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  • 1
    Electronic Resource
    Electronic Resource
    On transformations of graded matrices, with applications to stiff ODE's (1985)
    Springer
    Numerische Mathematik 47 (1985), S. 363-385 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: G1.3,G1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Estimates concerning the spectrum of a graded matrix and other information useful for a reliable and efficient handling of certain complications in the numerical treatment of some stiff ODE's, can be inexpensively obtained from the factorized Jacobian. The validity of the estimates is studied by considering them as the first step in a block LR algorithm, which may be of interest in its own right. Its convergence properties are examined.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389585
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  • 2
    Electronic Resource
    Electronic Resource
    On the uniform power-boundedness of a family of matrices and the applications to one-leg and linear multistep methods (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 1-13 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We study the difference equations obtained when a linear multistep method is applied to the scalar test equationdy/dt=λy and constant stepsizeh. LetS be the region of the absolute stability of the method, and letD be a closed subset ofS (on the Riemann sphere $$\mathbb{C}$$ ). It is shown that the solutions of these difference equations are bounded forn≧0, uniformly for λh∈D.S is itself closed in $$\mathbb{C}$$ iff ∂S is free of cusps. The question is studed by means of contractivity analysis and a matrix theorem, derived from the matrix theorem of Kreiss.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400914
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Convergence acceleration from the point of view of linear programming (1965)
    Springer
    BIT 5 (1965), S. 1-16 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The best possible bounds for the sum of an alternating series with completely monotonic terms, when 2N terms have been computed, are determined. It is shown that their difference decreases exponentially withN. Various generalizations are indicated. The optimal application of Euler's transformation is also discussed. The error of that method also decreases exponentially, though the logarithmic decrement is only about 2/3 compared with the best possible error bounds.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01975719
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    A “multigrid” extension of theFFT for the numerical inversion of Fourier and Laplace Transforms (1993)
    Springer
    BIT 33 (1993), S. 85-112 
    Details
    ISSN: 1572-9125
    Keywords: 65R10 ; 42A04 ; 65B20 ; 65D07
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An algorithm for accurate numerical inversion of slowly convergent Fourier and Laplace Transforms is studied. It makes use of several equidistant grids with the same number of points, covering different symmetric intervals of the time and frequency axes. Typically, the number of operations per computed function value is about twice as large as for an ordinary FFT. The distribution of points is, however, for many applications much more adequate because, globally, the union of the grids is an approximately equidistant point set on a logarithmic scale.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01990346
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  • 5
    Electronic Resource
    Electronic Resource
    On Summation Formulas Due to Plana, Lindelöf and Abel, and Related Gauss-Christoffel Rules, III (1999)
    Springer
    BIT 39 (1999), S. 51-78 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This is the last part of a work that has been divided into three parts. It contains Sections 4 and 5. Part I and Part II were published in BIT, vol. 37. Some amendments to the earlier parts are given in an appendix of this part. The reader is referred to Part I for the abstract, the contents and the references of the whole work. A short list of references for this part is found at the end of this part. A few new references of interest for the earlier parts are included.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022317224199
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  • 6
    Electronic Resource
    Electronic Resource
    Extrapolation of asymptotic expansions by a modified Aitken δ2-formula (1981)
    Springer
    BIT 21 (1981), S. 56-65 
    Details
    ISSN: 1572-9125
    Keywords: Primary 65B05 ; Secondary 65B10 ; 65D30 ; 65G99 ; Extrapolation ; error analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A modified Aitken formula permits iterated extrapolations to efficiently estimates 221E; froms n when an asymptotic expansion $$\int_0^x {(T_k^* (t)/(x^p - t^p )^\alpha ) } dt, 0 \mathbin{\lower.3ex\hbox{$\buildrel〈\over{\smash{\scriptstyle=}\vphantom{_x}}$}} x \mathbin{\lower.3ex\hbox{$\buildrel〈\over{\smash{\scriptstyle=}\vphantom{_x}}$}} 1,$$ holds for some (unknown) coefficientsc j . We study the truncation and irregular error and compare the method with other forms of extrapolation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01934071
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  • 7
    Electronic Resource
    Electronic Resource
    A randomly generated program for automatic identity checking (1975)
    Springer
    BIT 15 (1975), S. 381-384 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A method for making secret programs for automatic identity checking with very high demands for security, is described. The consistency of the sequence of digits, which identifies the person, is tested by a sequence of transformations, chosen by means of a stored secret random table. The method has been implemented for a system of unattended cash dispenser terminals, equipped with microprocessors.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01931676
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    Paper (German National Licenses)
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  • 8
    Electronic Resource
    Electronic Resource
    On accuracy and unconditional stability of linear multistep methods for second order differential equations (1978)
    Springer
    BIT 18 (1978), S. 133-136 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Linear multistep methods for solution of the equationy″=f(t, y) are studied by means of the test equationy″=−ω2 y, with ω real. It is shown that the order of accuracy cannot exceed 2 for an unconditionally stable method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01931689
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  • 9
    Electronic Resource
    Electronic Resource
    G-stability is equivalent toA-stability (1978)
    Springer
    BIT 18 (1978), S. 384-401 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In 1975 the author showed that a norm (Liapunov function) can be constructed for the stability and error analysis of a linear multistep method (and the related one-leg method) for the solution of stiff non-linear systems, provided that the system satisfies a monotonicity condition and the method possesses a property calledG-stability. In this paper it is shown thatG-stability is equivalent toA-stability. More generally, a Liapunov function exists if the stability region of the method contains a circle (half-plane), provided that the system satisfies a monotonicity condition related to this circle (half-plane). In the general case this condition depends on the stepsize.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01932018
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  • 10
    Electronic Resource
    Electronic Resource
    33 years of numerical instability, Part I (1985)
    Springer
    BIT 25 (1985), S. 188-204 
    Details
    ISSN: 1572-9125
    Keywords: Numerical instability ; history ; differential equations ; difference equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract BIT has played and plays a great role in the development of concepts concerning numerical (in)stability in initial value problems forODE's and related questions. This development is here seen through the looking-glass of the author, who experienced much of its pains and pleasures. The article is based on a talk given in 1981 at the Zürich symposium to commemorate the tenth anniversary of the death of the eminent Swiss numerical analyst, Heinz Rutishauser. The presentation is mainly chronological with a few digressions. Part I ends at the beginning of the stiff epoch.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01934997
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