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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 the ratios of spherical Bessel functions of consecutive order. This method was an improvement compared to other methods because it started from the beginning of the continued fraction rather than the tail, had a built-in check for convergence, and was numerically stable. The original algorithm uses algebra to bypass a zero in either the numerator or denominator.<ref name="Lentz 668–671">{{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=668–671|doi=10.1364/ao.15.000668|pmid=20165036 |bibcode=1976ApOpt..15..668L |issn=0003-6935}}</ref> Simpler Improvements to overcome unwanted zero terms 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 ==
This theory was initially motivated by Lentz's need for accurate calculation of ratios of spherical Bessel function necessary for [[Mie scattering]]. He created a new continued fraction algorithm that starts from the beginning of the continued fraction and not at the tail-end. This eliminates guessing how many terms of the continued fraction are needed for convergence. In addition, continued fraction representations for both ratios of Bessel functions and spherical Bessel functions of consecutive order themselves can be computed with Lentz's algorithm.<ref>{{Citejournal|lastname="Lentz|first=WilliamJ.|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=668–671|doi=10.1364/ao.15.000668|pmid=20165036 |bibcode=1976ApOpt..15..668L |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 book|last1=Masmoudi|first1=Atef|last2=Bouhlel|first2=Med Salim|last3=Puech|first3=William|title=2012 6th International Conference on Sciences of Electronics, Technologies of Information and Telecommunications (SETIT) |chapter=Image encryption using chaotic standard map and engle continued fractions map |date=March 2012|chapter-url=http://dx.doi.org/10.1109/setit.2012.6481959|pages=474–480 |publisher=IEEE|doi=10.1109/setit.2012.6481959|isbn=978-1-4673-1658-3 |s2cid=15380706 }}</ref>
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 at least as good as the other methods.<ref>{{Cite book |last1=Press |first1=W.H. |title=Numerical Recipes in Fortran, The Art of Scientific Computing|last2=Teukolsky |first2=S.A. |last3=Vetterling |first3=W.T. |last4=Flannery |first4=B. P. |publisher=Cambridge University Press |year=1992 |edition=2nd |page=165}}</ref> As an example, it was applied to evaluate exponential integral functions. This application was then called modified Lentz algorithm.<ref>{{Cite journal|last1=Press|first1=William H.|last2=Teukolsky|first2=Saul A.|date=1988|title=Evaluating Continued Fractions and Computing Exponential Integrals|journal=Computers in Physics|volume=2|issue=5|pages=88|doi=10.1063/1.4822777|bibcode=1988ComPh...2...88P |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|last1=Wand|first1=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|pmid=22533111 |s2cid=119636237 |issn=2049-1573}}</ref>
