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  • 1
    Electronic Resource
    Electronic Resource
    Uniform rational approximation on large data sets (1982)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 18 (1982), S. 1569-1575 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: We report on a very effective program for uniform approximation of functions (or discrete data) by generalized rational functions of any number of variables over point sets so large that conventional methods are inapplicable. The points need not be arranged in a regular pattern, but if they are, the number of points is normally limited only by the computer time available. The program extends the differential correction algorithm discussed in an earlier article in this journal by combining it with a first Remes-type exchange procedure, while retaining the flexibility and robustness of differential correction. Furthermore, this algorithm is guaranteed to converge in theory. Testing has shown that, even when the point set is small enough for differential correction to be applied directly, this algorithm is usually faster. We discuss the algorithm and program along with several numerical examples, one of which involves more than three million data points. A FORTRAN listing with illustrative output can be obtained from the first author.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620181011
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Uniform rational approximation of functions of several variables (1975)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 9 (1975), S. 297-323 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: We present a program which has given excellent results for uniform approximation of functions by polynomials, rational functions, generalized polynomials, and generalized rational functions. The algorithm is described in detail and several examples are discussed. The approximation is done over a finite point set, which is commonly a set of real numbers or points in the plane (in the latter case we are doing what is often known as surface fitting). Input to and output from the program is in tabular form. The method used is a linear programming approach known as the differential correction algorithm, which has been shown by several authors to always converge in theory (quadratically in some situations). In practice, we have obtained convergence in nearly every case, and quadratic convergence in most cases. The program can also be used for simultaneous approximation of several functions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620090204
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Adaptive l1-l2 curve fitting (1979)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 14 (1979), S. 871-903 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In References 1 and 2, algorithms were introduced for adaptively computing smooth piecewise polynomial approximations using uniform, least-squars (l2) and restricted-range uniform approximations. This present paper introduces a FORTRAN program for computing smooth piecewise polynomial approximations to data and/or precise mathematical functions (in discrete form) which allows the user the option of using best l1 of best l2 approximations. (The l2 option of this newly developed l1-l2 adaptive curve-fitting package supersedes the code listed in Reference 1 for the old least-squares program.) In this paper, adaptive curve-fitting algorithm used in this l1-l2 package is described in detail, the FORTRAN program is the given (in the Appendix) and discussed, the numerical results are presented in an effort to illustrate how the l1 and l2 versions of this algorithm may be used most effectively.
    Additional Material: 6 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620140608
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  • 4
    Electronic Resource
    Electronic Resource
    Restricted range adaptive curve fitting (1979)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 14 (1979), S. 379-408 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In this paper we present an algorithm for adaptively computing smooth piecewise polynomial approximations using restricted range uniform approximations on a finite point set. We also present several numerical examples and offer suggestions for the effective use of this algorithm. We have found the algorithm to be effective for approximating a wide class of functions, either with or without significant levels of noise. Furthermore, since the user of this algorithm actually defines tolerance bands within which the approximation will lie, the algorithm allows the user a great deal of flexibility and control over the shape of the resulting approximations.
    Additional Material: 7 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620140306
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    Uniform rational approximation by differential correction and remes-differential correlation (1981)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 17 (1981), S. 1273-1278 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: We present two self-contained programs which have given excellent results for uniform approximation of functions (or discrete data) by generalized rational functions. The first program is an extension of a program published in 1975 which uses a linear programming approach known as the differential correction algorithm. The present version is more robust, and allows for the use of a multiplicative weight function and restrictions on the values of the approximating function. These features make the program more suitable for applications such as digital filter approximation. The second program uses a combination of the Remes and differential correction algorithms which combines some of the good features of both algorithms. Given here is a discussion of the algorithms together with several examples. A FORTRAN listing can be obtained from the first author.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620170810
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    New basis functions and computational procedures for p-version finite element analysis (1993)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 36 (1993), S. 3759-3779 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: New basis functions and solution procedures for p-version finite element analysis are described. They are used in a highly efficient p-version finite element solver for linear elastostatics and dynamics, which has been used in an industrial environment for over two years. Using two sample applications it is shown that, using the techniques proposed here, p-version finite element analysis can have a substantially lower computational cost, for given accuracy, than standard finite element methods. This makes the industrial applicability of p-version finite element analysis much wider than is commonly believed.
    Additional Material: 11 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620362202
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    Paper (German National Licenses)
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