Lentz's algorithm

The idea was introduced in 1973 by William J. Lentz[1] and was simplified by him in 1982.[2] 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[3] 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.[4]

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.[5] The algorithm suggested that it is possible to terminate the evaluation of continued fractions when ${\displaystyle |f_{j}-f_{j-1}|}$ is relatively small.[6]

Lentz's algorithm is based on the Wallis-Euler relations. If

${\displaystyle {f}_{0}={b}_{0}}$

${\displaystyle {f}_{1}={b}_{0}+{\frac {{a}_{1}}{{b}_{1}}}}$

${\displaystyle {f}_{2}={b}_{0}+{\frac {{a}_{1}}{{b}_{1}+{\frac {{a}_{2}}{{b}_{2}}}}}}$

${\displaystyle {f}_{3}={b}_{0}+{\frac {{a}_{1}}{{b}_{1}+{\frac {{a}_{2}}{{b}_{2}+{\frac {{a}_{3}}{{b}_{3}}}}}}}}$

etc., or using the big-K notation, if

${\displaystyle {f}_{n}={b}_{0}+{\underset {j=1}{\overset {n}{\operatorname {K} }}}{\frac {{a}_{j}}{{b}_{j}+}}}$

is the ${\displaystyle n}$th convergent to ${\displaystyle f}$ then

${\displaystyle {f}_{n}={\frac {{A}_{n}}{{B}_{n}}}}$

where ${\displaystyle {A}_{n}}$ and ${\displaystyle {B}_{n}}$ are given by the Wallis-Euler recurrence relations

${\displaystyle {A}_{-1}=1}$

${\displaystyle {B}_{-1}=0}$

${\displaystyle {A}_{0}={b}_{0}}$

${\displaystyle {B}_{0}=1}$

${\displaystyle {A}_{n}={b}_{n}{A}_{n-1}+{a}_{n}{A}_{n-2}}$

${\displaystyle {B}_{n}={b}_{n}{B}_{n-1}+{a}_{n}{B}_{n-2}}$

Lenz's method defines

${\displaystyle {C}_{n}={\frac {{A}_{n}}{{A}_{n-1}}}}$

${\displaystyle {D}_{n}={\frac {{B}_{n-1}}{{B}_{n}}}}$

so that the ${\displaystyle n}$th convergent is

${\displaystyle {f}_{n}={C}_{n}{D}_{n}{f}_{n-1}}$

and uses the recurrence relations

${\displaystyle {C}_{0}={\frac {{A}_{0}}{{A}_{-1}}}={b}_{0}}$

${\displaystyle {D}_{0}={\frac {{B}_{-1}}{{B}_{0}}}=0}$

${\displaystyle {f}_{0}={b}_{0}}$

${\displaystyle {C}_{n}={b}_{n}+{\frac {{a}_{n}}{{C}_{n-1}}}}$

${\displaystyle {D}_{n}={\frac {1}{{b}_{n}+{a}_{n}{D}_{n-1}}}}$

When the product ${\displaystyle {C}_{n}{D}_{n}}$ approaches unity with increasing ${\displaystyle n}$, it is hoped that ${\displaystyle {f}_{n}}$ has converged to ${\displaystyle f}$.[7]

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.[8] 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.[9]