Accurate evaluation of a polynomial and its derivative in Bernstein form

Jiang, Hao; Li, Shengguo; Cheng, Lizhi; Su, Fang

doi:10.1016/j.camwa.2010.05.021

Cited by 21 publications

(25 citation statements)

References 11 publications

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“…But, we need a more accurate algorithm when the problem is ill-conditioned. We consider a bivariate polynomial in area [0, 1] × [0, 1] proposed by [14] ( , )…”

Section: Resultsmentioning

confidence: 99%

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Accurate and Efficient Evaluation of Chebyshev Tensor Product Surface

He¹,

Du²,

Jiang³

et al. 2017

Mathematical Problems in Engineering

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A Chebyshev tensor product surface is widely used in image analysis and numerical approximation. This article illustrates an accurate evaluation for the surface in form of Chebyshev tensor product. This algorithm is based on the application of error-free transformations to improve the traditional Clenshaw Chebyshev tensor product algorithm. Our error analysis shows that the error bound is + O( 2 ) × cond( , , ) in contrast to classic scheme × cond( , , ), where is working precision and cond( , , ) is a condition number of bivariate polynomial ( , ), which means that the accuracy of the computed result is similar to that produced by classical approach with twice working precision. Numerical experiments verify that the proposed algorithm is stable and efficient.

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“…But, we need a more accurate algorithm when the problem is ill-conditioned. We consider a bivariate polynomial in area [0, 1] × [0, 1] proposed by [14] ( , )…”

Section: Resultsmentioning

confidence: 99%

“…Compensated algorithms to evaluate the univariate polynomials in different basis have been proposed in [12][13][14][15]. Inspired by their work, we extend the univariate compensated algorithm to tensor product case using the compensated Clenshaw algorithm [16] for evaluation of Chebyshev series [15].…”

Section: Introductionmentioning

confidence: 99%

Accurate and Efficient Evaluation of Chebyshev Tensor Product Surface

He¹,

Du²,

Jiang³

et al. 2017

Mathematical Problems in Engineering

Self Cite

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“…We already mentioned that double-double algorithms are easy to derive. On the contrary, compensated algorithms have been, up to now, defined case by case and by experts of rounding error analysis [7], [9], [10], [11], [22]. For example the compensated Algorithm 9, SUM2 [22], returns a twice more accurate sum.…”

Section: Double-double and Compensated Algorithmsmentioning

confidence: 99%

“…and COMPDECASTELJAU-DER [11] for evaluating pD(x) = (x − 0.75) 7 (x − 1) and its derivative, written in the Bernstein basis, by means of deCasteljau's scheme. 4) COMPCLENSHAWI and COMPCLENSHAWII [9] for evaluating pC (x) = (x − 0.75) 7 (x − 1) 10 written in the Chebyshev basis, by means of Clenshaw's scheme.…”

Section: ) Compdecasteljaumentioning

confidence: 99%

Automatic Source-to-Source Error Compensation of Floating-Point Programs

Thévenoux

Langlois

Martel

2015

2015 IEEE 18th International Conference on Computational Science and Engineering

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International audienceNumerical programs with IEEE 754 floating-point computations may suffer from inaccuracies since finite precision arithmetic is an approximation of real arithmetic. Solutions that reduce the loss of accuracy are available as, for instance, compensated algorithms, more precise computation with double-double or similar libraries. Our objective is to automatically improve the numerical quality of a numerical program with the smallest impact on its performances. We define and implement source code transformation to derive automatically compensated programs. We present several experimental results to compare the transformed programs and existing solutions. The transformed programs are as accurate and efficient than the implementations of compensated algorithms when the latter exist

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“…Graillat also presented accurate floating-point product and exponentiation algorithms in [15]. The compensated de Casteljau algorithms to evaluate the univariate polynomial and its first order derivative in Bernstein form were proposed in [16]. The algorithms above, applying error-free transformations [17][18][19], can yield a full precision accuracy.…”

Section: Introductionmentioning

confidence: 96%