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PrimeHunter (talk | contribs) remove bad non-displayed character after "sin": U+2061 FUNCTION APPLICATION |
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:<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 \textrm{sinc}(\frac{x}{h-k})</math> |
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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 |
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:<math>\textrm{sinc}(x)=\frac{\ |
:<math>\textrm{sinc}(x)=\frac{\sin(\pi x)}{\pi x}</math> |
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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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Revision as of 14:32, 17 November 2012
In numerical analysis and applied mathematics, sinc numerical methods are numerical techniques[1] 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
where the step size h>0 and where the sinc function is defined by
Sinc approximation methods excel for problems whose solutions may have singularities, or infinite domains, or boundary layers.
Sinc numerical methods cover
- function approximation,
- approximation of derivatives,
- approximate definite and indefinite integration,
- approximate solution of initial and boundary value ordinary differential equation (ODE) problems,
- approximation and inversion of Fourier and Laplace transforms,
- approximation of Hilbert transforms,
- approximation of definite and indefinite convolution,
- approximate solution of partial differential equations,
- approximate solution of integral equations,
- construction of conformal maps.
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 with some c>0, where n is the number of nodes or bases used in the methods. However, Sugihara[2] has recently found that the errors in the Sinc numerical methods based on double exponential transformation are 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
- ^ 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)00348-4instead. - ^ 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.2003.09.016instead.