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  • Articles  (14)
  • CR: 5.13  (14)
  • Springer  (14)
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  • Articles  (14)
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  • Springer  (14)
  • Annual Reviews
  • Blackwell Publishing Ltd
  • Elsevier
  • Periodicals Archive Online (PAO)
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  • 2005-2009
  • 1990-1994
  • 1980-1984  (14)
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  • 1
    Electronic Resource
    Electronic Resource
    Fehlerabschätzungen für Koeffizienten von Exponentialsummen und Polynomen (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 293-307 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS):65D15 ; CR: 5.13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Summary Equivalence constants for the norms on parameter and function space are considered for both polynomials and exponential sums. In the latter case “Chebyshev” exponential sums are introduced as generalizations of the Chebyshev polynomials, providing a practical method for error estimation in parameter space. For theoretical purposes a “Markoff-type” inequality is proved. In the case of polynomials asymptotically optimal constants are derived, thus improving on earlier results of Gautschi. Furthermore, a simple construction of the equivalence constants for the monomial basis is included.
    Notes: Zusammenfassung Für Polynome und Exponentialsummen mit festen Frequenzen werden die Normäquivalenzkonstanten zwischen Parameterraum und Funktionenraum untersucht. Dies führt im Exponentialsummenfall auf “Tschebyscheff-Exponentialsummen” als Verallgemeinerung der Tschebyscheff-Polynome, wenn man nach numerisch praktikablen Strategien zur Fehlerabschätzung im Parameterraum sucht; für theoretische Zwecke wird eine Ungleichung von Markoff-Typ für Exponentialsummen hergeleitet. Im Falle der Polynome ergeben sich asymptotisch optimale Konstanten als Verschärfungen von Resultaten von Gautschi. Ferner wird eine elementare Herleitung der Normäquivalenzkonstanten für den Fall der Monombasis angegeben.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01408701
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  • 2
    Electronic Resource
    Electronic Resource
    The ɛ-algorithm and multivariate Padé-approximants (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 39-46 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D15 ; CR: 5.13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In the univariate case the ɛ-algorithm of Wynn is closely related to the Padé-table in the following sense: if we apply the ɛ-algorithm to the partial sums of the power series $$f(x) = \sum\limits_{i = 0}^\infty {c_i x^i } $$ then ε 2m l−m is the (l, m) Padé-approximant tof(x) wherel is the degree of the numerator andm is the degree of the denominator [1 pp. 66–68]. Several generalizations of the ɛ-algorithm exist but without any connection with a theory of Padé-approximants. Also several definitions of the Padé-approximant to a multivariate function exist, but up till now without any connection with the ɛ-algorithm. In this paper, we see that the multivariate Padé-approximants introduced in [3], satisfy the same property as the univariate Padé-approximants: if we apply the ɛ-algorithm to the partial sums of the power series $$f\left( {x_1 ,...,x_n } \right) = \sum\limits_{i_1 + ... + i_n = 0}^\infty {c_{i_1 ...i_n } x_1^{i_1 } ...x_n^{i_n } } $$ then ε 2m (l−m) is the (l, m) multivariate Padé-approximant tof(x 1, ...,x n ).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01459073
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  • 3
    Electronic Resource
    Electronic Resource
    Discrete cubic spline interpolation (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 71-78 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS) 65D05 ; CR: 5.13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In the present paper we study the existence, uniqueness and convergence of discrete cubic spline which interpolate to a given function at one interior point of each mesh interval. Our result in particular, includes the interpolation problems concerning continuous periodic cubic splines and discrete cubic splines with boundary conditions considered respectively in Meir and Sharma (1968) and Lyche (1976) for the case of equidistant knots.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01459076
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  • 4
    Electronic Resource
    Electronic Resource
    An alternative computational method for the approximation of random functions (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 361-368 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) 65D15 ; CR: 5.13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper shows that a computational procedure for approximation of random functions can be accomplished using purely linear programming techniques. This contrasts with previous results which use a twostage approach for the computation, one of which requires linear programming techniques. Computational results are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01399005
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  • 5
    Electronic Resource
    Electronic Resource
    An algorithm for discrete linearL p approximation (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 129-139 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) 65D15 ; CR: 5.13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper duality theory is used to derive an algorithm for the solution of the discrete linearL p approximation problem (for 1〈p〈2). This algorithm turns out to be similar to existing iteratively reweighted least squares algorithms, but can be shown to be globally andQ-superlinearly convergent.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01395812
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  • 6
    Electronic Resource
    Electronic Resource
    Solving complex approximation problems by semiinfinite - finite optimization techniques: A study on convergence (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 411-420 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D15 ; CR: 5.13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We transform a complex approximation problem into an equivalent semiinfinite optimization problem whose constraints are described in terms of a quantity ϕ∈[0,2π[=I. We study the effect of disturbing the problem by replacingI by a compact subsetM⊂I which includes as special case the discrete case whereM consists only of finitely many points. We introduce a measure ɛ for the deviation ofM fromI and show that in any complex approximation problem the minimal distance of the disturbed problem converges quadratically with ɛ→0 to the minimal distance of the undisturbed problem which is a generalization of a result by Streit and Nuttall. We also show that in a linear finite dimensional approximation problem the convergence of the coefficients of the disturbed problem is in general at most linear. There are some graphical representations of best complex approximations computed with the described method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01407872
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  • 7
    Electronic Resource
    Electronic Resource
    Unicity of best one-sidedL 1-approximations (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 229-243 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D15 ; CR: 5.13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper deals with the problem of uniqueness in one-sidedL 1-approximation. The chief purpose is to characterize finite dimensional subspacesG of the space of continuous or differentiable functions which have a unique best one-sidedL 1-approximation. In addition, we study a related problem in moment theory. These considerations have an important application to the uniqueness of quadrature formulae of highest possible degree of precision.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400541
