Mathematics, Vol. 86, No. 1 | 1 Nov 1997
- On Krylov Subspace Approximations to the Matrix Exponential OperatorSIAM Journal on Numerical Analysis, Vol. 34, No. 5 | 25 July 2006
- An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problemLinear Algebra and its Applications, Vol. 263 | 1 Sep 1997
- Avoiding breakdown in variants of the bi-cgstab algorithmLinear Algebra and its Applications, Vol. 263 | 1 Sep 1997
- An Efficient Implementation of the Nonsymmetric Lanczos AlgorithmSIAM Journal on Matrix Analysis and Applications, Vol. 18, No. 3 | 31 July 2006
- Breakdowns and stagnation in iterative methodsBIT Numerical Mathematics, Vol. 37, No. 2 | 1 Jun 1997
- A Stabilized QMR Version of Block BiCGSIAM Journal on Matrix Analysis and Applications, Vol. 18, No. 2 | 31 July 2006
- Homotopy Method for the Large, Sparse, Real Nonsymmetric Eigenvalue ProblemSIAM Journal on Matrix Analysis and Applications, Vol. 18, No. 2 | 31 July 2006
- A block QMR algorithm for non-Hermitian linear systems with multiple right-hand sidesLinear Algebra and its Applications, Vol. 254, No. 1-3 | 1 Mar 1997
- A Parallel implementation of the general Lanczos method on the CRAY T3DVector and Parallel Processing — VECPAR'96 | 5 August 2005
- A variant of the biconjugate gradient method suitable for massively parallel computingSolving Irregularly Structured Problems in Parallel | 8 June 2005
- Efficient implementation of the improved quasi-minimal residual method on massively distributed memory computersSolving Irregularly Structured Problems in Parallel | 8 June 2005
- Efficient implementation of the improved unsymmetric Lanczos process on massively distributed memory computersParallel Computing Technologies | 9 June 2005
- Circuit Simulation Techniques Based on Lanczos-Type AlgorithmsSystems and Control in the Twenty-First Century | 1 Jan 1997
- Reduced-Order Electromagnetic Modeling for Design-Driven Simulations of Complex Integrated Electronic SystemsComputational Electromagnetics and Its Applications | 1 Jan 1997
- A QMR-based interior-point algorithm for solving linear programsMathematical Programming, Vol. 76, No. 1 | 1 Jan 1997
- The improved quasi-minimal residual method on massively distributed memory computersHigh-Performance Computing and Networking | 25 June 2005
- Breakdowns in the implementation of the Lánczos method for solving linear systemsComputers & Mathematics with Applications, Vol. 33, No. 1-2 | 1 Jan 1997
- Lanczos-type solvers for nonsymmetric linear systems of equationsActa Numerica, Vol. 6 | 7 November 2008
- Fast identification of impulse response modes via Krylov space methodsProceedings of the 36th IEEE Conference on Decision and Control | 1 Jan 1997
- Practical issues of interconnect analysis in deep submicron integrated circuitsProceedings International Conference on Computer Design VLSI in Computers and Processors | 1 Jan 1997
- Recent advances in Krylov-subspace solvers for linear systems and applications in device simulationSISPAD '97. 1997 International Conference on Simulation of Semiconductor Processes and Devices. Technical Digest | 1 Jan 1997
- Iterative methods for solvingAx=b, GMRES/FOM versus QMR/BiCGAdvances in Computational Mathematics, Vol. 6, No. 1 | 1 Dec 1996
- Composite Step Product Methods for Solving Nonsymmetric Linear SystemsSIAM Journal on Scientific Computing, Vol. 17, No. 6 | 17 February 2012
- Arnoldi versus nonsymmetric Lanczos algorithms for solving matrix eigenvalue problemsBIT Numerical Mathematics, Vol. 36, No. 3 | 1 Sep 1996
