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Lentz's algorithm

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In mathematics, Lentz's algorithm is an <a href="Algorithm" title="Algorithm">algorithm</a> to evaluate <a href="Generalized_continued_fraction" class="mw-redirect" title="Generalized continued fraction">continued fractions</a>, and was originally devised to compute tables of spherical <a href="Bessel_function" title="Bessel function">Bessel functions</a>.<a href="#cite_note-:0-1">[1]</a><a href="#cite_note-numerical-recipes-c++-2">[2]</a>

The version usually employed now is due to Thompson and Barnett.<a href="#cite_note-Thompson-and-Barnett-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

The idea was introduced in 1973 by William J. Lentz<a href="#cite_note-:0-1">[1]</a> and was simplified by him in 1982.<a href="#cite_note-4">[4]</a> 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.<a href="#cite_note-Lentz_668–671-5">[5]</a> Simpler Improvements to overcome unwanted zero terms include an altered recurrence relation<a href="#cite_note-6">[6]</a> 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.<a href="#cite_note-Thompson-and-Barnett-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Initial_work">Initial work</h2></div>

This theory was initially motivated by Lentz's need for accurate calculation of ratios of spherical Bessel function necessary for <a href="Mie_scattering" title="Mie scattering">Mie scattering</a>. 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.<a href="#cite_note-Lentz_668–671-5">[5]</a> The algorithm suggested that it is possible to terminate the evaluation of continued fractions when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f_{j}-f_{j-1}|}">

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<annotation encoding="application/x-tex">{\displaystyle |f_{j}-f_{j-1}|}</annotation>

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</math><img src="./15a5e6904e89a603cf69359221d4badcc9a9e6f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.332ex; height:3.009ex;" alt="{\displaystyle |f_{j}-f_{j-1}|}" loading="lazy"> is relatively small.<a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading2"><h2 id="Algorithm">Algorithm</h2></div>

Lentz's algorithm is based on the <a href="Wallis-Euler_relations" class="mw-redirect" title="Wallis-Euler relations">Wallis-Euler relations</a>. If

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<annotation encoding="application/x-tex">{\displaystyle {f}_{0}={b}_{0}}</annotation>

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</math><img src="./e05f94f73c21b2fb61884517224e46c97bce5e42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.344ex; height:2.676ex;" alt="{\displaystyle {f}_{0}={b}_{0}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {f}_{1}={b}_{0}+{\frac {{a}_{1}}{{b}_{1}}}}</annotation>

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</math><img src="./dbf518b4d0eed29ab67b6055c632d2e7d1fa144d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.304ex; height:5.176ex;" alt="{\displaystyle {f}_{1}={b}_{0}+{\frac {{a}_{1}}{{b}_{1}}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {f}_{2}={b}_{0}+{\frac {{a}_{1}}{{b}_{1}+{\frac {{a}_{2}}{{b}_{2}}}}}}</annotation>

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</math><img src="./5f241cce72a37649a42d753f62c9ba3fb7df450d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:18.45ex; height:6.509ex;" alt="{\displaystyle {f}_{2}={b}_{0}+{\frac {{a}_{1}}{{b}_{1}+{\frac {{a}_{2}}{{b}_{2}}}}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {f}_{3}={b}_{0}+{\frac {{a}_{1}}{{b}_{1}+{\frac {{a}_{2}}{{b}_{2}+{\frac {{a}_{3}}{{b}_{3}}}}}}}}</annotation>

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</math><img src="./cd7ecc75a5248639285a62ea5396c24bce06a221.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:21.907ex; height:8.509ex;" alt="{\displaystyle {f}_{3}={b}_{0}+{\frac {{a}_{1}}{{b}_{1}+{\frac {{a}_{2}}{{b}_{2}+{\frac {{a}_{3}}{{b}_{3}}}}}}}}" loading="lazy"></dd></dl>

etc., or using the <a href="Generalized_continued_fraction" class="mw-redirect" title="Generalized continued fraction">big-K notation</a>, if

