Padé table

In complex analysis, a Padé table is an array, possibly of infinite extent, of the rational Padé approximants R_{m, n} to a formal power series. Certain sequences of approximants lying within a Padé table can often be shown to correspond with successive convergents of a continued fraction representation of a holomorphic or meromorphic function.

History

Although earlier mathematicians had obtained sporadic results involving sequences of rational approximations to transcendental functions, Frobenius (in 1881) was apparently the first to organize the approximants in the form of a table. Henri Padé further expanded this notion in his doctoral thesis Sur la representation approchee d'une fonction par des fractions rationelles, 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.

Modern interest in Padé tables was revived by H. S. Wall and Oskar Perron, who were primarily interested in the connections between the tables and certain classes of continued fractions. Daniel Shanks and Peter Wynn 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.

Notation

A function f(z) is represented by a formal power series:

$f(z)=c_{0}+c_{1}z+c_{2}z^{2}+\dots =\sum _{n=0}^{\infty }c_{n}z^{n}.$

The (m, n)th entry R_{m, n} in the Padé table for f(z) is then given by

$R_{m,n}(z)={\frac {P_{m}(z)}{Q_{n}(z)}}={\frac {a_{0}+a_{1}z+a_{2}z^{2}+\dots +a_{m}z^{m}}{b_{0}+b_{1}z+b_{2}z^{2}+\dots +b_{n}z^{n}}}$

where P_m(z) and Q_n(z) are polynomials of degrees m and n, respectively. The coefficients {a_i} and {b_i} can always be found by considering the expression

$Q_{n}(z)\left(c_{0}+c_{1}z+c_{2}z^{2}+\dots +c_{m+n}z^{m+n}\right)=P_{m}(z)$

and equating coefficients of like powers of z. The resulting system of linear equations contains a homogeneous system of n equations in n + 1 unknowns, and so admits of infinitely many solutions. However, it can be shown that the generated rational functions R_m, n are all the same, so that the (m, n)th entry in the Padé table is unique. Alternatively, we may require that b₀ = 1, thus putting the table in a standard form.

Although the entries in the Padé table can always be generated by solving this system of equations, that approach is computationally expensive. More efficient methods have been devised, including the epsilon algorithm.

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 e^z

	0	1	2	3
0	${\frac {1}{1}}$	${\frac {1+z}{1}}$	${\frac {1+z+{\scriptstyle {\frac {1}{2}}}z^{2}}{1}}$	${\frac {1+z+{\scriptstyle {\frac {1}{2}}}z^{2}+{\scriptstyle {\frac {1}{6}}}z^{3}}{1}}$
1	${\frac {1}{1-z}}$	${\frac {1+{\scriptstyle {\frac {3}{2}}}z}{1-{\scriptstyle {\frac {1}{2}}}z}}$	${\frac {1+{\scriptstyle {\frac {2}{3}}}z+{\scriptstyle {\frac {1}{6}}}z^{2}}{1-{\scriptstyle {\frac {1}{3}}}z}}$	${\frac {1+{\scriptstyle {\frac {3}{4}}}z+{\scriptstyle {\frac {1}{4}}}z^{2}+{\scriptstyle {\frac {1}{24}}}z^{3}}{1-{\scriptstyle {\frac {1}{4}}}z}}$
2	${\frac {1}{1-z+{\scriptstyle {\frac {1}{2}}}z^{2}}}$	${\frac {1+{\scriptstyle {\frac {1}{3}}}z}{1-{\scriptstyle {\frac {2}{3}}}z+{\scriptstyle {\frac {1}{6}}}z^{2}}}$	${\frac {1+{\scriptstyle {\frac {1}{2}}}z+{\scriptstyle {\frac {1}{12}}}z^{2}}{1-{\scriptstyle {\frac {1}{2}}}z+{\scriptstyle {\frac {1}{12}}}z^{2}}}$	${\frac {1+{\scriptstyle {\frac {3}{5}}}z+{\scriptstyle {\frac {3}{20}}}z^{2}+{\scriptstyle {\frac {1}{60}}}z^{3}}{1-{\scriptstyle {\frac {2}{5}}}z+{\scriptstyle {\frac {1}{20}}}z^{2}}}$
3	${\frac {1}{1-z+{\scriptstyle {\frac {1}{2}}}z^{2}-{\scriptstyle {\frac {1}{6}}}z^{3}}}$	${\frac {1+{\scriptstyle {\frac {1}{4}}}z}{1-{\scriptstyle {\frac {3}{4}}}z+{\scriptstyle {\frac {1}{4}}}z^{2}-{\scriptstyle {\frac {1}{24}}}z^{3}}}$	${\frac {1+{\scriptstyle {\frac {2}{5}}}z+{\scriptstyle {\frac {1}{20}}}z^{2}}{1-{\scriptstyle {\frac {3}{5}}}z+{\scriptstyle {\frac {3}{20}}}z^{2}-{\scriptstyle {\frac {1}{60}}}z^{3}}}$	${\frac {1+{\scriptstyle {\frac {1}{2}}}z+{\scriptstyle {\frac {1}{10}}}z^{2}+{\scriptstyle {\frac {1}{120}}}z^{3}}{1-{\scriptstyle {\frac {1}{2}}}z+{\scriptstyle {\frac {1}{10}}}z^{2}-{\scriptstyle {\frac {1}{120}}}z^{3}}}$

Several interesting features are immediately apparent.

• The first row of the table consists of the successive truncations of the Taylor series for e^z.
• Similarly, the first column contains the reciprocals of successive truncations of the series expansion of e^−z.
• The approximants R_m,n and R_n,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.
• Computations involving the R_n,n (on the main diagonal) can be done quite efficiently. For example, R_3,3 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.

References

• Jones, William B. (1980). Continued Fractions: Theory and Applications. Reading, Massachusetts: Addison-Wesley Publishing Company. pp. 185–197. ISBN 0-201-13510-8. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)