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Padé table
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<h2 id="history">History</h2> <p>Although earlier mathematicians had obtained sporadic results involving sequences of rational approximations to <a href="/facts/Transcendental_function/yMkeq4Eo">transcendental functions</a>, <a href="/facts/Ferdinand_Georg_Frobenius/4cXXQyu5">Frobenius</a> (in 1881) was apparently the first to organize the approximants in the form of a table. <a href="/facts/Henri_Pad%C3%A9/H5AH3Uhe">Henri Padé</a> further expanded this notion in his doctoral thesis <i>Sur la representation approchee d'une fonction par des fractions rationelles</i>, in 1892. Over the ensuing 16 years Padé published 28 additional papers exploring the properties of his table, and relating the table to analytic continued fractions.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p><p>Modern interest in Padé tables was revived by <a href="/facts/Hubert_Stanley_Wall/9XDNKUnP">H. S. Wall</a> and <a href="/facts/Oskar_Perron/FwVtMp2T">Oskar Perron</a>, who were primarily interested in the connections between the tables and certain classes of continued fractions. <a href="/facts/Daniel_Shanks/d1k1JNAE">Daniel Shanks</a> and <a href="/facts/Peter_Wynn_(mathematician)/1tWmX59w">Peter Wynn</a> published influential papers about 1955, and W. B. Gragg obtained far-reaching convergence results during the '70s. More recently, the widespread use of electronic computers has stimulated a great deal of additional interest in the subject.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <h2 id="notation">Notation</h2> <p>A function <i>f</i>(<i>z</i>) is represented by a formal <a href="/facts/Power_series/vYh6sTps">power series</a>: </p> f ( z ) = c 0 + c 1 z + c 2 z 2 + ⋯ = ∑ l = 0 ∞ c l z l , {\displaystyle f(z)=c_{0}+c_{1}z+c_{2}z^{2}+\cdots =\sum _{l=0}^{\infty }c_{l}z^{l},} <p>where <i>c</i>0 ≠ 0, by convention. The (<i>m</i>, <i>n</i>)th entry<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> <i>Rm, n</i> in the Padé table for <i>f</i>(<i>z</i>) is then given by </p> R m , n ( z ) = P m ( z ) Q n ( z ) = a 0 + a 1 z + a 2 z 2 + ⋯ + a m z m b 0 + b 1 z + b 2 z 2 + ⋯ + b n z n {\displaystyle R_{m,n}(z)={\frac {P_{m}(z)}{Q_{n}(z)}}={\frac {a_{0}+a_{1}z+a_{2}z^{2}+\cdots +a_{m}z^{m}}{b_{0}+b_{1}z+b_{2}z^{2}+\cdots +b_{n}z^{n}}}} <p>where <i>Pm</i>(<i>z</i>) and <i>Qn</i>(<i>z</i>) are polynomials of degrees not more than <i>m</i> and <i>n</i>, respectively. The coefficients {<i>ai</i>} and {<i>bi</i>} can always be found by considering the expression </p> f ( z ) ≈ ∑ l = 0 m + n c l z l =: f a p p r o x ( z ) {\displaystyle f(z)\approx \sum _{l=0}^{m+n}c_{l}z^{l}=:f_{\mathrm {approx} }(z)} Q n ( z ) f a p p r o x ( z ) = P m ( z ) {\displaystyle Q_{n}(z)f_{\mathrm {approx} }(z)=P_{m}(z)} Q n ( z ) ( c 0 + c 1 z + c 2 z 2 + ⋯ + c m + n z m + n ) = P m ( z ) {\displaystyle Q_{n}(z)\left(c_{0}+c_{1}z+c_{2}z^{2}+\cdots +c_{m+n}z^{m+n}\right)=P_{m}(z)} <p>and equating coefficients of like powers