:Lentz's algorithm

The idea was introduced in 1973 by William J. Lentz and was simplified by him in 1982.{{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}} 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.{{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}} Simpler Improvements to overcome unwanted zero terms include an altered recurrence relation{{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}} 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.{{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}}

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. The algorithm suggested that it is possible to terminate the evaluation of continued fractions when $|f\_j-f\_\{j-1\}\; |$ is relatively small.{{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 }}

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

:$\{f\}\_\{0\}\; =\; \{b\}\_\{0\}$

:$\{f\}\_\{1\}\; =\; \{b\}\_\{0\}\; +\; \backslash frac\{\{a\}\_\{1\}\}\{\{b\}\_\{1\}\}$

:$\{f\}\_\{2\}\; =\; \{b\}\_\{0\}\; +\; \backslash frac\{\{a\}\_\{1\}\}\{\{b\}\_\{1\}\; +\; \backslash frac\{\{a\}\_\{2\}\}\{\{b\}\_\{2\}\}\}$

:$\{f\}\_\{3\}\; =\; \{b\}\_\{0\}\; +\; \backslash frac\{\{a\}\_\{1\}\}\{\{b\}\_\{1\}\; +\; \backslash frac\{\{a\}\_\{2\}\}\{\{b\}\_\{2\}\; +\; \backslash frac\{\{a\}\_\{3\}\}\{\{b\}\_\{3\}\}\}\}$

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

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

is the $n$th convergent to $f$ then

:$\{f\}\_\{n\}\; =\; \backslash frac\{\{A\}\_\{n\}\}\{\{B\}\_\{n\}\}$

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

:$$

\begin{align}

{A}_{-1} & = 1 & {B}_{-1} & = 0\\

{A}_{0} & = {b}_{0} & {B}_{0} & = 1\\

{A}_{n} & = {b}_{n} {A}_{n-1} + {a}_{n} {A}_{n-2} & {B}_{n} & = {b}_{n} {B}_{n-1} + {a}_{n} {B}_{n-2}

\end{align}

Lentz's method defines

:$\{C\}\_\{n\}\; =\; \backslash frac\{\{A\}\_\{n\}\}\{\{A\}\_\{n\; -\; 1\}\}$

:$\{D\}\_\{n\}\; =\; \backslash frac\{\{B\}\_\{n\; -\; 1\}\}\{\{B\}\_\{n\}\}$

so that the $n$th convergent is $\{f\}\_\{n\}\; =\; \{C\}\_\{n\}\; \{D\}\_\{n\}\; \{f\}\_\{n\; -\; 1\}$ with $\{f\}\_\{0\}\; =\; \backslash frac\{\{A\}\_\{0\}\}\{\{B\}\_\{0\}\}\; =\; \{b\}\_\{0\}$ and uses the recurrence relations

:$$

\begin{align}

{C}_{0} & = \frac{{A}_{0}}{{A}_{-1}} = {b}_{0} & {D}_{0} & = \frac{{B}_{-1}}{{B}_{0}} = 0\\

{C}_{n} & = {b}_{n} + \frac{{a}_{n}}{{C}_{n-1}} & {D}_{n} & = \frac{1}{{b}_{n} + {a}_{n} {D}_{n-1}}

\end{align}

When the product $\{C\}\_\{n\}\; \{D\}\_\{n\}$ approaches unity with increasing $n$, it is hoped that $\{f\}\_\{n\}$ has converged to $f$.{{Cite book |last1=Press |first1=W.H. |title=Numerical Recipes: 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=2007 |edition=3rd |pages=207–208}}

Lentz's algorithm has the advantage of side-stepping an inconvenience of the Wallis-Euler relations, namely that the numerators $A\_n$ and denominators $B\_n$ are prone to grow or diminish very rapidly with increasing $n$. In direct numerical application of the Wallis-Euler relations, this means that $A\_\{n-1\}$, $A\_\{n-2\}$, $B\_\{n-1\}$, $B\_\{n-2\}$ must be periodically checked and rescaled to avoid floating-point overflow or underflow.

In Lentz's original algorithm, it can happen that $\{C\}\_\{n\}\; =\; 0$, resulting in division by zero at the next step. The problem can be remedied simply by setting $\{C\}\_\{n\}\; =\; \backslash varepsilon$ for some sufficiently small $\backslash varepsilon$. This gives $\{C\}\_\{n\; +\; 1\}\; =\; \{b\}\_\{n\; +\; 1\}\; +\; \backslash frac\{\{a\}\_\{n\; +\; 1\}\}\{\backslash varepsilon\}\; =\; \backslash frac\{\{a\}\_\{n\; +\; 1\}\}\{\backslash varepsilon\}$ to within floating-point precision, and the product $\{C\}\_\{n\}\; \{C\}\_\{n\; +\; 1\}\; =\; \{a\}\_\{n\; +\; 1\}$ irrespective of the precise value of ε. Accordingly, the value of $\{f\}\_\{0\}\; =\; \{C\}\_\{0\}\; =\; \{b\}\_\{0\}$ is also set to $\backslash varepsilon$ in the case of $\{b\}\_\{0\}\; =\; 0$.

Similarly, if the denominator in $\{D\}\_\{n\}\; =\; \backslash frac\{1\}\{\{b\}\_\{n\}\; +\; \{a\}\_\{n\}\; \{D\}\_\{n\; -\; 1\}\}$ is zero, then setting $\{D\}\_\{n\}\; =\; \backslash frac\{1\}\{\backslash varepsilon\}$ for small enough $\backslash varepsilon$ gives $\{D\}\_\{n\}\; \{D\}\_\{n\; +\; 1\}\; =\; \backslash frac\{1\}\{\{a\}\_\{n\; +\; 1\}\}$ irrespective of the value of $\backslash varepsilon$.

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.{{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}} As an example, it was applied to evaluate exponential integral functions. This application was then called modified Lentz algorithm.{{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}} 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.{{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}}

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Category:Special hypergeometric functions