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<annotation encoding="application/x-tex">{\displaystyle {f}_{n}={b}_{0}+{\underset {j=1}{\overset {n}{\operatorname {K} }}}{\frac {{a}_{j}}{{b}_{j}}}}</annotation>

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</math><img src="./8e0ff3e70d1587eb3b99c81c605094fd02c1aab7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.102ex; height:6.009ex;" alt="{\displaystyle {f}_{n}={b}_{0}+{\underset {j=1}{\overset {n}{\operatorname {K} }}}{\frac {{a}_{j}}{{b}_{j}}}}" loading="lazy"></dd></dl>

is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">th convergent to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> then

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<annotation encoding="application/x-tex">{\displaystyle {f}_{n}={\frac {{A}_{n}}{{B}_{n}}}}</annotation>

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</math><img src="./3822f0e9f77eb3f82465f116b767fdcbd4804389.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.275ex; height:5.843ex;" alt="{\displaystyle {f}_{n}={\frac {{A}_{n}}{{B}_{n}}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {A}_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle {A}_{n}}</annotation>

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</math><img src="./f8e79bb17c03f6c7a65c8c5f94b806278dc427b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle {A}_{n}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {B}_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle {B}_{n}}</annotation>

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</math><img src="./289124e7c837841043aa0a314986f148f4bcb517.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle {B}_{n}}" loading="lazy"> are given by the Wallis-Euler recurrence relations

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{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{aligned}}}</annotation>

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</math><img src="./675f5c69362dc2b6b32cd9fced70f7f052988b2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:55.066ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}{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{aligned}}}" loading="lazy"></dd></dl>

Lentz's method defines

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<annotation encoding="application/x-tex">{\displaystyle {C}_{n}={\frac {{A}_{n}}{{A}_{n-1}}}}</annotation>

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</math><img src="./fbee79339399549d05b5d80951d457a34bc91983.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.877ex; height:6.009ex;" alt="{\displaystyle {C}_{n}={\frac {{A}_{n}}{{A}_{n-1}}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {D}_{n}={\frac {{B}_{n-1}}{{B}_{n}}}}</annotation>

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</math><img src="./89d060cc327b47ccb0d744a0ded4e9816668c41f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.16ex; height:5.843ex;" alt="{\displaystyle {D}_{n}={\frac {{B}_{n-1}}{{B}_{n}}}}" loading="lazy"></dd></dl>

so that the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">th convergent is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {f}_{n}={C}_{n}{D}_{n}{f}_{n-1}}">

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<annotation encoding="application/x-tex">{\displaystyle {f}_{n}={C}_{n}{D}_{n}{f}_{n-1}}</annotation>

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</math><img src="./23b6a97914c5fff84663c1ec06d174933b8907ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.937ex; height:2.843ex;" alt="{\displaystyle {f}_{n}={C}_{n}{D}_{n}{f}_{n-1}}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {f}_{0}={\frac {{A}_{0}}{{B}_{0}}}={b}_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle {f}_{0}={\frac {{A}_{0}}{{B}_{0}}}={b}_{0}}</annotation>

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</math><img src="./b92a37ff3b38f97d6f250f2defc37ee054e501be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.097ex; height:5.843ex;" alt="{\displaystyle {f}_{0}={\frac {{A}_{0}}{{B}_{0}}}={b}_{0}}" loading="lazy"> and uses the recurrence relations

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{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{aligned}}}</annotation>

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</math><img src="./927b59d996603ec1f4491111af4357febedcd7c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.15ex; margin-bottom: -0.188ex; width:42.074ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}{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{aligned}}}" loading="lazy"></dd></dl>

When the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {C}_{n}{D}_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle {C}_{n}{D}_{n}}</annotation>

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</math><img src="./bc231803c323982fd3beb05a0abbfe814ed689c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.023ex; height:2.509ex;" alt="{\displaystyle {C}_{n}{D}_{n}}" loading="lazy"> approaches unity with increasing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, it is hoped that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {f}_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle {f}_{n}}</annotation>

