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In mathematics, '''Lentz's algorithm''' is an [[algorithm]] to evaluate [[generalized continued fraction|continued fraction]]s and compute tables of spherical [[Bessel function]]s.<ref name=":0">{{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|doi=10.21236/ad0767223 }}</ref>
== History ==
The idea was introduced in 1973 by William J. Lentz<ref name=":0" /> and was simplified by him in 1982.<ref>{{Cite book|last=J.|first=Lentz, W.|url=http://worldcat.org/oclc/227549426|title=A Simplification of Lentz's Algorithm.|date=August 1982|publisher=Defense Technical Information Center|oclc=227549426}}</ref> 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. The original algorithm assumes that the denominators occurring during execution remain non-zero throughout. Improvements to overcome this limitation include an altered recurrence relation<ref>{{Cite journal|
== Initial work ==
This theory was initially motivated by Lentz's other 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, continued fraction representations for both ratios of Bessel functions and spherical Bessel functions of consecutive order can be computed with Lentz's 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=
== Algorithm ==
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:<math>{D}_{n} = \frac{1}{{b}_{n} + {a}_{n} {D}_{n - 1}}</math>
When the product <math>{C}_{n} {D}_{n}</math> approaches unity with increasing <math>n</math>, it is hoped that <math>{f}_{n}</math> has converged to <math>f</math>.<ref>{{Cite book |
== Applications ==
Lentz's algorithm was used widely in the late twentieth century. 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|
==References==
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