Lentz's algorithm: Difference between revisions

Revision as of 22:01, 3 September 2021

Revision as of 11:26, 7 September 2021

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In mathematics, '''Lentz's Algorithm''' is used to calculate continued fractions and present tables of spherical [[Bessel function|Bessel functions]]s.<ref>{{Cite journal|last=Lentz|first=W. J.|date=1973-09-01|title=A Method of Computing Spherical Bessel Functions of Complex Argument with Tables|url=http://dx.doi.org/10.21236/ad0767223|___location=Fort Belvoir, VA}}</ref>

== History ==

The idea was introduced more than thirty years ago by W.J. Lentz. Lentz suggested that calculating ratios of spherical Bessel functions of complex arguments can be difficult. He developed a new continued fraction technique for calculating them. This method was an improvement compared to other methods because it eliminated errors on certain terms or provided zero as a result.<ref>{{Cite book|last=J.|first=Lentz, W.|url=http://worldcat.org/oclc/227549426|title=A Simplification of Lentz's Algorithm.|date=August 1982-08|publisher=Defense Technical Information Center|oclc=227549426}}</ref>

== Initial work ==

This theory was initially implemented in Lentz's another research when he calculated ratios of Bessel function necessary for [[Mie scattering]]. He demonstrated that the algorithm uses a technique involving the evaluation continued fractions that starts from the beginning and not at the tail. In addition, that continued fraction representations for both ratios of Bessel functions and spherical Bessel functions of consecutive order can be presented with the Lentz algorithm.<ref>{{Cite journal|last=Lentz|first=William J.|date=1976-03-01|title=Generating Bessel functions in Mie scattering calculations using continued fractions|url=http://dx.doi.org/10.1364/ao.15.000668|journal=Applied Optics|volume=15|issue=3|pages=668|doi=10.1364/ao.15.000668|issn=0003-6935}}</ref> The algorithm suggested that it is possible to terminate the evaluation of continued fractions when <math>|f_j-f_{j-1} |</math> is relatively small.<ref>{{Cite journal|last=Masmoudi|first=Atef|last2=Bouhlel|first2=Med Salim|last3=Puech|first3=William|date=March 2012-03|title=Image encryption using chaotic standard map and engle continued fractions map|url=http://dx.doi.org/10.1109/setit.2012.6481959|journal=2012 6th International Conference on Sciences of Electronics, Technologies of Information and Telecommunications (SETIT)|publisher=IEEE|doi=10.1109/setit.2012.6481959}}</ref>

== Applications ==

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{{[[Category: Mathematics]]}}