of <i>z</i> up through <i>m</i> + <i>n</i>. For the coefficients of powers <i>m</i> + 1 to <i>m</i> + <i>n</i>, the right hand side is 0 and the resulting <a href="/facts/System_of_linear_equations/fk9CFdgp">system of linear equations</a> contains a homogeneous system of <i>n</i> equations in the <i>n</i> + 1 unknowns <i>bi</i>, and so admits of infinitely many solutions each of which determines a possible <i>Qn</i>. <i>Pm</i> is then easily found by equating the first <i>m</i> coefficients of the equation above. However, it can be shown that, due to cancellation, the generated rational functions <i>Rm, n</i> are all the same, so that the (<i>m</i>, <i>n</i>)th entry in the Padé table is unique.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> Alternatively, we may require that <i>b</i>0 = 1, thus putting the table in a standard form. </p><p>Although the entries in the Padé table can always be generated by solving this system of equations, that approach is computationally expensive. Usage of the Padé table has been extended to meromorphic functions by newer, timesaving methods such as the epsilon algorithm.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> <h2 id="the-block-theorem-and-normal-approximants">The block theorem and normal approximants</h2> <p>Because of the way the (<i>m</i>, <i>n</i>)th approximant is constructed, the difference </p> <i>Qn</i>(<i>z</i>)<i>f</i>(<i>z</i>) − <i>Pm</i>(<i>z</i>) <p>is a power series whose first term is of degree no less than </p> <i>m</i> + <i>n</i> + 1. <p>If the first term of that difference is of degree </p> <i>m</i> + <i>n</i> + <i>r</i> + 1, <i>r</i> > 0, <p>then the rational function <i>Rm, n</i> occupies </p> (<i>r</i> + 1)2 <p>cells in the Padé table, from position (<i>m</i>, <i>n</i>) through position (<i>m</i>+<i>r</i>, <i>n</i>+<i>r</i>), inclusive. In other words, if the same rational function appears more than once in the table, that rational function occupies a square block of cells within the table. This result is known as the block theorem. </p><p>If a particular rational function occurs exactly once in the Padé table, it is called a normal approximant to <i>f</i>(<i>z</i>). If every entry in the complete Padé table is normal, the table itself is said to be normal. Normal Padé approximants can be characterized using <a href="/facts/Determinant/eGYqJ2TF">determinants</a> of the coefficients <i>cn</i> in the Taylor series expansion of <i>f</i>(<i>z</i>), as follows. Define the (<i>m</i>, <i>n</i>)th determinant by </p> D m , n = | c m c m − 1 … c m − n + 2 c m − n + 1 c m + 1 c m … c m − n + 3 c m − n + 2 ⋮ ⋮ ⋮ ⋮ c m + n − 2 c m + n − 3 … c m c m − 1 c m + n − 1 c m + n − 2 … c m + 1 c m | {\displaystyle D_{m,n}=\left|{\begin{matrix}c_{m}&c_{m-1}&\ldots &c_{m-n+2}&c_{m-n+1}\\c_{m+1}&c_{m}&\ldots &c_{m-n+3}&c_{m-n+2}\\\vdots &\vdots &&\vdots &\vdots \\c_{m+n-2}&c_{m+n-3}&\ldots &c_{m}&c_{m-1}\\c_{m+n-1}&c_{m+n-2}&\ldots &c_{m+1}&c_{m}\\\end{matrix}}\right|} <p>with <i>D</i><i>m</i>,0 = 1, <i>D</i><i>m</i>,1 = <i>cm</i>, and <i>ck</i> = 0 for <i>k</i> < 0. Then </p> <ul><li>the (<i>m</i>, <i>n</i>)th approximant to <i>f</i>(<i>z</i>) is normal if and only if none of the four determinants <i>D</i><i>m</i>,<i>n</i>−1, <i>Dm,n</i>, <i>D</i><i>m</i>+1,<i>n</i>, and <i>D</i><i>m</i>+1,<i>n</i>+1 vanish; and</li> <li>the Padé table is normal if and only if none of the determinants <i>Dm,n</i> are equal to zero (note in particular that this means none of the coefficients <i>ck</i> in the series representation of <i>f</i>(<i>z</i>) can be zero).<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li></ul> <h2 id="connection-with-continued-fractions">Connection with continued fractions</h2> <p>One of the most important forms in which an analytic continued fraction can appear is as a regular <a href="/facts/Continued_fraction/Wg1cMPxN">continued fraction</a>, which is a continued fraction of the form </p> f ( z ) = b 0 + a 1 z 1 − a 2 z 1 − a 3 z 1 − a 4 z 1 − ⋱ . {\displaystyle f(z)=b_{0}+{\cfrac {a_{1}z}{1-{\cfrac {a_{2}z}{1-{\cfrac {a_{3}z}{1-{\cfrac {a_{4}z}{1-\ddots }}}}}}}}.} <p>where the <i>ai</i> ≠ 0 are complex constants, and <i>z</i> is a complex variable. </p><p>There is an intimate connection between regular continued fractions and Padé tables with normal approximants along the main diagonal: the "stairstep" sequence of Padé approximants <i>R</i>0,0, <i>R</i>1,0, <i>R</i>1,1, <i>R</i>2,1, <i>R</i>2,2, ... is normal if and only if that sequence coincides with the successive <a href="/facts/Convergent_(continued_fraction)/Wg1cMPxN">convergents</a> of a regular continued fraction. In other words, if the Padé table is normal along the main diagonal, it can be used to construct a regular continued fraction, and if a regular continued fraction representation for the function <i>f</i>(<i>z</i>) exists, then the main diagonal of the Padé table representing <i>f</i>(<i>z</i>) is normal.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> <h2 id="an-example--the-exponential-function">An example – the exponential function</h2> <p>Here is an example of a Padé table, for the <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a>. </p> A portion of the Padé table for the exponential function <i>ez</i><table><tbody><tr><th><i>n</i><i>m</i></th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th></tr><tr><th>0</th><td> 1 1 {\displaystyle {\frac {1}{1}}} </td><td> 1 1 − z {\displaystyle {\frac {1}{1-z}}} </td><td> 1 1 − z + 1 2 z 2 {\displaystyle {\frac {1}{1-z+{\scriptstyle {\frac {1}{2}}}z^{2}}}} </td><td> 1 1 − z + 1 2 z 2 − 1 6 z 3 {\displaystyle {\frac {1}{1-z+{\scriptstyle {\frac {1}{2}}}z^{2}-{\scriptstyle {\frac {1}{6}}}z^{3}}}} </td><td> 1 1 − z + 1 2 z 2 − 1 6 z 3 + 1 24 z 4 {\displaystyle {\frac {1}{1-z+{\frac {1}{2}}z^{2}-{\frac {1}{6}}z^{3}+{\frac {1}{24}}z^{4}}}} </td></tr><tr><th>1</th><td> 1 + z 1 {\displaystyle {\frac {1+z}{1}}} </td><td> 1 + 1 2 z 1 − 1 2 z {\displaystyle {\frac {1+{\scriptstyle {\frac {1}{2}}}z}{1-{\scriptstyle {\frac {1}{2}}}z}}} </td><td> 1 + 1 3 z 1 − 2 3 z + 1 6 z 2 {\displaystyle {\frac {1+{\scriptstyle {\frac {1}{3}}}z}{1-{\scriptstyle {\frac {2}{3}}}z+{\scriptstyle {\frac {1}{6}}}z^{2}}}} </td><td> 1 + 1 4 z 1 − 3 4 z + 1 4 z 2 − 1 24 z 3 {\displaystyle {\frac {1+{\scriptstyle {\frac {1}{4}}}z}{1-{\scriptstyle {\frac {3}{4}}}z+{\scriptstyle {\frac {1}{4}}}z^{2}-{\scriptstyle {\frac {1}{24}}}z^{3}}}} </td><td> 1 + 1 5 z 1 − 4 5 z + 3 10 z 2 − 1 15 z 3 + 1 120 z 4 {\displaystyle {\frac {1+{\frac {1}{5}}z}{1-{\frac {4}{5}}z+{\frac {3}{10}}z^{2}-{\frac {1}{15}}z^{3}+{\frac {1}{120}}z^{4}}}} </td></tr><tr><th>2</th><td> 1 + z + 1 2 z 2 1 {\displaystyle {\frac {1+z+{\scriptstyle {\frac {1}{2}}}z^{2}}{1}}} </td><td> 1 + 2 3 z + 1 6 z 2 1 − 1 3 z {\displaystyle {\frac {1+{\scriptstyle {\frac {2}{3}}}z+{\scriptstyle {\frac {1}{6}}}z^{2}}{1-{\scriptstyle {\frac {1}{3}}}z}}} </td><td> 1 + 1 2 z + 1 12 z 2 1 − 1 2 z + 1 12 z 2 {\displaystyle {\frac {1+{\scriptstyle {\frac {1}{2}}}z+{\scriptstyle {\frac {1}{12}}}z^{2}}{1-{\scriptstyle {\frac {1}{2}}}z+{\scriptstyle {\frac {1}{12}}}z^{2}}}} </td><td> 1 + 2 5 z + 1 20 z 2 1 − 3 5 z + 3 20 z 2 − 1 60 z 3 {\displaystyle {\frac {1+{\scriptstyle {\frac {2}{5}}}z+{\scriptstyle {\frac {1}{20}}}z^{2}}{1-{\scriptstyle {\frac {3}{5}}}z+{\scriptstyle {\frac {3}{20}}}z^{2}-{\scriptstyle {\frac {1}{60}}}z^{3}}}} </td><td> 1 + 1 3 z + 1 30 z 2 1 − 2 3 z + 1 5 z 2 − 1 30 z 3 + 1 360 z 4 {\displaystyle {\frac {1+{\frac {1}{3}}z+{\frac {1}{30}}z^{2}}{1-{\frac {2}{3}}z+{\frac {1}{5}}z^{2}-{\frac {1}{30}}z^{3}+{\frac {1}{360}}z^{4}}}} </td></tr><tr><th>3</th><td> 1 + z + 1 2 z 2 + 1 6 z 3 1 {\displaystyle {\frac {1+z+{\scriptstyle {\frac {1}{2}}}z^{2}+{\scriptstyle {\frac {1}{6}}}z^{3}}{1}}} </td><td> 1 + 3 4 z + 1 4 z 2 + 1 24 z 3 1 − 1 4 z {\displaystyle {\frac {1+{\scriptstyle {\frac {3}{4}}}z+{\scriptstyle {\frac {1}{4}}}z^{2}+{\scriptstyle {\frac {1}{24}}}z^{3}}{1-{\scriptstyle {\frac {1}{4}}}z}}} </td><td> 1 + 3 5 z + 3 20 z 2 + 1 60 z 3 1 − 2 5 z + 1 20 z 2 {\displaystyle {\frac {1+{\scriptstyle {\frac {3}{5}}}z+{\scriptstyle {\frac {3}{20}}}z^{2}+{\scriptstyle {\frac {1}{60}}}z^{3}}{1-{\scriptstyle {\frac {2}{5}}}z+{\scriptstyle {\frac {1}{20}}}z^{2}}}} </td><td> 1 + 1 2 z + 1 10 z 2 + 1 120 z 3 1 − 1 2 z + 1 10 z 2 − 1 120 z 3 {\displaystyle {\frac {1+{\scriptstyle {\frac {1}{2}}}z+{\scriptstyle {\frac {1}{10}}}z^{2}+{\scriptstyle {\frac {1}{120}}}z^{3}}{1-{\scriptstyle {\frac {1}{2}}}z+{\scriptstyle {\frac {1}{10}}}z^{2}-{\scriptstyle {\frac {1}{120}}}z^{3}}}} </td><td> 1 + 3 7 z + 1 14 z 2 + 1 210 z 3 1 − 4 7 z + 1 7 z 2 − 2 105 z 3 + 1 840 z 4 {\displaystyle {\frac {1+{\frac {3}{7}}z+{\frac {1}{14}}z^{2}+{\frac {1}{210}}z^{3}}{1-{\frac {4}{7}}z+{\frac {1}{7}}z^{2}-{\frac {2}{105}}z^{3}+{\frac {1}{840}}z^{4}}}} </td></tr><tr><th>4</th><td> 1 + z + 1 2 z 2 + 1 6 z 3 + 1 24 z 4 1 {\displaystyle {\frac {1+z+{\scriptstyle {\frac {1}{2}}}z^{2}+{\scriptstyle {\frac {1}{6}}}z^{3}+{\scriptstyle {\frac {1}{24}}}z^{4}}{1}}} </td><td> 1 + 4 5 z + 3 10 z 2 + 1 15 z 3 + 1 120 z 4 1 − 1 5 z {\displaystyle {\frac {1+{\scriptstyle {\frac {4}{5}}}z+{\scriptstyle {\frac {3}{10}}}z^{2}+{\scriptstyle {\frac {1}{15}}}z^{3}+{\scriptstyle {\frac {1}{120}}}z^{4}}{1-{\scriptstyle {\frac {1}{5}}}z}}} </td><td> 1 + 2 3 z + 1 5 z 2 + 1 30 z 3 + 1 360 z 4 1 − 1 3 z + 1 30 z 2 {\displaystyle {\frac {1+{\scriptstyle {\frac {2}{3}}}z+{\scriptstyle {\frac {1}{5}}}z^{2}+{\scriptstyle {\frac {1}{30}}}z^{3}+{\scriptstyle {\frac {1}{360}}}z^{4}}{1-{\scriptstyle {\frac {1}{3}}}z+{\scriptstyle {\frac {1}{30}}}z^{2}}}} </td><td> 1 + 4 7 z + 1 7 z 2 + 2 105 z 3 + 1 840 z 4 1 − 3 7 z + 1 14 z 2 − 1 210 z 3 {\displaystyle {\frac {1+{\scriptstyle {\frac {4}{7}}}z+{\scriptstyle {\frac {1}{7}}}z^{2}+{\scriptstyle {\frac {2}{105}}}z^{3}+{\scriptstyle {\frac {1}{840}}}z^{4}}{1-{\scriptstyle {\frac {3}{7}}}z+{\scriptstyle {\frac {1}{14}}}z^{2}-{\scriptstyle {\frac {1}{210}}}z^{3}}}} </td><td> 1 + 1 2 z + 3 28 z 2 + 1 84 z 3 + 1 1680 z 4 1 − 1 2 z + 3 28 z 2 − 1 84 z 3 + 1 1680 z 4 {\displaystyle {\frac {1+{\frac {1}{2}}z+{\frac {3}{28}}z^{2}+{\frac {1}{84}}z^{3}+{\frac {1}{1680}}z^{4}}{1-{\frac {1}{2}}z+{\frac {3}{28}}z^{2}-{\frac {1}{84}}z^{3}+{\frac {1}{1680}}z^{4}}}} </td></tr></tbody></table> <p>Several features are immediately apparent. </p> <ul><li>The first column of the table consists of the successive truncations of the <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> for <i>e</i><i>z</i>.</li> <li>Similarly, the first row contains the reciprocals of successive truncations of the series expansion of <i>e</i>−<i>z</i>.</li> <li>The approximants <i>Rm,n</i> and <i>Rn,m</i> are quite symmetrical – the numerators and denominators are interchanged, and the patterns of plus and minus signs are different, but the same coefficients appear in both of these approximants. They can be expressed in terms of special functions as</li></ul> R m , n = 1 F 1 ( − m ; − m − n ; z ) 1 F 1 ( − n ; − m − n ; − z ) = n ! 2 m θ m ( z 2 ; n − m + 2 , 2 ) m ! 2 n θ n ( − z 2 ; m − n + 2 , 2 ) {\displaystyle R_{m,n}={\frac {{}_{1}F_{1}(-m;-m-n;z)}{{}_{1}F_{1}(-n;-m-n;-z)}}={\frac {n!\,2^{m}\theta _{m}\left({\frac {z}{2}};n-m+2,2\right)}{m!\,2^{n}\theta _{n}\left(-{\frac {z}{2}};m-n+2,2\right)}}} , where 1 F 1 ( a ; b ; z ) {\displaystyle {}_{1}F_{1}(a;b;z)} is a <a href="/facts/Generalized_hypergeometric_series/fqXXDHdR">generalized hypergeometric series</a> and θ n ( x ; α , β ) {\displaystyle \theta _{n}(x;\alpha ,\beta )} is a <a href="/facts/Bessel_polynomials/rX8pb2HS">generalized reverse Bessel polynomial</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> The expressions on the main diagonal reduce to R n , n = θ n ( z / 2 ) / θ n ( − z / 2 ) {\displaystyle R_{n,n}=\theta _{n}(z/2)/\theta _{n}(-z/2)} , where θ n ( x ) {\displaystyle \theta _{n}(x)} is a <a href="/facts/Bessel_polynomials/rX8pb2HS">reverse Bessel polynomial</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> <ul><li>Computations involving the <i>R</i><i>n</i>,<i>n</i> (on the main diagonal) can be done quite efficiently. For example, <i>R</i>3,3 reproduces the power series for the exponential function perfectly up through 1/720 <i>z</i>6, but because of the symmetry of the two cubic polynomials, a very fast evaluation algorithm can be devised.