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  • 1
    ISSN: 1432-0940
    Keywords: Convergence analysis ; Differential correction algorithm ; Generalized rational approximation ; Rational approximation ; Primary ; 41A20 ; Secondary ; 65D99
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The differential correction algorithm for generalized rational functions is described, and two theorems on convergence and order of convergence are given. An example shows that the order of convergence may deteriorate from superlinear to linear when a best generalized rational approximation does not exist.
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1432-0940
    Keywords: 41A25 ; 41A30 ; 65D10 ; Multiquadric approximation ; Order of convergence ; Quasi-interpolation ; Radial basis functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The univariate multiquadric function with centerx j ∈R has the form {ϕ j (x)=[(x−x j )2+c 2]1/2, x∈R} wherec is a positive constant. We consider three approximations, namely, ℒ A f, ℒℬ f, and ℒ C f, to a function {f(x),x 0≤x≤x N } from the space that is spanned by the multiquadrics {ϕ j :j=0, 1, ...,N} and by linear polynomials, the centers {x j :j=0, 1,...,N} being given distinct points of the interval [x 0,x N ]. The coefficients of ℒ A f and ℒℬ f depend just on the function values {f(x j ):j=0, 1,...,N}. while ℒ A f, ℒ C f also depends on the extreme derivativesf′(x 0) andf′(x N ). These approximations are defined by quasi-interpolation formulas that are shown to give good accuracy even if the distribution of the centers in [x 0,x N ] is very irregular. Whenf is smooth andc=O(h), whereh is the maximum distance between adjacent centers, we find that the error of each quasi-interpolant isO(h 2|logh|) away from the ends of the rangex 0≤x≤x N. Near the ends of the range, however, the accuracy of ℒ A f and ℒℬ f is onlyO(h), because the polynomial terms of these approximations are zero and a constant, respectively. Thus, some of the known accuracy properties of quasiinterpolation when there is an infinite regular grid of centers {x j =jh:j ∈F} given by Buhmann (1988), are preserved in the case of a finite rangex 0≤x≤x N , and there is no need for the centers {x j :j=0, 1, ...,N} to be equally spaced.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Numerical algorithms 5 (1993), S. 99-120 
    ISSN: 1572-9265
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Thin plate splines are highly useful for the approximation of functions of two variables, partly because they provide the interpolant to scattered function values that minimizes a 2-norm of second derivatives. On the other hand, they have the severe disadvantage that the explicit calculation of a thin plate spline approximation requires a log function to be evaluatedm times, wherem is the number of “r 2logr 2” terms that occur. Therefore we consider a recent technique that saves much work whenm is large by forming sets of terms, and then the total contribution to the thin plate spline from the terms of each set is estimated by a single truncated Laurent expansion. In order to apply this technique, one has to pick the sets, one has to generate the coefficients of the expansions, and one has to decide which expansions give enough accuracy when the value of the spline is required at a general point of ℓ2. Our answers to these questions are different from those that are given elsewhere, as we prefer to refine sets of terms recursively by splitting them into two rather than four subsets. Some theoretical properties and several numerical results of our method are presented. They show that the work to calculate all the Laurent coefficients is usuallyO(m logm), and then onlyO(logm) operations are needed to estimate the value of the thin plate spline at a typical point of ℓ2. Thus substantial gains over direct methods are achieved form⩾200.
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 1 (1971), S. 26-57 
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We survey the development of algorithms and theory for the unconstrained optimization problem during the years 1967–1970. Therefore (except for one remark) the material is taken from papers that have already been published. This exception is an explanation of some numerical difficulties that can occur when using Davidon's (1959) variable metric algorithm.
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 11 (1976), S. 42-49 
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract It has been conjectured that the conjugate gradient method for minimizing functions of several variables has a superlinear rate of convergence, but Crowder and Wolfe show by example that the conjecture is false. Now the stronger result is given that, if the objective function is a convex quadratic and if the initial search direction is an arbitrary downhill direction, then either termination occurs or the rate of convergence is only linear, the second possibility being more usual. Relations between the starting point and the initial search direction that are necessary and sufficient for termination in the quadratic case are studied.
    Type of Medium: Electronic Resource
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 20 (1981), S. 353-357 
    ISSN: 1436-4646
    Keywords: Active Set ; Cycling ; Feasible Point ; Linear Constraints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract An example shows that, if in Rosen's [4] algorithm for calculating a point that satisfies linear constraints a natural criterion is used to select constraints to delete from active sets, then the algorithm may cycle instead of converging. There are no degeneracies in the example.
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 12 (1977), S. 241-254 
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The conjugate gradient method is particularly useful for minimizing functions of very many variables because it does not require the storage of any matrices. However the rate of convergence of the algorithm is only linear unless the iterative procedure is “restarted” occasionally. At present it is usual to restart everyn or (n + 1) iterations, wheren is the number of variables, but it is known that the frequency of restarts should depend on the objective function. Therefore the main purpose of this paper is to provide an algorithm with a restart procedure that takes account of the objective function automatically. Another purpose is to study a multiplying factor that occurs in the definition of the search direction of each iteration. Various expressions for this factor have been proposed and often it does not matter which one is used. However now some reasons are given in favour of one of these expressions. Several numerical examples are reported in support of the conclusions of this paper.
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 42 (1988), S. 171-180 
    ISSN: 1436-4646
    Keywords: Active sets ; bound constraints ; conjugate gradients ; maximum entropy ; nonlinear optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract An unusual form of the maximum entropy problem is considered, that includes simple bound constraints on the Fourier coefficients of the required image, as well as nonnegativity conditions on the image intensities. The algorithm avoids mixing these constraints by introducing a parameter into the objective function that is adjusted by an outer iteration. For each parameter value an inner iteration solves a large optimization calculation, whose constraints are just the simple bounds, by a combination of the conjugate gradient procedure and an active set method. An important feature is the ability to make many changes to the active set at once. The outer iteration includes a test for inconsistency of all the given constraints. The algorithm is described, a proof of convergence is given, and there are some second-hand remarks on numerical results.
    Type of Medium: Electronic Resource
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 4 (1973), S. 193-201 
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Some examples are given of differentiable functions of three variables, having the property that if they are treated by the minimization algorithm that searches along the coordinate directions in sequence, then the search path tends to a closed loop. On this loop the gradient of the objective function is bounded away from zero. We discuss the relevance of these examples to the problem of proving general convergence theorems for minimization algorithms that use search directions.
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  • 10
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 42 (1988), S. 3-3 
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Type of Medium: Electronic Resource
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