Lentz's algorithm: Difference between revisions

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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><ref>Numerical Receipes in C++ page 177-179 ISBN 0 521 75033 4.</ref>

The nowversion usualusually methodemployed now is bydue to Thompson IJ and Barnett.<ref ARname=":1" (1986) Journal of Computational Physics, vol 64, pp 490-509./>

== 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|last1=Jaaskelainen|first1=T.|last2=Ruuskanen|first2=J.|date=1981-10-01|title=Note on Lentz's algorithm|url=http://dx.doi.org/10.1364/ao.20.003289|journal=Applied Optics|volume=20|issue=19|pages=3289–3290|doi=10.1364/ao.20.003289|pmid=20333144 |bibcode=1981ApOpt..20.3289J |issn=0003-6935}}</ref> suggested by Jaaskelainen and Ruuskanen in 1981 or a simple shift of the denominator by a very small number as suggested by Thompson and Barnett in 1986.<ref name=":1">{{Cite journal|last1=Thompson|first1=I.J.|last2=Barnett|first2=A.R.|date=1986|title=Coulomb and Bessel functions of complex arguments and order|url=http://dx.doi.org/10.1016/0021-9991(86)90046-x|journal=Journal of Computational Physics|volume=64|issue=2|pages=490–509|doi=10.1016/0021-9991(86)90046-x|bibcode=1986JCoPh..64..490T |issn=0021-9991}}</ref>

== Initial work ==