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Parallel Numerical Linear Algebra
, 1993
"... We survey general techniques and open problems in numerical linear algebra on parallel architectures. We first discuss basic principles of parallel processing, describing the costs of basic operations on parallel machines, including general principles for constructing efficient algorithms. We illust ..."
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Cited by 766 (23 self)
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We survey general techniques and open problems in numerical linear algebra on parallel architectures. We first discuss basic principles of parallel processing, describing the costs of basic operations on parallel machines, including general principles for constructing efficient algorithms. We illustrate these principles using current architectures and software systems, and by showing how one would implement matrix multiplication. Then, we present direct and iterative algorithms for solving linear systems of equations, linear least squares problems, the symmetric eigenvalue problem, the nonsymmetric eigenvalue problem, and the singular value decomposition. We consider dense, band and sparse matrices.
PETSc users manual
 ANL95/11  Revision 2.1.0, Argonne National Laboratory
, 2001
"... tract W31109Eng38. 2 This manual describes the use of PETSc for the numerical solution of partial differential equations and related problems on highperformance computers. The Portable, Extensible Toolkit for Scientific Computation (PETSc) is a suite of data structures and routines that provid ..."
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Cited by 278 (20 self)
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tract W31109Eng38. 2 This manual describes the use of PETSc for the numerical solution of partial differential equations and related problems on highperformance computers. The Portable, Extensible Toolkit for Scientific Computation (PETSc) is a suite of data structures and routines that provide the building blocks for the implementation of largescale application codes on parallel (and serial) computers. PETSc uses the MPI standard for all messagepassing communication. PETSc includes an expanding suite of parallel linear, nonlinear equation solvers and time integrators that may be used in application codes written in Fortran, C, and C++. PETSc provides many of the mechanisms needed within parallel application codes, such as parallel matrix and vector assembly routines. The library is organized hierarchically, enabling users to employ the level of abstraction that is most appropriate for a particular problem. By using techniques of objectoriented programming, PETSc provides enormous flexibility for users. PETSc is a sophisticated set of software tools; as such, for some users it initially has a much steeper
Continuation and Path Following
, 1992
"... CONTENTS 1 Introduction 1 2 The Basics of PredictorCorrector Path Following 3 3 Aspects of Implementations 7 4 Applications 15 5 PiecewiseLinear Methods 34 6 Complexity 41 7 Available Software 44 References 48 1. Introduction Continuation, embedding or homotopy methods have long served as useful ..."
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Cited by 95 (6 self)
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CONTENTS 1 Introduction 1 2 The Basics of PredictorCorrector Path Following 3 3 Aspects of Implementations 7 4 Applications 15 5 PiecewiseLinear Methods 34 6 Complexity 41 7 Available Software 44 References 48 1. Introduction Continuation, embedding or homotopy methods have long served as useful theoretical tools in modern mathematics. Their use can be traced back at least to such venerated works as those of Poincar'e (18811886), Klein (1882 1883) and Bernstein (1910). Leray and Schauder (1934) refined the tool and presented it as a global result in topology, viz., the homotopy invariance of degree. The use of deformations to solve nonlinear systems of equations Partially supported by the National Science Foundation via grant # DMS9104058 y Preprint, Colorado State University, August 2 E. Allgower and K. Georg may be traced back at least to Lahaye (1934). The classical embedding methods were the
Recent computational developments in Krylov subspace methods for linear systems
 NUMER. LINEAR ALGEBRA APPL
, 2007
"... Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are metho ..."
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Cited by 86 (12 self)
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Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters.
Reducedorder modeling techniques based on Krylov subspaces and their use in circuit simulation
 in Applied and Computational Control, Signals, and Circuits
, 1999
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An Iterative Method for Nonsymmetric Systems with Multiple RightHand Sides
 SIAM J. Sci. Comput
, 1995
"... . We propose a method for the solution of linear systems AX = B where A is a large, possibly sparse, nonsymmetric matrix of order n, and B is an arbitrary rectangular matrix of order n \Theta s with s of moderate size. The method uses a single Krylov subspace per step as a generator of approximation ..."
