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183
Calculation Of Pseudospectra By The Arnoldi Iteration
, 1996
"... The Arnoldi iteration, usually viewed as a method for calculating eigenvalues, can also be used to estimate pseudospectra. This possibility may be of practical importance, for in applications involving highly nonnormal matrices or operators, such as hydrodynamic stability, pseudospectra may be phys ..."
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Cited by 41 (5 self)
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The Arnoldi iteration, usually viewed as a method for calculating eigenvalues, can also be used to estimate pseudospectra. This possibility may be of practical importance, for in applications involving highly nonnormal matrices or operators, such as hydrodynamic stability, pseudospectra may be physically more significant than spectra.
A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature
, 2003
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Approximating the exponential from a Lie algebra to a Lie group
 Math. Comp
, 1998
"... Abstract. Consider a differential equation y ′ = A(t, y)y, y(0) = y0 with y0 ∈ GandA: R + × G → g, wheregis a Lie algebra of the matricial Lie group G. Every B ∈ g canbemappedtoGbythematrixexponentialmap exp (tB) witht∈R. Most numerical methods for solving ordinary differential equations (ODEs) on ..."
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Cited by 41 (8 self)
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Abstract. Consider a differential equation y ′ = A(t, y)y, y(0) = y0 with y0 ∈ GandA: R + × G → g, wheregis a Lie algebra of the matricial Lie group G. Every B ∈ g canbemappedtoGbythematrixexponentialmap exp (tB) witht∈R. Most numerical methods for solving ordinary differential equations (ODEs) on Lie groups are based on the idea of representing the approximation yn of the exact solution y(tn), tn ∈ R +, by means of exact exponentials of suitable elements of the Lie algebra, applied to the initial value y0. This ensures that yn ∈ G. When the exponential is difficult to compute exactly, as is the case when the dimension is large, an approximation of exp (tB) plays an important role in the numerical solution of ODEs on Lie groups. In some cases rational or polynomial approximants are unsuitable and we consider alternative techniques, whereby exp (tB) is approximated by a product of simpler exponentials. In this paper we present some ideas based on the use of the Strang splitting for the approximation of matrix exponentials. Several cases of g and G are considered, in tandem with general theory. Order conditions are discussed, and a number of numerical experiments conclude the paper. 1.
Error estimation and evaluation of matrix functions via the Faber transform
 SIAM J. Numer. Anal
"... Abstract. The need to evaluate expressions of the form f(A) orf(A)b, wheref is a nonlinear function, A is a large sparse n × n matrix, and b is an nvector, arises in many applications. This paper describes how the Faber transform applied to the field of values of A can be used to determine improved ..."
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Cited by 39 (13 self)
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Abstract. The need to evaluate expressions of the form f(A) orf(A)b, wheref is a nonlinear function, A is a large sparse n × n matrix, and b is an nvector, arises in many applications. This paper describes how the Faber transform applied to the field of values of A can be used to determine improved error bounds for popular polynomial approximation methods based on the Arnoldi process. Applications of the Faber transform to rational approximation methods and, in particular, to the rational Arnoldi process also are discussed.
The Magnus expansion and some of its applications
, 2008
"... Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an ..."
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Cited by 35 (6 self)
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Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as TimeDependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial resummation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related nonperturbative
ROWMAP  a ROWcode with Krylov techniques for large stiff ODEs
 Appl. Numer. Math
, 1997
"... We present a KrylovWcode ROWMAP for the integration of stiff initial value problems. It is based on the ROWmethods of the code ROS4 of Hairer and Wanner and uses Krylov techniques for the solution of linear systems. A special multiple Arnoldi process ensures order p = 4 already for fairly low dim ..."
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Cited by 26 (5 self)
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We present a KrylovWcode ROWMAP for the integration of stiff initial value problems. It is based on the ROWmethods of the code ROS4 of Hairer and Wanner and uses Krylov techniques for the solution of linear systems. A special multiple Arnoldi process ensures order p = 4 already for fairly low dimensions of the Krylov subspaces independently of the dimension of the differential equations. Numerical tests and comparisons with the multistep code VODPK illustrate the efficiency of ROWMAP for large stiff systems. Furthermore, the application to nonautonomous systems is discussed in more detail. Key words. ROWmethods, stiff initial value problems, Krylov subspaces, multiple Arnoldi process AMS(MOS) subject classifications. 65L06, 65F10 1 Introduction For the numerical solution of stiff initial value problems y 0 (t) = f(t; y(t)) y(t 0 ) = y 0 2 R n ; (1.1) implicit or linearly implicit methods have to be used due to stability requirements. For large dimensions n these methods sp...
A variational splitting integrator for quantum molecular dynamics
 Appl. Numer. Math
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Evaluating matrix functions for exponential integrators via Carathéodory–Fejér approximation and contour integrals
 ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS
, 2007
"... Among the fastest methods for solving stiff PDE are exponential integrators, which require the evaluation of, where is a negative semidefinite matrix and is the exponential function or one of the related “ functions ” such as. Building on previous work by Trefethen and Gutknecht, Minchev, and Lu, w ..."
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Cited by 25 (1 self)
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Among the fastest methods for solving stiff PDE are exponential integrators, which require the evaluation of, where is a negative semidefinite matrix and is the exponential function or one of the related “ functions ” such as. Building on previous work by Trefethen and Gutknecht, Minchev, and Lu, we propose two methods for the fast evaluation of that are especially useful when shifted systems #"$& % can be solved efficiently, e.g. by a sparse direct solver. The first method is based on best rational approximations to on the negative real axis computed via the CarathéodoryFejér procedure. Rather than using optimal poles we approximate the functions in a set of common poles, which speeds up typical computations by a factor ' of (*) + to. The second method is an application of the trapezoid rule on a Talbottype contour.