Lentz's algorithm: Difference between revisions

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In mathematics, '''Lentz's Algorithm''' is used to calculate continued fractions and present tables of spherical [[Bessel function|Bessel functions]].<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=1982-08|publisher=Defense Technical Information Center|oclc=227549426}}</ref>.

== Initial Working ==

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=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 ==

Lentz's algorithm was used widely in the late 1900s. It was suggested that it doesn't have any rigorous analysis of error propagation. However, a few empirical tests suggest that it's almost as good as the other methods. As an example, it was applied to evaluate exponential integral functions. This application was then called modified Lentz algorithm.<ref>{{Cite journal|last=Press|first=William H.|last2=Teukolsky|first2=Saul A.|date=1988|title=Evaluating Continued Fractions and Computing Exponential Integrals|url=http://dx.doi.org/10.1063/1.4822777|journal=Computers in Physics|volume=2|issue=5|pages=88|doi=10.1063/1.4822777|issn=0894-1866|doi-access=free}}</ref>. It's also stated that the Lentz algorithm is not applicable for every calculation, and convergence can be quite rapid for some continued fractions and vice versa for others.<ref>{{Cite journal|last=Wand|first=Matt P.|last2=Ormerod|first2=John T.|date=2012-09-18|title=Continued fraction enhancement of Bayesian computing|url=http://dx.doi.org/10.1002/sta4.4|journal=Stat|volume=1|issue=1|pages=31–41|doi=10.1002/sta4.4|issn=2049-1573}}</ref>.

==References==