Accurate evaluation of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math>-th derivative of a polynomial and its application

Jiang, Hao; Graillat, Stef; Hu, Canbin; Li, Shengguo; Liao, Xiangke; Cheng, Lizhi; Su, Fang

doi:10.1016/j.cam.2012.11.008

Cited by 14 publications

(16 citation statements)

References 14 publications

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“…The approach pursued by the compensated schemes proposed in [21,15,22] consists of computing the correcting termŷ =…”

Section: Gemignani / Computers and Mathematics With Applications (mentioning

confidence: 99%

Accurate polynomial root-finding methods for symmetric tridiagonal matrix eigenproblems

Gemignani¹

2016

Computers & Mathematics with Applications

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In this paper we consider the application of polynomial root-finding methods to the\ud solution of the tridiagonal matrix eigenproblem. All considered solvers are based on\ud evaluating the Newton correction. We show that the use of scaled three-term recurrence\ud relations complemented with error free transformations yields some compensated\ud schemes which significantly improve the accuracy of computed results at a modest increase\ud in computational cost. Numerical experiments illustrate that under some restriction on\ud the conditioning the novel iterations can approximate and/or refine the eigenvalues of a\ud tridiagonal matrix with high relative accuracy

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“…The approach pursued by the compensated schemes proposed in [21,15,22] consists of computing the correcting termŷ =…”

Section: Gemignani / Computers and Mathematics With Applications (mentioning

confidence: 99%

Accurate polynomial root-finding methods for symmetric tridiagonal matrix eigenproblems

Gemignani¹

2016

Computers & Mathematics with Applications

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“…From [9,15], we show two inequations to get the error bound. Suppose , its accuracy will be the same as computed in original Clenshaw algorithm with normal condition numbers.…”

Section: Theorem 2 Let Be a Laguerre Series Of Degree And X Is A Flomentioning

confidence: 99%

“…When the problem is ill-conditioned, numerous compensated algorithms [8,9,10] based on error-free transformations [11] are proposed to evaluate different polynomials.…”

Section: Introductionmentioning

confidence: 99%

Accurate Evaluation of Finite Laguerre Series

2017

The 5th International Conference on Advanced Computer Science Applications and Technologies (ACSAT 2017)

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Abstract.A compensated algorithm is presented to evaluate finite Laguerre series. We record round-off error in Clenshaw algorithm by using error-free transformations (EFTs). In this work, we establish a valid error estimate that is as accurate as the Clenshaw scheme using twice the working precision. Numerical experiments illustrate that our compensated algorithm is more accurate than the Clenshaw algorithm and faster than the DDClenshaw algorithm (Clenshaw in double-double arithmetic).

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“…To demonstrate the efficiency of this approach, we compare our automatically transformed algorithms to existing compensated ones such as floating-point summation [22] and polynomial evaluation [7], [10]. The goal of this demonstration is to recover automatically the same results in terms of accuracy and execution time.…”

Section: Introductionmentioning

confidence: 98%

“…Several techniques have been introduced to improve the accuracy of numerical algorithms, as for instance expansions [4], [23], compensations [7], [10], differential methods [14] or extended precision arithmetic using multiple-precision libraries [5], [8]. Nevertheless, bugs from numerical failures are numerous and well known [2], [18].…”

Section: Introductionmentioning

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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Accurate evaluation of thek-th derivative of a polynomial and its application

Cited by 14 publications

References 14 publications

Accurate polynomial root-finding methods for symmetric tridiagonal matrix eigenproblems

Accurate polynomial root-finding methods for symmetric tridiagonal matrix eigenproblems

Accurate Evaluation of Finite Laguerre Series

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

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