- Reliable updated residuals in hybrid Bi-CG methodsComputing, Vol. 56, No. 2 | 1 Jun 1996
- Relations between Galerkin and Norm-Minimizing Iterative Methods for Solving Linear SystemsSIAM Journal on Matrix Analysis and Applications, Vol. 17, No. 2 | 17 February 2012
- APPLICATION OF THE COLLOCATION METHOD IN THREE DIMENSIONS TO A MODEL SEMICONDUCTOR PROBLEMInternational Journal for Numerical Methods in Engineering, Vol. 39, No. 6 | 30 Mar 1996
- A COMPARISON OF COUPLED AND SEGREGATED ITERATIVE SOLUTION TECHNIQUE FOR INCOMPRESSIBLE SWIRLING FLOWInternational Journal for Numerical Methods in Fluids, Vol. 22, No. 5 | 15 Mar 1996
- A rational Lanczos algorithm for model reductionNumerical Algorithms, Vol. 12, No. 1 | 1 Mar 1996
- An adaptive block Lanczos algorithmNumerical Algorithms, Vol. 12, No. 1 | 1 Mar 1996
- A linear system solver based on a modified Krylov subspace method for breakdown recoveryNumerical Algorithms, Vol. 12, No. 1 | 1 Mar 1996
- QMRPACKACM Transactions on Mathematical Software, Vol. 22, No. 1 | 1 March 1996
- The methods of Vorobyev and LanczosLinear Algebra and its Applications, Vol. 234 | 1 Feb 1996
- Efficient Parasitic Substrate Modeling for Monolithic Mixed-A/D Circuit Design and VerificationModeling and Simulation of Mixed Analog-Digital Systems | 1 Jan 1996
- Efficient parasitic substrate modeling for monolithic mixed-A/D circuit design and verificationAnalog Integrated Circuits and Signal Processing, Vol. 10, No. 1-2 | 1 Jan 1996
- Iterative methods for unsymmetric linear systemsSOFSEM'96: Theory and Practice of Informatics | 26 June 2005
- Navier-Stokes equations for incompressible flowsHandbook of Computational Fluid Mechanics | 1 Jan 1996
- Small-signal circuit analysis and sensitivity computations with the PVL algorithmIEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, Vol. 43, No. 8 | 1 Jan 1996
- Reduced-order modeling of large passive linear circuits by means of the SyPVL algorithmProceedings of International Conference on Computer Aided Design | 1 Jan 1996
- Quantum time evolution in time-dependent fields and time-independent reactive-scattering calculations via an efficient Fourier grid preconditionerThe Journal of Chemical Physics, Vol. 103, No. 23 | 15 Dec 1995
- A theoretical overview of Krylov subspace methodsApplied Numerical Mathematics, Vol. 19, No. 3 | 1 Dec 1995
- An overview of approaches for the stable computation of hybrid BiCG methodsApplied Numerical Mathematics, Vol. 19, No. 3 | 1 Dec 1995
- Software for simplified Lanczos and QMR algorithmsApplied Numerical Mathematics, Vol. 19, No. 3 | 1 Dec 1995
- Low-rank modification of the unsymmetric Lanczos algorithmMathematics of Computation, Vol. 64, No. 212 | 1 January 1995
- The quasi-minimal residual algorithm applied to complex symmetric linear systems in quantum reactive scatteringThe Journal of Chemical Physics, Vol. 103, No. 12 | 22 Sep 1995
- Maintaining convergence properties of BiCGstab methods in finite precision arithmeticNumerical Algorithms, Vol. 10, No. 2 | 1 Sep 1995
- Bi-Lanczos with partial orthogonalizationComputers & Structures, Vol. 56, No. 4 | 1 Aug 1995
- QMR methods in computational fluid dynamics12th Computational Fluid Dynamics Conference | 22 August 1995
- Look-ahead in Bi-CGSTAB and other product methods for linear systemsBIT Numerical Mathematics, Vol. 35, No. 2 | 1 Jun 1995
- Efficient linear circuit analysis by Pade approximation via the Lanczos processIEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, Vol. 14, No. 5 | 1 May 1995