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</math><img src="./76a7ec98160e69214bbddb1853284a068a50768c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.358ex; height:2.676ex;" alt="{\displaystyle {f}_{n}}" loading="lazy"> has converged to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">.<a href="#cite_note-Numerical-Recipes-8">[8]</a>

Lentz's algorithm has the advantage of side-stepping an inconvenience of the Wallis-Euler relations, namely that the numerators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle A_{n}}</annotation>

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</math><img src="./730f6906700685b6d52f3958b1c2ae659d2d97d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle A_{n}}" loading="lazy"> and denominators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation>

</semantics>

</math><img src="./2f568bf6d34e97b9fdda0dc7e276d6c4501d2045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}" loading="lazy"> are prone to grow or diminish very rapidly with increasing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">. In direct numerical application of the Wallis-Euler relations, this means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n-1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{n-1}}</annotation>

</semantics>

</math><img src="./b576e75f0336d126580faaad6039e2e84f6f3ee2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.062ex; height:2.509ex;" alt="{\displaystyle A_{n-1}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n-2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{n-2}}</annotation>

</semantics>

</math><img src="./d06b40157dd6c080023f50cbc7754140a552d90f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.062ex; height:2.509ex;" alt="{\displaystyle A_{n-2}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n-1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n-1}}</annotation>

</semantics>

</math><img src="./35a38fdd982d39522072e328364b4ccec132f6dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.083ex; height:2.509ex;" alt="{\displaystyle B_{n-1}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n-2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n-2}}</annotation>

</semantics>

</math><img src="./ef9fccee51766d46ca6e7ca6e427f9019bd332d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.083ex; height:2.509ex;" alt="{\displaystyle B_{n-2}}" loading="lazy"> must be periodically checked and rescaled to avoid floating-point overflow or underflow.<a href="#cite_note-Numerical-Recipes-8">[8]</a>

<div class="mw-heading mw-heading2"><h2 id="Thompson_and_Barnett_modification">Thompson and Barnett modification</h2></div>

In Lentz's original algorithm, it can happen that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {C}_{n}=0}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {C}_{n}=0}</annotation>

</semantics>

</math><img src="./0c6dc979df0e4e1b0192e733db13c96419980088.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.141ex; height:2.509ex;" alt="{\displaystyle {C}_{n}=0}" loading="lazy">, resulting in division by zero at the next step. The problem can be remedied simply by setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {C}_{n}=\varepsilon }">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {C}_{n}=\varepsilon }</annotation>

</semantics>

</math><img src="./c9dc00f2641ceaad6c7e2dfb6c5c56daf0514e3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.062ex; height:2.509ex;" alt="{\displaystyle {C}_{n}=\varepsilon }" loading="lazy"> for some sufficiently small <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

</semantics>

</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy">. This gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {C}_{n+1}={b}_{n+1}+{\frac {{a}_{n+1}}{\varepsilon }}={\frac {{a}_{n+1}}{\varepsilon }}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

<msub>

</mrow>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {C}_{n+1}={b}_{n+1}+{\frac {{a}_{n+1}}{\varepsilon }}={\frac {{a}_{n+1}}{\varepsilon }}}</annotation>

</semantics>

</math><img src="./95573493387e6cdb265b1f1815590a46783d6e4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.105ex; height:4.843ex;" alt="{\displaystyle {C}_{n+1}={b}_{n+1}+{\frac {{a}_{n+1}}{\varepsilon }}={\frac {{a}_{n+1}}{\varepsilon }}}" loading="lazy"> to within floating-point precision, and the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {C}_{n}{C}_{n+1}={a}_{n+1}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {C}_{n}{C}_{n+1}={a}_{n+1}}</annotation>

