Monotonicity and error bounds in schemes of Romberg type for the computation off(n) (0) by central differences and extrapolation

Ström, Torsten

doi:10.1007/bf01436182

Cited by 4 publications

(6 citation statements)

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“…In a recent paper [1] Str6m has derived some results relating to an extrapolation algorithm for numerical differentiation. We summarize the algorithm and the results in this section.…”

Section: Previous Resultsmentioning

confidence: 99%

On numerical differentiation

Ström

Lyness

1975

BIT

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Abstract.In a recent paper Str6m analyzed a simple extrapola~ion algorithm for numerical differentiation and derived certain properties abou~ the kernel function of the integral representation of the remainder term. These properties are useful for placing bounds on the error in cases when specified higher order derivatives are known not 1be change sign. The algorithm involves a separate Romberg table for each derivative and is rather inconvenien~ from the poin~ of view of economizing the number of function values required.In this paper we generalize Str6m's results in two stages. First we show that they are valid for a very wide choice of definitions of the initial column of each Romberg table. Then we show that one such choice, making full use of the computed function values, gives results identical to those that can be obtained using an algorithm suggested by Lyness and Moler with a particular choice of sequence of function evaluations.There is no delb~iled discussion of the effect of round-off error.

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“…In a recent paper [1] Str6m has derived some results relating to an extrapolation algorithm for numerical differentiation. We summarize the algorithm and the results in this section.…”

Section: Previous Resultsmentioning

confidence: 99%

On numerical differentiation

Ström

Lyness

1975

BIT

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“…The use of a geometric sequence has of course other advantages, such as the simplification of the Neville extrapolation algorithms (3b) and (7b), and this is the probable reason for their being favoured by earlier authors such as Rutishauser [13] and Fflippi and Engels [5]. The applicability of the error estimation techniques first suggested for general extrapolation processes by Bulirsch and Stoer [2], and later discussed in the particular context of differentiation by Engels [3,4] and StrSm [17,18], constitutes a further reason for employing geometric sequences, usually with fl < 1.…”

Section: Finally We Note Frommentioning

confidence: 96%

“…We shall denote the minimum values of S~ which result, by S~*, and they are given to one decimal place in Table la for n < 10. For symmetric approximations (7) it is convenient ¢o write, recalling (6) and (14), (17) ~ -: r2i_ 1 = -r~,…”

Section: Optimum Sequences {Rl¢")}mentioning

confidence: 99%

The selection of interpolation points in numerical differentiation

Oliver

Ruffhead

1975

BIT

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Abstract.The relationships between various numerical methods for obtaining polynomial approximations to the first derivative of a known function are investigated, and their computational advantages discussed. Optimum sequences of interpolation points are then selected with the objective of minimising the relative contribution of rounding errors to the total error, and geometric sequences, though non-optimal in this sense, are considered for computational reasons.

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“…Classical estimates were given by BulirschStoer [2]. Non-asymptotic examples may be found in StrSm [7,8] where, however, a majorant concept introduces constraints and extra computational labour. Schmidt [6] presents an interesting approach which does not explicitly involve the stepsizes.…”

Section: T~"mentioning

confidence: 98%

“…Our basic argument for (8) was "later columns converge at least as fast as earlier ones". This is true very generally although the actual speed of convergence may deviate considerably from any a priori estimate.…”

Section: An Elementary Argumentmentioning

confidence: 99%