- Preconditioned Krylov Subspace Methods for Lyapunov Matrix EquationsSIAM Journal on Matrix Analysis and Applications, Vol. 16, No. 1 | 17 July 2006
- Numerical impluimentation of a new algorithm for polynomials with multiple rootsJournal of Difference Equations and Applications, Vol. 1, No. 2 | 1 Jan 1995
- Krylov space methods on state-space control modelsCircuits, Systems, and Signal Processing, Vol. 13, No. 6 | 1 Nov 1994
- Region-Dependent optimal m-stage Runge-Kutta schemes for solving a class of nonsymmetric linear systemsLinear Algebra and its Applications, Vol. 212-213 | 1 Nov 1994
- Thermodynamics and spin gap of the Heisenberg ladder calculated by the look-ahead Lanczos algorithmPhysical Review B, Vol. 50, No. 18 | 1 November 1994
- On the use of two QMR algorithms for solving singular systems and applications in Markov chain modelingNumerical Linear Algebra with Applications, Vol. 1, No. 4 | 8 July 2005
- An Implementation of the QMR Method Based on Coupled Two-Term RecurrencesSIAM Journal on Scientific Computing, Vol. 15, No. 2 | 13 July 2006
- A composite step bi-conjugate gradient algorithm for nonsymmetric linear systemsNumerical Algorithms, Vol. 7, No. 1 | 1 Mar 1994
- Treatment of near-breakdown in the CGS algorithmNumerical Algorithms, Vol. 7, No. 1 | 1 Mar 1994
- A Completed Theory of the Unsymmetric Lanczos Process and Related Algorithms. Part IISIAM Journal on Matrix Analysis and Applications, Vol. 15, No. 1 | 31 July 2006
- Stable Look-Ahead Versions of the Euclidean and Chebyshev AlgorithmsApproximation and Computation: A Festschrift in Honor of Walter Gautschi | 1 Jan 1994
- A breakdown-free variation of the nonsymmetric Lanczos algorithmsMathematics of Computation, Vol. 62, No. 205 | 1 January 1994
- The multipoint Pad� table and general recurrences for rational interpolationActa Applicandae Mathematicae, Vol. 33, No. 2-3 | 1 Dec 1993
- Variants of BICGSTAB for Matrices with Complex SpectrumSIAM Journal on Scientific Computing, Vol. 14, No. 5 | 13 July 2006
- Rational interpolation via orthogonal polynomialsComputers & Mathematics with Applications, Vol. 26, No. 5 | 1 Sep 1993
- Formally biorthogonal polynomials and a look-ahead Levinson algorithm for general Teoplitz systemsLinear Algebra and its Applications, Vol. 188-189 | 1 Jul 1993
- Stable row recurrences for the Padé table and generically superfast lookahead solvers for non-Hermitian Toeplitz systemsLinear Algebra and its Applications, Vol. 188-189 | 1 Jul 1993
- Lanczos-type algorithms for solving systems of linear equationsApplied Numerical Mathematics, Vol. 11, No. 6 | 1 Apr 1993
- Bifurcations of corank greater than twoComputers & Mathematics with Applications, Vol. 25, No. 7 | 1 Apr 1993
- A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear SystemsSIAM Journal on Scientific Computing, Vol. 14, No. 2 | 13 July 2006
- The Look-Ahead Lanczos Process for Large Nonsymmetric Matrices and Related AlgorithmsLinear Algebra for Large Scale and Real-Time Applications | 1 Jan 1993
- BibliographyScientific Computing | 1 Jan 1993
- Quasi-kernel polynomials and their use in non-Hermitian matrix iterationsJournal of Computational and Applied Mathematics, Vol. 43, No. 1-2 | 1 Nov 1992
- Efficient Small-signal Circuit Analysis And Sensitivity Computations With The Pvl AlgorithmIEEE/ACM International Conference on Computer-Aided Design | 1 Jan 1984
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].