Error Bounds Based on Approximation Theory

Brass, Helmut

doi:10.1007/978-94-011-2646-5_12

Cited by 5 publications

(2 citation statements)

References 20 publications

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“…which use the Clenshaw-Curtis points x i j , see (6), and are exact for all polynomials of degree less than m i . Sloan and Smith (1978) show that…”

Section: Remark 4 Our Methods Is Easily Modified To Work For Weightedmentioning

confidence: 99%

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High dimensional integration of smooth functions over cubes

Novak

Ritter

1996

Numerische Mathematik

323

249

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We construct a new algorithm for the numerical integration of functions that are defined on a d -dimensional cube. It is based on the Clenshaw-Curtis rule for d = 1 and on Smolyak's construction. This way we make the best use of the smoothness properties of any (nonperiodic) function. We prove error bounds showing that our algorithm is almost optimal (up to logarithmic factors) for different classes of functions with bounded mixed derivative. Numerical results show that the new method is very competitive, in particular for smooth integrands and d ≥ 8. Classification (1991): 41A55, 41A63, 65D30, 65Y20 Mathematics Subject

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“…which use the Clenshaw-Curtis points x i j , see (6), and are exact for all polynomials of degree less than m i . Sloan and Smith (1978) show that…”

Section: Remark 4 Our Methods Is Easily Modified To Work For Weightedmentioning

confidence: 99%

“…This property holds, for instance, for any sequence of interpolatory formulas U i with positive weights, see Brass (1992). We suggest to use the Clenshaw-Curtis method with a suitable choice of the sequence m i .…”

Section: The Clenshaw-curtis Rulementioning

confidence: 99%

High dimensional integration of smooth functions over cubes

Novak

Ritter

1996

Numerische Mathematik

323

249

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A Note On Variable Knot, Variable Order Composite Quadrature For Integrands With Power Singularities

Schwab

1992

Numerical Integration

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Verified computed Peano constants and applications in numerical quadrature

Chen

2007

Bit Numer Math

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Peano kernels are critical for the error estimates of numerical approximation of linear functionals. In the literature, considerable efforts have been devoted to the numerical computation or estimation of Peano constants, however, their discussions have mainly focused on the Peano kernels of some specific higher orders related to the degrees of the underlying error functionals. Limiting to such higher orders either requires certain smoothness of the approximated functions, which is not always fulfilled, or is not always the optimal choice for error estimates, even if the approximated functions are sufficiently smooth. Practical considerations in computing Peano constants for the full range of orders are given in this paper. Numerical examples are also given to demonstrate that much better error bounds can be derived while Peano constants of lower orders are considered. (2000): 41-04, 41A44, 41A55, 41A80, 65-04, 65A05, 65D20, 65D30, 65G20, 65G30. AMS subject classification

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Error Bounds Based on Approximation Theory

Cited by 5 publications

References 20 publications

High dimensional integration of smooth functions over cubes

High dimensional integration of smooth functions over cubes

A Note On Variable Knot, Variable Order Composite Quadrature For Integrands With Power Singularities

Verified computed Peano constants and applications in numerical quadrature

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