Validating Computation in a Function Space

Kaucher, E.; Miranker, Willard L.

doi:10.1016/b978-0-12-505630-4.50028-6

Cited by 4 publications

(5 citation statements)

References 4 publications

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“…3.4, we assume that the outer functions F : X ⊆ R → R is s-times continuously differentiable, and we consider the scheme of qth-order (25), (26) [resp. from (26), (28)] if s ≥ q + 1; or otherwise constructed from (25), (27) [resp. from (28), (29)] if F (s) has a finite total variation on X .…”

Section: Local Convergence Rate Of Univariate Outer-composition Operamentioning

confidence: 99%

“…These results are summarized in the following theorem: (25), (26) [resp. from (26), (28)] if s ≥ q + 1; or otherwise constructed from (25), (27) [resp. from (28), (29) …”

Section: Lemma 3 the Scheme (Pmentioning

confidence: 99%

“…Otherwise, the following bound may be used instead of (27) if F and its derivatives through F (s−1) are absolutely continuous on W and…”

Section: Univariate Outer-composition Operationsmentioning

confidence: 99%

“…Regarding univariate functions, improved variants based on the Chebyshev polynomials, Bernstein polynomials and others were widely used during the early 1980s under the name of ultra-arithmetic [26]. A comparison of univariate Taylor forms and Chebyshev forms in [27] showed that expansions in Chebyshev series may be orders of magnitude more accurate than expansions in Taylor series. The use of Chebyshev series expansion as an arithmetic is also the philosophy behind the package Chebfun by Trefethen et al [6,57], which has recently been extended to functions in two variables as well [56].…”

Section: Introductionmentioning

confidence: 99%

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Chebyshev model arithmetic for factorable functions

et al. 2016

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This article presents an arithmetic for the computation of Chebyshev models for factorable functions and an analysis of their convergence properties. Similar to Taylor models, Chebyshev models consist of a pair of a multivariate polynomial approximating the factorable function and an interval remainder term bounding the actual gap with this polynomial approximant. Propagation rules and local convergence bounds are established for the addition, multiplication and composition operations with Chebyshev models. The global convergence of this arithmetic as the polynomial expansion order increases is also discussed. A generic implementation of Chebyshev model arithmetic is available in the library MC++. It is shown through several numerical case studies that Chebyshev models provide tighter bounds than their Taylor model counterparts, but this comes at the price of extra computational burden.

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Section: Local Convergence Rate Of Univariate Outer-composition Operamentioning

confidence: 99%

“…These results are summarized in the following theorem: (25), (26) [resp. from (26), (28)] if s ≥ q + 1; or otherwise constructed from (25), (27) [resp. from (28), (29) …”

Section: Lemma 3 the Scheme (Pmentioning

confidence: 99%

“…Otherwise, the following bound may be used instead of (27) if F and its derivatives through F (s−1) are absolutely continuous on W and…”

Section: Univariate Outer-composition Operationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Chebyshev model arithmetic for factorable functions

et al. 2016

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“…This is one simple instance of the functional enclosure methods mentioned in Section 1.1 and widely used in interval analysis [44]. In most cases (with the notable exception of [21,30,31]), these methods are based on interval Taylor series expansions such as so-called Taylor models [37,45]. Here, we describe an adaptation designed to work in linear time with polynomial approximations written on the Chebyshev basis.…”

Section: Validationmentioning

confidence: 99%

Rigorous uniform approximation of D-finite functions using Chebyshev expansions

Benoit¹,

Joldeş²,

Mezzarobba³

2016

Math. Comp.

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Abstract. A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of such methods in the context of rigorous computing (where we need guarantees on the accuracy of the result), and from the complexity point of view.It is well-known that the order-truncation of the Chebyshev expansion of a function over a given interval is a near-best uniform polynomial approximation of the function on that interval. In the case of solutions of linear differential equations with polynomial coefficients, the coefficients of the expansions obey linear recurrence relations with polynomial coefficients. Unfortunately, these recurrences do not lend themselves to a direct recursive computation of the coefficients, owing among other things to a lack of initial conditions.We show how they can nevertheless be used, as part of a validated process, to compute good uniform approximations of D-finite functions together with rigorous error bounds, and we study the complexity of the resulting algorithms. Our approach is based on a new view of a classical numerical method going back to Clenshaw, combined with a functional enclosure method.

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A Symbolic-Numeric Validation Algorithm for Linear ODEs with Newton–Picard Method

Bréhard

2021

Math.Comput.Sci.

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Validating Computation in a Function Space

Cited by 4 publications

References 4 publications

Chebyshev model arithmetic for factorable functions

Chebyshev model arithmetic for factorable functions

Rigorous uniform approximation of D-finite functions using Chebyshev expansions

A Symbolic-Numeric Validation Algorithm for Linear ODEs with Newton–Picard Method

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