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  • 1
    Book
    Book
    Stochastic analysis with applications to mathematical finance /; 460.2004,2041 (2004)
    London,
    Details
    Title: Stochastic analysis with applications to mathematical finance /; 460.2004,2041
    Contributer: Cash, J. R. , National Science Foundation , Workshop at the American National Science Foundation 2002. , NSF 2002.
    Publisher: London,
    Year of publication: 2004
    Pages: 402 S. : , graph. Darst.
    Series Statement: Proceedings of the Royal Society of London 460.2004,2041
    Type of Medium: Book
    Language: Undetermined
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  • 2
    Electronic Resource
    Electronic Resource
    Implementation issues in solving nonlinear equations for two-point boundary value problems (1990)
    Springer
    Computing 45 (1990), S. 17-37 
    Details
    ISSN: 1436-5057
    Keywords: 65L10 (boundary value problems) ; 68N99 (mathematical software) ; Boundary value problems ; nonlinear algebraic equations ; merit functions ; mathematical software ; watchdog strategy
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Komplizierte numerische Methoden enthalten oft Teilprobleme, die sich leicht mathematisch formulieren lassen, die aber schwierig in Computerprogramme umgewandelt werden können. Mehrere solche Fragestellungen traten bei der Erstellung eines Newton-Verfahrens als Bestandteil eines Differenzenverfahrens für 2-Punkt Randwert-probleme auf. Wir beschreiben die praktischen und theoretischen Überlegungen, welche den Entscheidungen zugrunde liegen, die schließlich zum Computerprogramm führten. Insbesondere betonen wir dabei zwei “Watchdog”-Strategien, welche die Zuverlässigkeit verbessern und ein frühes Abbrechen der Newton-Iteration ermöglichen.
    Notes: Abstract Complex numerical methods often contain subproblems that are easy to state in mathematical form, but difficult to translate into software. Several algorithmic isues of this nature arise in implementing a Newton iteration scheme as part of a finite-difference method for two-point boundary value problems. We describe the practical as well as theoretical considerations behind the decisions included in the final code, with special emphasis on two “watchdog” strategies designed to improve reliability and allow early termination of the Newton iterates.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02250582
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    Paper (German National Licenses)
    Fulltext
  • 3
    Electronic Resource
    Electronic Resource
    A block method for the numerical integration of stiff systems of ordinary differential equations (1979)
    Springer
    BIT 19 (1979), S. 429-447 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new approach to the approximate numerical integration of stiff systems of first order ordinary differential equations is developed. In this approach several different formulae are applied in a well defined cyclic order to produce highly accurate integration schemes with infinite regions of absolute stability. The efficiency of these new algorithms, compared with that of certain existing ones, is demonstrated for some particular test problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01931259
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    Paper (German National Licenses)
    Fulltext
  • 4
    Electronic Resource
    Electronic Resource
    A high order method for the numerical solution of two-point boundary value problems (1980)
    Springer
    BIT 20 (1980), S. 44-52 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A one step finite difference scheme of order 4 for the numerical solution of the general two-point boundary value problemy′=f(t,y),a ≦t ≦b, withg(y(a),y(b))=0 is presented. The global discretization error of the scheme is shown, in sufficiently smooth cases, to have an asymptotic expansion containing even powers of the mesh size only. This justifies the use of Richardson extrapolation (or deferred correction) to obtain high orders of accuracy. A theoretical examination of the new scheme for large systems of equations shows that for a given mesh size it generally requires about twice as much work as the Keller box scheme. However, the expectation of higher accuracy usually justifies this extra computational effort. Some numerical results are given which confirm these expectations and show that the new scheme can be generally competitive with the box scheme.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01933584
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    On a class of cyclic methods for the numerical integration of stiff systems of O.D.E.s (1977)
    Springer
    BIT 17 (1977), S. 270-280 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A class of cyclic linear multistep methods suitable for the approximate numerical integration of stiff systems of first order ordinary differential equations is developed. Particular attention is paid to the problem of deriving schemes which are almostA-stable, self starting, have relatively high orders of accuracy and contain a built in error estimate. These requirements demand that the linear multistep methods which are used are solved iteratively rather than directly in the usual way and an efficient method for doing this is suggested. Finally the algorithms are illustrated by application to a particular test problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01932147
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    Paper (German National Licenses)
    Fulltext
  • 6
    Electronic Resource
    Electronic Resource
    A note on the exponential fitting of blended, extended linear multistep methods (1981)
    Springer
    BIT 21 (1981), S. 450-454 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A class of exponentially fitted blended, extended linear multistep methods is investigated andA-stable formulae of order 3, 4 and 5 are derived.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01932841
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    Paper (German National Licenses)
    Fulltext
  • 7
    Electronic Resource
    Electronic Resource
    High order methods for the numerical solution of two-point boundary value problems (1982)
    Springer
    BIT 22 (1982), S. 183-199 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In a recent paper, Cash and Moore have given a fourth order formula for the approximate numerical integration of two-point boundary value problems in O.D.E.s. The formula presented was in effect a “one-off” formula in that it was obtained using a trial and error approach. The purpose of the present paper is to describe a unified approach to the derivation of high order formulae for the numerical integration of two-point boundary value problems. It is shown that the formula derived by Cash and Moore fits naturally into this framework and some new formulae of orders 4, 6 and 8 are derived using this approach. A numerical comparison with certain existing finite difference methods is made and this comparison indicates the efficiency of the high order methods for problems having a suitably smooth solution.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01944476
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    Paper (German National Licenses)
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  • 8
    Electronic Resource
    Electronic Resource
    EfficientP-stable methods for periodic initial value problems (1984)
    Springer
    BIT 24 (1984), S. 248-252 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Efficient families ofP-stable formulae are developed for the numerical integration of periodic initial value problems where the required solution has an unknown period. Formulae of orders 4 and 6 requiring respectively 2 and 4 function evaluations per step are derived and some numerical results are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01937491
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    Paper (German National Licenses)
    Fulltext
  • 9
    Electronic Resource
    Electronic Resource
    Split linear multistep methods for the numerical integration of stiff differential systems (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 299-310 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A new approach to the problem of numerically integrating stiff differential systems is described. In this approach a linear multistep method (the basic method) is split into a kind of predictor-corrector scheme, where the predictor is also implicit. If this splitting is done in an appropriate manner, the modified method has considerably better stability properties than the basic method. As a result, splitting methods are particularly useful for problems where conventional integration methods experience stability difficulties. In particular some highly stable split linear multistep methods based on backward differentiation formulae are derived and a highly stable variable step implementation is proposed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389575
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    Paper (German National Licenses)
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  • 10
    Electronic Resource
    Electronic Resource
    A note on the iterative solution of recurrence relations (1977)
    Springer
    Numerische Mathematik 27 (1977), S. 165-170 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A class of Gauss-Seidel iteration schemes suitable for the stable generation of non-dominant solutions of certain third order linear recurrence relations is developed. The algorithms derived have two main advantages over existing algorithms which generally re-formulate the problem as the solution of a system of algebraic equations. Firstly, unlike existing algorithms, the algorithms developed in this paper automatically determine the size of the system to be solved in all cases and secondly they may be extended directly to an important class of nonlinear recurrence relations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396636
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    Paper (German National Licenses)
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