</li></ul> <p>The procedure used to derive <a href="/facts/Gauss%27s_continued_fraction/QQKz1vaQ">Gauss's continued fraction</a> can be applied to a certain <a href="/facts/Confluent_hypergeometric_series/D0FxgWUa">confluent hypergeometric series</a> to derive the following C-fraction expansion for the exponential function, valid throughout the entire complex plane: </p> e z = 1 + z 1 − 1 2 z 1 + 1 6 z 1 − 1 6 z 1 + 1 10 z 1 − 1 10 z 1 + − ⋱ . {\displaystyle e^{z}=1+{\cfrac {z}{1-{\cfrac {{\frac {1}{2}}z}{1+{\cfrac {{\frac {1}{6}}z}{1-{\cfrac {{\frac {1}{6}}z}{1+{\cfrac {{\frac {1}{10}}z}{1-{\cfrac {{\frac {1}{10}}z}{1+-\ddots }}}}}}}}}}}}.} <p>By applying the <a href="/facts/Fundamental_recurrence_formulas/kbj3glnP">fundamental recurrence formulas</a> one may easily verify that the successive convergents of this C-fraction are the stairstep sequence of Padé approximants <i>R</i>0,0, <i>R</i>1,0, <i>R</i>1,1, ... In this particular case a closely related continued fraction can be obtained from the identity </p> e z = 1 e − z ; {\displaystyle e^{z}={\frac {1}{e^{-z}}};} <p>that continued fraction looks like this: </p> e z = 1 1 − z 1 + 1 2 z 1 − 1 6 z 1 + 1 6 z 1 − 1 10 z 1 + 1 10 z 1 − + ⋱ . {\displaystyle e^{z}={\cfrac {1}{1-{\cfrac {z}{1+{\cfrac {{\frac {1}{2}}z}{1-{\cfrac {{\frac {1}{6}}z}{1+{\cfrac {{\frac {1}{6}}z}{1-{\cfrac {{\frac {1}{10}}z}{1+{\cfrac {{\frac {1}{10}}z}{1-+\ddots }}}}}}}}}}}}}}.} <p>This fraction's successive convergents also appear in the Padé table, and form the sequence <i>R</i>0,0, <i>R</i>0,1, <i>R</i>1,1, <i>R</i>1,2, <i>R</i>2,2, ... </p> <h2 id="generalizations">Generalizations</h2> <p>A formal Newton series <i>L</i> is of the form </p> L ( z ) = c 0 + ∑ n = 1 ∞ c n ∏ k = 1 n ( z − β k ) {\displaystyle L(z)=c_{0}+\sum _{n=1}^{\infty }c_{n}\prod _{k=1}^{n}(z-\beta _{k})} <p>where the sequence {β<i>k</i>} of points in the complex plane is known as the set of <i>interpolation points</i>. A sequence of rational approximants <i>Rm,n</i> can be formed for such a series <i>L</i> in a manner entirely analogous to the procedure described above, and the approximants can be arranged in a <i>Newton-Padé table</i>. It has been shown<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> that some "staircase" sequences in the Newton-Padé table correspond with the successive convergents of a Thiele-type continued fraction, which is of the form </p> a 0 + a 1 ( z − β 1 ) 1 − a 2 ( z − β 2 ) 1 − a 3 ( z − β 3 ) 1 − ⋱ . {\displaystyle a_{0}+{\cfrac {a_{1}(z-\beta _{1})}{1-{\cfrac {a_{2}(z-\beta _{2})}{1-{\cfrac {a_{3}(z-\beta _{3})}{1-\ddots }}}}}}.} <p>Mathematicians have also constructed <i>two-point Padé tables</i> by considering two series, one in powers of <i>z</i>, the other in powers of 1/<i>z</i>, which alternately represent the function <i>f</i>(<i>z</i>) in a neighborhood of zero and in a neighborhood of infinity.