</semantics>

</math><img src="./20b6224e5ea3e6116601a8d03dae8f5a6f1abf73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.508ex; height:2.509ex;" alt="{\displaystyle {C}_{n}{C}_{n+1}={a}_{n+1}}" loading="lazy"> irrespective of the precise value of ε. Accordingly, the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {f}_{0}={C}_{0}={b}_{0}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {f}_{0}={C}_{0}={b}_{0}}</annotation>

</semantics>

</math><img src="./493a73b10262a44bcdfa5a31b112adcbccafaacf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.158ex; height:2.676ex;" alt="{\displaystyle {f}_{0}={C}_{0}={b}_{0}}" loading="lazy"> is also set to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

</semantics>

</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"> in the case of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {b}_{0}=0}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {b}_{0}=0}</annotation>

</semantics>

</math><img src="./e87ec6d0bc29cbac18ed6af6e9e3e94036ced7ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.313ex; height:2.509ex;" alt="{\displaystyle {b}_{0}=0}" loading="lazy">.

Similarly, if the denominator in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {D}_{n}={\frac {1}{{b}_{n}+{a}_{n}{D}_{n-1}}}}">

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {D}_{n}={\frac {1}{{b}_{n}+{a}_{n}{D}_{n-1}}}}</annotation>

</semantics>

</math><img src="./d30ec8be08148270369071188bd6a5418e72b718.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.825ex; height:5.676ex;" alt="{\displaystyle {D}_{n}={\frac {1}{{b}_{n}+{a}_{n}{D}_{n-1}}}}" loading="lazy"> is zero, then setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {D}_{n}={\frac {1}{\varepsilon }}}">

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {D}_{n}={\frac {1}{\varepsilon }}}</annotation>

</semantics>

</math><img src="./ba6edc7fc3d577cecf3c14833dcca9b31ee59910.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.24ex; height:5.176ex;" alt="{\displaystyle {D}_{n}={\frac {1}{\varepsilon }}}" loading="lazy"> for small enough <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

</semantics>

</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"> gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {D}_{n}{D}_{n+1}={\frac {1}{{a}_{n+1}}}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {D}_{n}{D}_{n+1}={\frac {1}{{a}_{n+1}}}}</annotation>

</semantics>

</math><img src="./826a90cce507b3faa8e1a571a7d35a8595d62662.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.869ex; height:5.676ex;" alt="{\displaystyle {D}_{n}{D}_{n+1}={\frac {1}{{a}_{n+1}}}}" loading="lazy"> irrespective of the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

</semantics>

</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy">.<a href="#cite_note-Thompson-and-Barnett-3">[3]</a><a href="#cite_note-Numerical-Recipes-8">[8]</a>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

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.<a href="#cite_note-9">[9]</a> As an example, it was applied to evaluate exponential integral functions. This application was then called modified Lentz algorithm.<a href="#cite_note-10">[10]</a> 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.<a href="#cite_note-11">[11]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFLentz1973" class="citation report cs1">Lentz, W. J. (September 1973). <a rel="nofollow" class="external text" href="https://apps.dtic.mil/sti/pdfs/AD0767223.pdf">A Method of Computing Spherical Bessel Functions of Complex Argument with Tables</a> (PDF) (Research and Development Technical Report ECOM-5509). White Sands Missile Range, New Mexico: Atmospheric Sciences Laboratory, US Army Electronics Command.</cite>

</li>

<li id="cite_note-numerical-recipes-c++-2"><a href="#cite_ref-numerical-recipes-c++_2-0">^</a> <cite class="citation book cs1">Numerical Recipes in C++. pp. 177–179. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0 521 75033 4</bdi>.</cite>

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<li id="cite_note-Thompson-and-Barnett-3">^ <a href="#cite_ref-Thompson-and-Barnett_3-0">a</a> <a href="#cite_ref-Thompson-and-Barnett_3-1">b</a> <a href="#cite_ref-Thompson-and-Barnett_3-2">c</a> <cite id="CITEREFThompsonBarnett1986" class="citation journal cs1">Thompson, I.J.; Barnett, A.R. (1986). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016/0021-9991(86)90046-x">"Coulomb and Bessel functions of complex arguments and order"</a>. Journal of Computational Physics. 64 (2): 490–509. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1986JCoPh..64..490T">1986JCoPh..64..490T</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0021-9991%2886%2990046-x">10.1016/0021-9991(86)90046-x</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0021-9991">0021-9991</a>.</cite>

