In mathematics, Lentz's algorithm is an algorithm to evaluate continued fractions, and was originally devised to compute tables of spherical Bessel functions. The version often employed now is the simplification due to Thompson and Barnett.
History The idea was introduced in 1973 by William J. Lentz and was simplified by him in 1982. Lentz suggested that calculating ratios of spherical Bessel functions of complex arguments over a wide range of values 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 starts from the beginning of the continued fraction rather than the tail, had a built-in check for convergence, and is numerically stable. The original algorithm uses algebra to bypass a zero or near-zero independently in either the numerator or denominator. Simpler Improvements to overcome unwanted zero terms include an altered recurrence relation 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., or Lentz's simplification.
Initial work This theory was initially motivated by Lentz's need for accurate calculation of ratios of spherical Bessel functions of consecutive order and complex argument 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. The algorithm suggested that it is possible to terminate the evaluation of continued fractions when | f j − f j − 1 | {\displaystyle |f_{j}-f_{j-1}|} is relatively small.
Algorithm Lentz's algorithm is based on the Wallis-Euler relations. John Wallis independently rediscovered the recursion relations about 500 years after the Indian mathematician Bhas-Cara II. For continued fractions, the definitive standard notation is found under "Elementary Analytical Methods", page 19 and throughout the text for each function.
f 0 = b 0 {\displaystyle f_{0}=b_{0}}
f 1 = b 0 + a 1 b 1 {\displaystyle f_{1}=b_{0}+{\frac {a_{1}}{b_{1}}}}
f 2 = b 0 + a 1 b 1 + a 2 b 2 {\displaystyle f_{2}=b_{0}+{\frac {a_{1}}{b_{1}+{\frac {a_{2}}{b_{2}}}}}}
f 3 = b 0 + a 1 b 1 + a 2 b 2 + a 3 b 3 {\displaystyle f_{3}=b_{0}+{\frac {a_{1}}{b_{1}+{\frac {a_{2}}{b_{2}+{\frac {a_{3}}{b_{3}}}}}}}}
etc., or using the big-K notation, if
f n = b 0 + K n j = 1 a j b j {\displaystyle f_{n}=b_{0}+{\underset {j=1}{\overset {n}{\operatorname {K} }}}{\frac {a_{j}}{b_{j}}}}
is the n {\displaystyle n} th convergent to f {\displaystyle f} then
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