An Error Analysis for Numerical Differentiation

Johnson, L. Clark; Riess, R. D.

doi:10.1093/imamat/11.1.115

Cited by 9 publications

(2 citation statements)

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“…75-81] and Rutishauser [13]. Here a sequence of second-order central difference approximations, Johnson and Riess [7] proved that the final result T Although thus mathematically equivalent to the algorithm described earlier, this extrapolation (7b) of the symmetric second-order approximations (7a) clearly involves fewer arithmetic operations and would seem to be preferable from this standpoint. If f(x) can be evaluated only on one side of x 0, however, then we must revert to extrapolating the one-sided approximations (3a), as Fflippi and Engels [5] pointed out.…”

Section: Relationships Between Methodsmentioning

confidence: 95%

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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Section: Relationships Between Methodsmentioning

confidence: 95%

The selection of interpolation points in numerical differentiation

Oliver

Ruffhead

1975

BIT

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“…[45], [65], [198]- [201], [217]- [220] -and of course mimics several of its ascendants -see e.g. [2], [3], [7]- [8], [9], [13], [19], [38], [44], [47], [48], [51], [52], [59], [73], [79], [97], [103], [110], [111], [133], [134], [146], [155]- [156], [157], [158], [162], [180]- [181], [182], [188], [193], [203], [204], [216], [225].…”

Section: Behavior Near the Causticmentioning

confidence: 99%

On Complex-valued 2D Eikonals. Part Four: Continuation Past a Caustic

Magnanini

Talenti

2009

Milan J. Math.

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Abstract. Theories of monochromatic high-frequency electromagnetic fields have been designed by Felsen, Kravtsov, Ludwig and others with a view to portraying features that are ignored by geometrical optics. These theories have recourse to eikonals that encode information on both phase and amplitude -in other words, are complex-valued. The following mathematical principle is ultimately behind the scenes: any geometric optical eikonal, which conventional rays engender in some light region, can be consistently continued in the shadow region beyond the relevant caustic, provided an alternative eikonal, endowed with a non-zero imaginary part, comes on stage.In the present paper we explore such a principle in dimension 2. We investigate a partial differential system that governs the real and the imaginary parts of complex-valued two-dimensional eikonals, and an initial value problem germane to it. In physical terms, the problem in hand amounts to detecting waves that rise beside, but on the dark side of, a given caustic. In mathematical terms, such a problem shows two main peculiarities: on the one hand, degeneracy near the initial curve; on the other hand, ill-posedness in the sense of Hadamard. We benefit from using a number of technical devices: hodograph transforms, artificial viscosity, and a suitable discretization. Approximate differentiation and a parody of the quasi-reversibility method are also involved. We offer an algorithm that restrains instability and produces effective approximate solutions. Mathematics Subject Classification (2000). Primary 35F25, 35Q60, 78A05; Secondary 65D25.

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Numerical Differentiation

Murio¹

1993

The Mollification Method and the Numerical Solution of Ill‐Posed Problems

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Abstract.A new approach to the construction of finite-difference methods is presented. It is shown how the multi-point differentiators can generate regularizing algorithms with a stepsize h being a regularization parameter. The explicitly computable estimation constants are given. Also an iteratively regularized scheme for solving the numerical differentiation problem in the form of Volterra integral equation is developed.

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An Error Analysis for Numerical Differentiation

Cited by 9 publications

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The selection of interpolation points in numerical differentiation

The selection of interpolation points in numerical differentiation

On Complex-valued 2D Eikonals. Part Four: Continuation Past a Caustic

Numerical Differentiation

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