Padé Table - An Example – The Exponential Function

An Example – The Exponential Function

Here is an example of a Padé table, for the exponential function.

A portion of the Padé table for the exponential function ez
m \ n 0 1 2 3
0
1 \frac{1 + {\scriptstyle\frac{1}{3}}z}
{1 - {\scriptstyle\frac{2}{3}}z + {\scriptstyle\frac{1}{6}}z^2} \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}
2 \frac{1 + {\scriptstyle\frac{2}{3}}z + {\scriptstyle\frac{1}{6}}z^2}
{1 - {\scriptstyle\frac{1}{3}}z} \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} \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}
3 \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} \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} \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}
4 \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} \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} \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}

Several interesting features are immediately apparent.

  • The first column of the table consists of the successive truncations of the Taylor series for ez.
  • Similarly, the first row contains the reciprocals of successive truncations of the series expansion of e−z.
  • The approximants Rm,n and Rn,m 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. In fact, using the notation of generalized hypergeometric series,
  • Computations involving the Rn,n (on the main diagonal) can be done quite efficiently. For example, R3,3 reproduces the power series for the exponential function perfectly up through 1/720 z6, but because of the symmetry of the two cubic polynomials, a very fast evaluation algorithm can be devised.

The procedure used to derive Gauss's continued fraction can be applied to a certain confluent hypergeometric series to derive the following C-fraction expansion for the exponential function, valid throughout the entire complex plane:


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}}}}}}.

By applying the fundamental recurrence formulas one may easily verify that the successive convergents of this C-fraction are the stairstep sequence of Padé approximants R0,0, R1,0, R1,1, … Interestingly, in this particular case a closely related continued fraction can be obtained from the identity


e^z = \frac{1}{e^{-z}};

that continued fraction looks like this:


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}}}}}}}.

This fraction's successive convergents also appear in the Padé table, and form the sequence R0,0, R0,1, R1,1, R1,2, R2,2, …

Read more about this topic:  Padé Table

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