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  • Articles  (15)
  • AMS(MOS): 65D15  (15)
  • Springer  (15)
  • American Chemical Society
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  • 1980-1984  (15)
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  • Mathematics  (15)
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  • Articles  (15)
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  • Springer  (15)
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  • Mathematics  (15)
  • 1
    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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  • 2
    Electronic Resource
    Electronic Resource
    A stable method for the evaluation of a polynomial and of a rational function of one variable (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 309-319 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D15 ; 65G05 ; CR: 5.11
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary It is shown that if one uses a carefully defined concept of stability (Stewart, G.W., Introduction to matrix computations, Academic Press, New York and London, 1973) then Horner's rule for the evaluation of a polynomial and some other evaluation methods are not always stable. A method is presented which is always stable. The operations count for this method is the same as that for Horner's rule. The method is generalized to apply to all rational functions of one variable.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01418328
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  • 3
    Electronic Resource
    Electronic Resource
    Padé-type approximants of exp (−z) whose denominators are (1+z/n) n (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 283-292 
    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, we show that the sequences of Padé-type approximants (k−1/k) and (k/k) converge to exp (−z), uniformly and geometrically on every compact subset of the plane. A numerical study has been done, which discriminates these sequences from the point of view ofA-acceptability.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390128
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  • 4
    Electronic Resource
    Electronic Resource
    Best constants occuring with the modulus of continuity in the error estimate for spline interpolants of odd degree on equidistant grids (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 407-415 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D15 ; CR: 5.13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We deal with spline interpolants of odd degree on equidistant grids. The main interest is directed on cardinal and onn-periodic interpolation, wheren is odd. By the aid of an explicit representation of the interpolant by Euler-Frobenius-polynomials, we calculate the minimal constant occuring in the error estimate with the modulus of continuity.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01405571
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  • 5
    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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  • 6
    Electronic Resource
    Electronic Resource
    Quotienten-Differenzen-Algorithmus: Beweis der Regeln von Rutishauser (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 93-98 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D15 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Zusammenfassung Der Quotienten-Differenzen-Algorithmus nach Rutishauser ist geeignet zur Bestimmung von Polen meromorpher Funktionen, gegeben durch eine Taylorreihe. Sind mehrere Pole betragsgleich, so kann eine Polynomfolge bestimmt werden, deren Grenzpolynom diese Pole als Nullstellen hat. Die Konvergenz wurde von Rutishauser jedoch nicht bewiesen. Ein Beweis wird in der vorliegenden Arbeit präsentiert.
    Notes: Summary The Quotient-Difference Algorithm of Rutishauser can be used for the determination of poles of a meromorphic function given by its power series. If some of the poles have same modulus, a sequence of polynomials can be determined such that the limiting polynomial has exactly these poles as zeros. The convergence has not been proved by Rutishauser, however. A proof is presented in this paper.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01459078
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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
    A Remez type algorithm for spline functions (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 117-146 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D15 ; CR: 5.13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A Remez type algorithm for computing best spline approximations of degreen withk fixed knots is developed. It is shown that the sequence constructed in the algorithm converges to a best approximation, ifk≦n+1, and at least to a nearly best approximation, ifk〉n+1. In contrast to the standard interpolation methods no restriction to the position of the knots is required. This fact may be important to treat approximation problems for splines with free knots.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396309
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  • 9
    Electronic Resource
    Electronic Resource
    The QD-algorithm and multivariate Padé-approximants (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 259-269 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D15 ; CR: 5.13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The quotient-difference (QD) algorithm can be used to construct univariate Padé-approximants [1]. In this paper we see that it can also be used to construct the multivariate Padé-approximants introduced in [3], just by reformulating the quotient-difference algorithm as in Sect. 1. The multivariate Padé-approximants and the multivariate QD-scheme are treated in the Sects. 2 and 3 respectively. Thus for this type of multivariate Padé-approximants a link with the theory of multivariate continued fractions is established.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389572
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  • 10
    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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  • 11
    Electronic Resource
    Electronic Resource
    General order Newton-Padé approximants for multivariate functions (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 293-307 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D15 ; CR: 5.13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Padé approximants are a frequently used tool for the solution of mathematical problems. One of the main drawbacks of their use for multivariate functions is the calculation of the derivatives off(x 1, ...,x p ). Therefore multivariate Newton-Padé approximants are introduced; their computation will only use the value off at some points. In Sect. 1 we shall repeat the univariate Newton-Padé approximation problem which is a rational Hermite interpolation problem. In Sect. 2 we sketch some problems that can arise when dealing with multivariate interpolation. In Sect. 3 we define multivariate divided differences and prove some lemmas that will be useful tools for the introduction of multivariate Newton-Padé approximants in Sect. 4. A numerical example is given in Sect. 5, together with the proof that forp=1 the classical Newton-Padé approximants for a univariate function are obtained.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390129
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  • 12
    Electronic Resource
    Electronic Resource
    Error estimates for spline interpolants on equidistant grids (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 417-424 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D15 ; CR: 5.13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary For oddm, the error of them-th-degree spline interpolant of power growth on an equidistant grid is estimated. The method is based on a decomposition formula for the spline function, which locally can be represented as an interpolation polynomial of degreem which is corrected by an (m+1)-st.-order difference term.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01405572
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  • 13
    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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  • 14
    Electronic Resource
    Electronic Resource
    Die Struktur der normalen rationalen Funktionen (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 379-388 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D15 ; 41A20 ; CR: 5.13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We determine the connected components of the set of normal elements of the family ℛ m n [a,b] of rational functions. Numerical difficulties occuring with the computation of the Chebyshev approximation via the Remez algorithm can be caused by its disconnectedness. In order to illustrate this we give numerical examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390180
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  • 15
    Electronic Resource
    Electronic Resource
    Discrete approximations to spherically symmetric distributions (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 53-60 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D15 ; CR: 5. 13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider the approximation of spherically symmetric distributions in ℝ d by linear combinations of Heaviside step functions or Dirac delta functions. The approximations are required to faithfully reproduce as many moments as possible. We discuss stable methods of computing such approximations, taking advantage of the close connection with Gauss-Christoffel quadrature. Numerical results are presented for the distributions of Maxwell, Bose-Einstein, and Fermi-Dirac.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389754
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