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFJ.1982" class="citation book cs1">J., Lentz, W. (August 1982). <a rel="nofollow" class="external text" href="http://worldcat.org/oclc/227549426">A Simplification of Lentz's Algorithm</a>. Defense Technical Information Center. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/227549426">227549426</a>.</cite><code class="cs1-code">{{cite book}}</code>: CS1 maint: multiple names: authors list (link)

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<li id="cite_note-Lentz_668–671-5">^ <a href="#cite_ref-Lentz_668–671_5-0">a</a> <a href="#cite_ref-Lentz_668–671_5-1">b</a> <cite id="CITEREFLentz1976" class="citation journal cs1">Lentz, William J. (1976-03-01). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1364/ao.15.000668">"Generating Bessel functions in Mie scattering calculations using continued fractions"</a>. Applied Optics. 15 (3): 668–671. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1976ApOpt..15..668L">1976ApOpt..15..668L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1364%2Fao.15.000668">10.1364/ao.15.000668</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-6935">0003-6935</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/20165036">20165036</a>.</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFJaaskelainenRuuskanen1981" class="citation journal cs1">Jaaskelainen, T.; Ruuskanen, J. (1981-10-01). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1364/ao.20.003289">"Note on Lentz's algorithm"</a>. Applied Optics. 20 (19): 3289–3290. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1981ApOpt..20.3289J">1981ApOpt..20.3289J</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1364%2Fao.20.003289">10.1364/ao.20.003289</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-6935">0003-6935</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/20333144">20333144</a>.</cite>

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFMasmoudiBouhlelPuech2012" class="citation book cs1">Masmoudi, Atef; Bouhlel, Med Salim; Puech, William (March 2012). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1109/setit.2012.6481959">"Image encryption using chaotic standard map and engle continued fractions map"</a>. 2012 6th International Conference on Sciences of Electronics, Technologies of Information and Telecommunications (SETIT). IEEE. pp. 474–480. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Fsetit.2012.6481959">10.1109/setit.2012.6481959</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4673-1658-3</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15380706">15380706</a>.</cite>

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<li id="cite_note-Numerical-Recipes-8">^ <a href="#cite_ref-Numerical-Recipes_8-0">a</a> <a href="#cite_ref-Numerical-Recipes_8-1">b</a> <a href="#cite_ref-Numerical-Recipes_8-2">c</a> <cite id="CITEREFPressTeukolskyVetterlingFlannery2007" class="citation book cs1">Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B. P. (2007). Numerical Recipes: The Art of Scientific Computing (3rd ed.). Cambridge University Press. pp. 207–208.</cite>

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<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFPressTeukolskyVetterlingFlannery1992" class="citation book cs1">Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B. P. (1992). Numerical Recipes in Fortran, The Art of Scientific Computing (2nd ed.). Cambridge University Press. p. 165.</cite>

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<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFPressTeukolsky1988" class="citation journal cs1">Press, William H.; Teukolsky, Saul A. (1988). <a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.4822777">"Evaluating Continued Fractions and Computing Exponential Integrals"</a>. Computers in Physics. 2 (5): 88. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1988ComPh...2...88P">1988ComPh...2...88P</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.4822777">10.1063/1.4822777</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0894-1866">0894-1866</a>.</cite>

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<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFWandOrmerod2012" class="citation journal cs1">Wand, Matt P.; Ormerod, John T. (2012-09-18). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1002/sta4.4">"Continued fraction enhancement of Bayesian computing"</a>. Stat. 1 (1): 31–41. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fsta4.4">10.1002/sta4.4</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2049-1573">2049-1573</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/22533111">22533111</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119636237">119636237</a>.</cite>

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