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  • 1
    Digitale Medien
    Digitale Medien
    The structured sensitivity of Vandermonde-like systems (1992)
    Springer
    Numerische Mathematik 62 (1992), S. 17-33 
    Details
    ISSN: 0945-3245
    Schlagwort(e): 65F99
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We consider a general class of structured matrices that includes (possibly confluent) Vandermonde and Vandermonde-like matrices. Here the entries in the matrix depend nonlinearly upon a vector of parameters. We define, condition numbers that measure the componentwise sensitivity of the associated primal and dual solutions to small componentwise perturbations in the parameters and in the right-hand side. Convenient expressions are derived for the infinity norm based condition numbers, and order-of-magnitude estimates are given for condition numbers defined in terms of a general vector norm. We then discuss the computation of the corresponding backward errors. After linearising the constraints, we derive an exact expression for the infinity norm dual backward error and show that the corresponding primal backward error is given by the minimum infinity-norm solution of an underdetermined linear system. Exact componentwise condition numbers are also derived for matrix inversion and the least squares problem, and the linearised least squares backward error is characterised.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396218
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  • 2
    Digitale Medien
    Digitale Medien
    A-Stability and Stochastic Mean-Square Stability (2000)
    Springer
    BIT 40 (2000), S. 404-409 
    Details
    ISSN: 1572-9125
    Schlagwort(e): Mean-square ; Milstein method ; multiplicative noise ; Theta method
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract This note extends and interprets a result of Saito and Mitsui [SIAM J. Numer. Anal., 33 (1996), pp. 2254–2267] for a method of Milstein. The result concerns mean-square stability on a stochastic differential equation test problem with multiplicative noise. The numerical method reduces to the Theta Method on deterministic problems. Saito and Mitsui showed that the deterministic A-stability property of the Theta Method does not carry through to the mean-square context in general, and gave a condition under which unconditional stability holds. The main purpose of this note is to emphasize that the approach of Saito and Mitsui makes it possible to quantify precisely the point where unconditional stability is lost in terms of the ratio of the drift (deterministic) and diffusion (stochastic) coefficients. This leads to a concept akin to deterministic A(α)-stability that may be useful in the stability analysis of more general methods. It is also shown that mean-square A-stability is recovered if the Theta Method parameter is increased beyond its normal range to the value 3/2.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1022355410570
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  • 3
    Digitale Medien
    Digitale Medien
    Stiffness of ODEs (1993)
    Springer
    BIT 33 (1993), S. 285-303 
    Details
    ISSN: 1572-9125
    Schlagwort(e): 65L05 ; stiffness ; stability ; pseudospectra
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract It is argued that even for a linear system of ODEs with constant coefficients, stiffness cannot properly be characterized in terms of the eigenvalues of the Jacobian, because stiffness is a transient phenomenon whereas the significance of eigenvalues is asymptotic. Recent theory from the numerical solution of PDEs is adapted to show that a more appropriate characterization can be based upon pseudospectra instead of spectra. Numerical experiments with an adaptive ODE solver illustrate these findings.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01989751
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  • 4
    Digitale Medien
    Digitale Medien
    Runge-Kutta stability on a Floquet problem (1994)
    Springer
    BIT 34 (1994), S. 88-98 
    Details
    ISSN: 1572-9125
    Schlagwort(e): Runge-Kutta ; absolute stability ; Floquet ; equilibrium ; steady state ; 65L05
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract This work examines the stability of explicit Runge-Kutta methods applied to a certain linear ordinary differential equation with periodic coefficients. On this problem naïve use of the eigenvalues of the Jacobian results in misleading conclusions about stable behaviour. It is shown, however, that a valid analogue of the classical absolute stability theory can be developed. Further, using a suitable generalisation of the equilibrium theory of Hall [ACM Trans. on Math. Soft. 11 (1985), pp. 289–301], accurate predictions are made about the performance of modern, adaptive algorithms.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01935018
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  • 5
    Buch
    Buch
    Learning LaTeX / (2016)
    Philadelphia, PA :SIAM,
    Details
    Titel: Learning LaTeX /
    Autor: Griffiths, David F.
    Beteiligte Person(en): Higham, Desmond J.
    Ausgabe: 2nd ed.
    Verlag: Philadelphia, PA :SIAM,
    Erscheinungsjahr: 2016
    Seiten: X, 103 S.
    ISBN: 978-1-611974-41-6
    Materialart: Buch
    Sprache: Englisch
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  • 6
    Buch
    Buch
    Matlab guide (2000)
    Philadelphia, PA :SIAM,
    Details
    Titel: Matlab guide
    Autor: Higham, Desmond J.
    Beteiligte Person(en): Higham, Nicholas J.
    Verlag: Philadelphia, PA :SIAM,
    Erscheinungsjahr: 2000
    Seiten: 283 S.
    Materialart: Buch
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  • 7
    Buch
    Buch
    Learning Latex (1997)
    Philadelphia, PA :SIAM,
    Details
    Titel: Learning Latex
    Autor: Griffiths, David F.
    Beteiligte Person(en): Higham, Desmond J.
    Verlag: Philadelphia, PA :SIAM,
    Erscheinungsjahr: 1997
    Seiten: 84 S.
    Materialart: Buch
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