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  • 8
    Electronic Resource
    Electronic Resource
    Strong uniqueness and second order convergence in nonlinear discrete approximation (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 439-455 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D15 ; CR: 5.13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Strong uniqueness has proved to be an important condition in demonstrating the second order convergence of the generalised Gauss-Newton method for discrete nonlinear approximation problems [4]. Here we compare strong uniqueness with the multiplier condition which has also been used for this purpose. We describe strong uniqueness in terms of the local geometry of the unit ball and properties of the problem functions at the minimum point. When the norm is polyhedral we are able to give necessary and sufficient conditions for the second order convergence of the generalised Gauss-Newton algorithm.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01403680
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  • 9
    Electronic Resource
    Electronic Resource
    Reliable rational interpolation (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 111-128 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D05 ; CR: 5.13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A modification of the Thacher-Tukey algorithm for rational interpolation is proposed. The method employed demonstrates the reliability of the proposed algorithm as well as the reliability of the Thacher-Tukey algorithm. Furthermore, the proposed algorithm eliminates almost all the array storage space required to implement the Thacher-Tukey algorithm.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396754
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  • 10
    Electronic Resource
    Electronic Resource
    An algorithm for a special case of a generalization of the Richardson extrapolation process (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 299-307 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 63B05 ; CR: 5.13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A special case of a generalization of the Richardson extrapolation process is considered, and its complete solution is given in closed form. Using this, an algorithm for implementing the extrapolation is devised. It is shown that this algorithm needs a very small amount of arithmetic operations and very little storage. Convergence and stability properties for some cases are also considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396434
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  • 11
    Electronic Resource
    Electronic Resource
    Optimal cycles in doubly weighted graphs and approximation of bivariate functions by univariate ones (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 65-84 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D15 ; CR: 5.13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The problem of finding optimal cycles in a doubly weighted directed graph (Problem A) is closely related to the problem of approximating bivariate functions by the sum of two univariate functions with respect to the supremum norm (Problem B). The close relationship between Problem A and Problem B is detected by the characterization (7.4) of the distance dist (f, t) of Problem B. In Part 1 we construct an algorithm for Problem A where the essential role is played by the minimal lengthsy j(k) defined by (2.3). If weight functiont≡1 then the minimum of Problem A is computed by equality (2.4). Ift≡1 then the minimum is obtained by a binary search procedure, Algorithm 3. In Part 2 we construct our algorithms for solving Problem B by following exactly the ideas of Part 1. By Algorithm 4 we compute the minimal pseudolengthsh k(y, M) defined by (7.5). If weight functiont≡1 then the infimum dist(f,t) of Problem B is obtained by equality (7.12) which is closely related to (2.4). Ift≢1 we compute the infimum dist(f,t) by the binary search procedure Algorithm 5. Additionally, Algorithm 4 leads to a constructive proof of the existence of continuous optimal solutions of Problem B (see Theorem 7.1e) which is already known in caset≡1 but unknown in caset≢1. Interesting applications to the steady-state behaviour of industrial processes with interference (Sect. 3) and the solution of integral equations (Problem C) are included.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01399312
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  • 12
    Electronic Resource
    Electronic Resource
    Procedures for kernel approximation and solution of fredholm integral equations of the second kind (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 125-141 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 41A15 ; 65R05 ; 68A10 ; CR: 5.13 ; 5.18
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The purpose of this paper is to present explicit ALGOL procedures for (1) the approximation of a kernel (surface) by tensor products of splines, and (2) the computation of approximate eigenvalues and eigenfunctions for Fredholm integral equations of the second kind.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396055
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  • 13
    Electronic Resource
    Electronic Resource
    Ein verallgemeinerter Kettenbruch-Algorithmus zur rationalen Hermite-Interpolation (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 99-107 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D05 ; CR: 5.13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper we consider rational interpolation for an Hermite Problem, i.e. prescribed values of functionf and its derivatives. The algorithm presented here computes a solutionp/q of the linearized equationsp−fq=0 in form of a generalized continued fraction. Numeratorp and denominatorq of the solution attain minimal degree compatible with the linearized problem. The main advantage of this algorithm lies in the fact that accidental zeros of denominator calculated during the algorithm cannot lead to an unexpected stop of the algorithm. Unattainable points are characterized.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01395991
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  • 14
    Electronic Resource
    Electronic Resource
    The Haar condition and multiplicity of zeros (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 139-153 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 41A52 ; 65H05 ; CR: 5.13 ; 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper the problem is investigated of how to take the (possibly noninteger) multiplicity of zeros into account in the Haar condition for a linear function space on a given interval. Therefore, a distinction is made between regular and singular points of the interval, and a notion of geometric multiplicity, which always is a positive integer, is introduced. It is pointed out that, for regular zeros (i.e., zeros situate at regular points), aq-fold zero (in the sense that its geometric multiplicity equalsq), counts forq distinct zeros in the Haar condition. For singular zeros (i.e., zeros situated at singular points), this geometric multiplicity has to be diminished by some well-determinable integer.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01399317
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