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Sinc numerical methods: Difference between revisions

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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 In ] and ], '''sinc numerical methods''' are numerical techniques<ref>{{Cite journal | last1 = Stenger | first1 = F. | doi = 10.1016/S0377-0427(00)00348-4 | title = Summary of sinc numerical methods | journal = Journal of Computational and Applied Mathematics | volume = 121 | pages = 379–420 | year = 2000 | issue = 1–2 | doi-access = free }}</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> :<math>C(f,h)(x)=\sum_{k=-\infty}^\infty f(kh) \, \textrm{sinc} \left(\dfrac{x}{h}-k \right)</math>
where the step size h>0 and where the sinc function is defined by 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> :<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.


The truncated Sinc expansion of f is defined by the following series:
⚫ ==Sinc numerical methods cover==
:<math> C_{M,N}(f,h)(x) = \displaystyle \sum_{k=-M}^{N} f(kh) \, \textrm{sinc} \left(\dfrac{x}{h}-k \right) </math> .


⚫ ==Sinc numerical methods cover==
*function approximation, *function approximation,
*approximation of ]s, *approximation of ]s,
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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 (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. 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 journal | last1 = Sugihara | first1 = M. | last2 = Matsuo | first2 = T. | doi = 10.1016/j.cam.2003.09.016 | title = Recent developments of the Sinc numerical methods | journal = Journal of Computational and Applied Mathematics | volume = 164-165 | pages = 673–689 | year = 2004 | doi-access = free }}</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 *{{cite book
⚫ |title=Handbook of Sinc Numerical Methods
|title=
⚫ |last1=Stenger |first1=Frank

⚫ |year= 2011
⚫ Handbook of Sinc Numerical Methods

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


Latest revision as of 17:12, 28 September 2024

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 = − ∞ ∞ f ( k h ) sinc ( x h − k ) {\displaystyle C(f,h)(x)=\sum _{k=-\infty }^{\infty }f(kh)\,{\textrm {sinc}}\left({\dfrac {x}{h}}-k\right)}

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

sinc ( x ) = sin ⁡ ( π x ) π x {\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.

The truncated Sinc expansion of f is defined by the following series:

C M , N ( f , h ) ( x ) = ∑ k = − M N f ( k h ) sinc ( x h − k ) {\displaystyle C_{M,N}(f,h)(x)=\displaystyle \sum _{k=-M}^{N}f(kh)\,{\textrm {sinc}}\left({\dfrac {x}{h}}-k\right)} .

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, Florida: CRC Press. ISBN 9781439821596.
  • Lund, John; Bowers, Kenneth (1992). Sinc Methods for Quadrature and Differential Equations. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). ISBN 9780898712988.

References

  1. Stenger, F. (2000). "Summary of sinc numerical methods". Journal of Computational and Applied Mathematics. 121 (1–2): 379–420. doi:10.1016/S0377-0427(00)00348-4.
  2. Sugihara, M.; Matsuo, T. (2004). "Recent developments of the Sinc numerical methods". Journal of Computational and Applied Mathematics. 164–165: 673–689. doi:10.1016/j.cam.2003.09.016.


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