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  • 1
    Electronic Resource
    Electronic Resource
    Rational approximants and numerical methods for initial-value problems
    Amsterdam : Elsevier
    Physica D: Nonlinear Phenomena 60 (1992), S. 280-292 
    Details
    ISSN: 0167-2789
    Source: Elsevier Journal Backfiles on ScienceDirect 1907 - 2002
    Topics: Physics
    Type of Medium: Electronic Resource
    URL: http://linkinghub.elsevier.com/retrieve/pii/0167-2789(92)90244-H
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  • 2
    Electronic Resource
    Electronic Resource
    Two-step numerical methods for parabolic differential equations (1981)
    Springer
    BIT 21 (1981), S. 80-96 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A family of two-stepA-stable methods of maximal order for the numerical solution of ordinary differential systems is developed. If these methods are applied to the stiff, large systems which originate from linear parabolic differential equations they yield a large, sparse set of linear algebraic equations of special form. This set is considerably easier to solve than the algebraic equations which are obtained when using diagonal Obrechkoff methods, which are one-step,A-stable and of maximal order
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01934073
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  • 3
    Electronic Resource
    Electronic Resource
    On the Implementation of the Method of Magnus Series for Linear Differential Equations (1999)
    Springer
    BIT 39 (1999), S. 281-304 
    Details
    ISSN: 1572-9125
    Keywords: Ordinary differential equations ; initial value problems ; numerical analysis ; geometric integration ; Lie groups ; manifolds
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The method of Magnus series has recently been analysed by Iserles and Nørsett. It approximates the solution of linear differential equations y′ = a(t)y in the form y(t) = e σ(t) y 0, solving a nonlinear differential equation for σ by means of an expansion in iterated integrals of commutators. An appealing feature of the method is that, whenever the exact solution evolves in a Lie group, so does the numerical solution. The subject matter of the present paper is practical implementation of the method of Magnus series. We commence by briefly reviewing the method and highlighting its connection with graph theory. This is followed by the derivation of error estimates, a task greatly assisted by the graph-theoretical connection. These error estimates have been incorporated into a variable-step fourth-order code. The concluding section of the paper is devoted to a number of computer experiments that highlight the promise of the proposed approach even in the absence of a Lie-group structure.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022393913721
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  • 4
    Electronic Resource
    Electronic Resource
    Equilibria of Runge-Kutta methods (1990)
    Springer
    Numerische Mathematik 58 (1990), S. 243-254 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: G1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary It is known that certain Runge-Kutta methods share the property that, in a constant-step implementation, if a solution trajectory converges to a bounded limit then it must be a fixed point of the underlying differential system. Such methods are calledregular. In the present paper we provide a recursive test to check whether given method is regular. Moreover, by examining solution trajectories of linear equations, we prove that the order of ans-stage regular method may not exceed 2[(s+2)/2] and that the maximal order of regular Runge-Kutta method with an irreducible stability function is 4.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385623
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  • 5
    Electronic Resource
    Electronic Resource
    Solving linear ordinary differential equations by exponentials of iterated commutators (1984)
    Springer
    Numerische Mathematik 45 (1984), S. 183-199 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65L05 ; CR: G1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A sequence of transformations of a linear system of ordinary differential equations is investigated. It is shown that these transformations produce new systems which represent progressively smaller perturbations of the original set of equations. The transformations are implemented as a basis of a numerical method. Order, stability and error control of this method are analyzed. Numerical examples demonstrate the potential of this approach.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389464
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  • 6
    Electronic Resource
    Electronic Resource
    Error control of rational approximations to the exponential function (1986)
    Springer
    Constructive approximation 2 (1986), S. 41-57 
    Details
    ISSN: 1432-0940
    Keywords: 41A20 ; Rational approximations ; Exponential function ; Padé approximants ; Stiff ordinary differential equations ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Several methods for the numerical solution of stiff ordinary differential equations require approximation of an exponential of a matrix. In the present paper we present a technique for estimating the error incurred in replacing a matrix exponential by a rational approximation. This estimation is done by introducing another approximation, of superior order, whose aposteriori evaluation is cheap. Properties of the new approximation pertaining to both its stability and the behavior of the error for matrices with negative eigenvalues are analyzed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01893416
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  • 7
    Electronic Resource
    Electronic Resource
    Christoffel-Darboux-type formulae and a recurrence for biorthogonal polynomials (1989)
    Springer
    Constructive approximation 5 (1989), S. 437-453 
    Details
    ISSN: 1432-0940
    Keywords: 42C05 ; Biorthogonality ; Orthogonal polynomials ; Recurrence relations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract It is proved that biorthogonal polynomials obey two different kinds of Christoffel-Darboux-type formulae, one linking polynomials with a different parameter and one combining polynomials with different degrees. This is used to produce a mixed recurrence relation, which is valid for all biorthogonal polynomials. This recurrence relation establishes several results on interlacing property of zeros of successive biorthogonal polynomials and leads to a new result on the interlace of zeros of orthogonal polynomials (of equal degrees) with respect to two distributionsdψ(x) andx p dψ(x), 0〈p≤1, with support in either [0, 1] or [1, ∞).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01889620
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  • 8
    Electronic Resource
    Electronic Resource
    On Cayley-Transform Methods for the Discretization of Lie-Group Equations (2001)
    Springer
    Foundations of computational mathematics 1 (2001), S. 129-160 
    Details
    ISSN: 1615-3383
    Keywords: AMS Classification. 65L05; 22E60.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. In this paper we develop in a systematic manner the theory of time-stepping methods based on the Cayley transform. Such methods can be applied to discretize differential equations that evolve in some Lie groups, in particular in the orthogonal group and the symplectic group. Unlike many other Lie-group solvers, they do not require the evaluation of matrix exponentials. Similarly to the theory of Magnus expansions in [13], we identify terms in a Cayley expansion with rooted trees, which can be constructed recursively. Each such term is an integral over a polytope but all such integrals can be evaluated to high order by using special quadrature formulas similar to the construction in [13]. Truncated Cayley expansions (with exact integrals) need not be time-symmetric, hence the method does not display the usual advantages associated with time symmetry, e.g., even order of approximation. However, time symmetry (with its attendant benefits) is attained when exact integrals are replaced by certain quadrature formulas.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s102080010003
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  • 9
    Book
    Book
    Order stars; 2 (1991)
    London u.a. :Chapman and Hall,
    Details
    Title: Order stars; 2
    Author: Iserles, A.
    Contributer: Norsett, Syvert Paul
    Publisher: London u.a. :Chapman and Hall,
    Year of publication: 1991
    Pages: 248 S.
    Series Statement: Applied mathematics & mathematical computation 2
    Type of Medium: Book
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  • 10
    Book
    Book
    ¬The¬ State of the art in numerical analysis. Proc. of the joint IMA/SIAM conf.; 9 (1987)
    Oxford u.a. :Clarendon Press,
    Details
    Title: ¬The¬ State of the art in numerical analysis. Proc. of the joint IMA/SIAM conf.; 9
    Contributer: Iserles, A. , Powell, M.J.D.-(Hg.)
    Publisher: Oxford u.a. :Clarendon Press,
    Year of publication: 1987
    Pages: 719 S.
    Series Statement: ¬The¬ Institute of Mathematics and its Applications Conference Series 9
    Type of Medium: Book
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