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In ] and ], '''sinc numerical methods''' are numerical techniques for finding approximate solutions of ] and ] based on the translates of ] function and Cardinal function C(f,h)which is an expansion of f defined by <math>C(f,h)(x)=\sum_{k=-\infty}^\infty \textrm{sinc}(\frac{x}{h-k})</math> where the step size h>0 and where the sinc function is defined by Sinc(x)=sin⁡(\pi x)/(\pi x) In ] and ], '''sinc numerical methods''' are numerical techniques<ref>{{cite doi|10.1016/S0377-0427(00)00348-4}}</ref> for finding approximate solutions of ] and ] based on the translates of ] function and Cardinal function C(f,h)which is an expansion of f defined by
:<math>C(f,h)(x)=\sum_{k=-\infty}^\infty \textrm{sinc}(\frac{x}{h-k})</math>
where the step size h>0 and where the sinc function is defined by
:<math>\textrm{sinc}(x)=\frac{\sin⁡(\pi x)}{\pi x}</math>
Sinc approximation methods excel for problems whose solutions may have singularities, or infinite domains, or boundary layers. Sinc approximation methods excel for problems whose solutions may have singularities, or infinite domains, or boundary layers.


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Indeed, Sinc are ubiquitous for approximating every operation of calculus Indeed, Sinc are ubiquitous for approximating every operation of calculus


In the standard setup of the sinc numerical methods, the errors are known to be O(exp(−c√n)) with some c>0, where n is the number of nodes or bases used in the methods. However, Sugihara has recently found that the errors in the Sinc numerical methods based on double exponential transformation are O(exp(−k n/log n)) with some k>0, in a setup that is also meaningful both theoretically and practically. It has also been found that the error bounds of O(exp(−k n/log n)) are best possible in a certain mathematical sense. In the standard setup of the sinc numerical methods, the errors (in ]) are known to be <math>O\left(e^{-c\sqrt{n}}\right)</math> with some c>0, where n is the number of nodes or bases used in the methods. However, Sugihara<ref>{{cite doi|10.1016/j.cam.2003.09.016}}</ref> has recently found that the errors in the Sinc numerical methods based on double exponential transformation are <math>O\left(e^{-\frac{k n}{\ln n}}\right)</math> with some k>0, in a setup that is also meaningful both theoretically and practically and are found to be best possible in a certain mathematical sense.


==Reading== ==Reading==
*{{cite book
⚫ J. Lund and K.L. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, Philadelphia, 1992.
|title=
⚫ F.Stenger ''Handbook of Sinc Numerical Methods'', Taylor & Francis,2010.

M. Sugihara and Takayasu Matsuo, ''Recent developments of the Sinc numerical methods'', Journal of Computational and Applied Mathematics 164–165, 2004.
⚫ Handbook of Sinc Numerical Methods
F. Stenger, Summary of Sinc numerical methods, Journal of Computational and Applied Mathematics (121) 379-420 (2000)

|last1=Stenger |first1=Frank |authorlink1=
|coauthors=
|editor1-last= |editor1-first= |editor1-link=
|year={{{year| 2011 }}}
|publisher=CRC Press
|location=Boca Raton, FL
|isbn=9781439821596
|url=
|page={{{page|}}}
|pages={{{pages|}}}
|ref={{{ref|}}}
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*{{cite book
⚫ |title=Sinc Methods for Quadrature and Differential Equations
|last1=Lund |first1=John |authorlink1=
|last2=Bowers | first2=Kenneth
|coauthors=
|editor1-last= |editor1-first= |editor1-link=
|year={{{year| 1992 }}}
|publisher=Society for Industrial and Applied Mathematics (SIAM)
|location=Philadelphia
|isbn=9780898712988
|url=
|page={{{page|}}}
|pages={{{pages|}}}
|ref={{{ref|}}}
}}

==References==
{{Reflist}}


] ]

Revision as of 17:02, 16 November 2012

In numerical analysis and applied mathematics, sinc numerical methods are numerical techniques for finding approximate solutions of partial differential equations and integral equations based on the translates of sinc function and Cardinal function C(f,h)which is an expansion of f defined by

C ( f , h ) ( x ) = ∑ k = − ∞ ∞ sinc ( x h − k ) {\displaystyle C(f,h)(x)=\sum _{k=-\infty }^{\infty }{\textrm {sinc}}({\frac {x}{h-k}})}

where the step size h>0 and where the sinc function is defined by

Failed to parse (syntax error): {\displaystyle \textrm{sinc}(x)=\frac{\sin⁡(\pi x)}{\pi x}}

Sinc approximation methods excel for problems whose solutions may have singularities, or infinite domains, or boundary layers.

Sinc numerical methods cover

Indeed, Sinc are ubiquitous for approximating every operation of calculus

In the standard setup of the sinc numerical methods, the errors (in big O notation) are known to be O ( e − c n ) {\displaystyle O\left(e^{-c{\sqrt {n}}}\right)} with some c>0, where n is the number of nodes or bases used in the methods. However, Sugihara has recently found that the errors in the Sinc numerical methods based on double exponential transformation are O ( e − k n ln ⁡ n ) {\displaystyle O\left(e^{-{\frac {kn}{\ln n}}}\right)} with some k>0, in a setup that is also meaningful both theoretically and practically and are found to be best possible in a certain mathematical sense.

Reading

  • Stenger, Frank (2011). Handbook of Sinc Numerical Methods. Boca Raton, FL: CRC Press. ISBN 9781439821596. {{cite book}}: Cite has empty unknown parameter: |coauthors= (help)
  • Lund, John; Bowers, Kenneth (1992). Sinc Methods for Quadrature and Differential Equations. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). ISBN 9780898712988. {{cite book}}: Cite has empty unknown parameter: |coauthors= (help)

References

  1. Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1016/S0377-0427(00)00348-4, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1016/S0377-0427(00)00348-4 instead.
  2. Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1016/j.cam.2003.09.016, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1016/j.cam.2003.09.016 instead.


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