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Shanks_transformation/RAS4mDuo">Shanks transformation</a></li></ul> <h2 id="notes">Notes</h2> <ul><li>Jones, William B.; Thron, W. J. (1980). <a href="https://archive.org/details/continuedfractio0000jone"><i>Continued Fractions: Theory and Applications</i></a>. Reading, Massachusetts: Addison-Wesley Publishing Company. pp. <a href="https://archive.org/details/continuedfractio0000jone/page/185">185–197</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-201-13510-8.</li> <li>Wall, H. S. (1973). <i>Analytic Theory of Continued Fractions</i>. <a href="/facts/Chelsea_Publishing_Company/KWAyXCyF">Chelsea Publishing Company</a>. pp. 377–415. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8284-0207-8.(This is a reprint of the volume originally published by <a href="/facts/D._Van_Nostrand_Company,_Inc./SBWcDeMm">D. Van Nostrand Company, Inc.</a>, in 1948.)</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>O'Connor, John J.; Robertson, Edmund F., "Padé table", MacTutor History of Mathematics Archive, University of St Andrews <a href="/wiki/Edmund_F._Robertson" target="_blank">/wiki/Edmund_F._Robertson</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Jones and Thron, 1980. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>The (m, n)th entry is considered to lie in row m and column n, and numbering of the rows and columns begins at (0, 0). <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Jones and Thron, 1980. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Wynn, Peter (Apr 1956). "On a Device for Computing the em(Sn) Transformation". Mathematical Tables and Other Aids to Computation. 10 (54). American Mathematical Society: 91–96. doi:10.2307/2002183. JSTOR 2002183. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Gragg, W.B. (Jan 1972). "The Padé Table and its Relation to Certain Algorithms of Numerical Analysis". SIAM Review. 14 (1): 1–62. doi:10.1137/1014001. ISSN 0036-1445. JSTOR 2028911. <a href="https://doi.org/10.1137%2F1014001" target="_blank">https://doi.org/10.1137%2F1014001</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Jones and Thron, 1980. <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>*Underhill, C. (1986). "Some Asymptotic Properties of Padé Approximants to e − x {\displaystyle e^{-x}} ". Mathematics of Computation. 47 (175): 253–263. JSTOR 2008092. <a href="/wiki/JSTOR_(identifier)" target="_blank">/wiki/JSTOR_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>*"The On-Line Encyclopedia of Integer Sequences (OEIS)". Founded in 1964 by Sloane, N. J. A. The OEIS Foundation Inc.{{cite web}}: CS1 maint: others (link) (See sequence OEIS: A113025.) <a href="https://oeis.org/" target="_blank">https://oeis.org/</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Thiele, T.N. (1909). Interpolationsrechnung. Leipzig: Teubner. ISBN 1-4297-0249-4. <a href="1-4297-0249-4" target="_blank">1-4297-0249-4</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Jones and Thron, 1980. <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> </ol>