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== 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>{{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> 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
== Algorithm ==
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