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Cited by 68 (3 self)
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. We propose a method for the solution of linear systems AX = B where A is a large, possibly sparse, nonsymmetric matrix of order n, and B is an arbitrary rectangular matrix of order n \Theta s with s of moderate size. The method uses a single Krylov subspace per step as a generator of approximations, a projection process, and a Richardson acceleration technique. It thus combines the advantages of recent hybrid methods with those for solving symmetric systems with multiple righthand sides. Numerical experiments indicate that in several cases the method has better practical performance and significantly lower memory requirements than block versions of nonsymmetric solvers and other proposed methods for the solution of systems with multiple righthand sides. Key words. nonsymmetric systems, iterative method, Krylov subspace, multiple righthand sides, GMRES, hybrid method, Richardson AMS subject classifications. 65F10, 65Y20 1. Introduction. We consider techniques for the solution of ...
Gmres/cr and Arnoldi/Lanczos as Matrix Approximation Problems
 SIAM J. SCI. COMPUT
"... The GMRES and Arnoldi algorithms, which reduce to the CR and Lanczos algorithms in the symmetric case, both minimize kp(A)bk over polynomials p of degree n. The difference is that p is normalized at z = 0 for GMRES and at z = 1 for Arnoldi. Analogous "ideal GMRES " and "ideal Arnoldi ..."
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Cited by 59 (7 self)
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The GMRES and Arnoldi algorithms, which reduce to the CR and Lanczos algorithms in the symmetric case, both minimize kp(A)bk over polynomials p of degree n. The difference is that p is normalized at z = 0 for GMRES and at z = 1 for Arnoldi. Analogous "ideal GMRES " and "ideal Arnoldi" problems are obtained if one removes b from the discussion and minimizes kp(A)k instead. Investigation of these true and ideal approximation problems gives insight into how fast GMRES converges and how the Arnoldi iteration locates eigenvalues.
NITSOL: A NEWTON ITERATIVE SOLVER FOR NONLINEAR SYSTEMS
, 1998
"... We introduce a welldeveloped Newton iterative (truncated Newton) algorithm for solving largescale nonlinear systems. The framework is an inexact Newton method globalized by backtracking. Trial steps are obtained using one of several Krylov subspace methods. The algorithm is implemented in a Fort ..."
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Cited by 50 (8 self)
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We introduce a welldeveloped Newton iterative (truncated Newton) algorithm for solving largescale nonlinear systems. The framework is an inexact Newton method globalized by backtracking. Trial steps are obtained using one of several Krylov subspace methods. The algorithm is implemented in a Fortran solver called NITSOL that is robust yet easy to use and provides a number of useful options and features. The structure offers the user great flexibility in addressing problem specicity through preconditioning and other means and allows easy adaptation to parallel environments. Features and capabilities are illustrated in numerical experiments.
Solution of shifted linear systems by quasiminimal residual iterations, in Numerical linear algebra
, 1992
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Reducedorder modeling of timevarying systems
 Circuits and Systems II: Analog and Digital Signal Processing
, 1999
"... We present a theory for reducedorder modelling of linear timevarying systems, together with efficient numerical methods for application to large systems. The technique, called TVP (TimeVarying Padk), is applicable to deterministic as well as noise analysis of many types of communication subsystem ..."
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Cited by 46 (8 self)
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We present a theory for reducedorder modelling of linear timevarying systems, together with efficient numerical methods for application to large systems. The technique, called TVP (TimeVarying Padk), is applicable to deterministic as well as noise analysis of many types of communication subsystems, such as mixers and switchedcapacitor filters, for which existing model reduction techniques cannot be used. TVP is therefore suitable for hierarchical verification of entire communication systems. We present practical applications in which TVP generates macromodels which are more than two orders of magnitude smaller, but still replicate the inputoutput behaviour of the original systems accurately. The size reduction results in a speedup of more than 500. 1