# Padé approximation

2020 Mathematics Subject Classification: *Primary:* 41A21 [MSN][ZBL]

The field concerned with best rational approximation to power series. Let

\begin{equation} \label{eq:1} f( z) = \sum _ {k=0}^ \infty f _ {k} z ^ {k} \end{equation}

be an arbitrary power series (formal or convergent), and let $ R _ {n,m}$ ($ n, m \geq 0 $) be the class of all rational functions of type $ p/q $ where $ p $ and $ q $ are polynomials in $ z $, $ \mathop{\rm deg} q \leq m $, $ \mathop{\rm deg} p \leq n $ and $ q \not\equiv 0 $. A Padé approximant of type $ ( n, m) $ to the power series \eqref{eq:1} (the function $ f $) is a rational function $ \pi _ {n,m} \in R _ {n,m} $ having the maximum possible order of contact in the class $ R _ {n,m} $ with the power series \eqref{eq:1} at the point $ z= 0 $. More precisely, the function $ \pi _ {n,m} $ is determined by the condition

$$ \sigma ( f - \pi _ {n,m} ) = \max \{ {\sigma ( f- r) } : {r \in R _ {n,m} } \}, $$

where $ \sigma ( \phi ) $ is the index of the first non-zero coefficient of the series

$$ \phi = \sum _ { k=0 }^ \infty \phi _ {k} z ^ {k} . $$

It is also possible to determine the function $ \pi _ {n,m} $ as the quotient $ p/q $ of arbitrary polynomials $ p $ and $ q $ $ ( q \not\equiv 0) $ satisfying the conditions

$$ \mathop{\rm deg} p \leq n ,\ \mathop{\rm deg} q \leq m , $$

\begin{equation} \label{eq:2} ( qf - p)( z) = A _ {n,m} z ^ {n+} m+ 1 + \dots . \end{equation}

For fixed $ n, m $, there exists a unique Padé approximant $ \pi _ {n,m} $ to the power series \eqref{eq:1}. The table $ \{ \pi _ {n,m} \} _ {n,m= 0} ^ \infty $ is called the Padé table of the series \eqref{eq:1}. The sequences of type $ \{ \pi _ {n,m} \} _ {n= 0} ^ \infty $ are called the rows of the Padé table (the zero row coincides with the sequence of Taylor polynomials of $ f $); $ \{ \pi _ {n,m} \} _ {m= 0} ^ \infty $ are called the columns of the Padé table; and $ \{ \pi _ {n+ j,n} \} _ {j= 0} ^ \infty $ are called the diagonals of the Padé table. The most important special case $ j= 0 $ is the principal diagonal.

The calculation of the functions $ \pi _ {n,m} $ reduces to the solution of a system of linear equations whose coefficients are expressed in terms of the coefficients $ f _ {k} $, $ k = 0 \dots n+ m $, of the given power series. If the Hankel matrix $ \Delta _ {n,m} $,

$$ \Delta _ {n,m} = \left | \begin{array}{cccc} f _ {n- m+ 1} &f _ {n- m+ 2} &\dots &f _ {n} \\ \dots &\dots &\dots &\dots \\ f _ {n} &f _ {n+ 1} &\dots &f _ {n+ m- 1} \\ \end{array} \right | , $$

has determinant non-zero, then the denominator $ q _ {n,m} $ of the function $ \pi _ {n,m} $ is given by the formula

$$ q _ {n,m} ( z) = \frac{1}{ \mathop{\rm det} ( \Delta _ {n,m} ) } \left | \begin{array}{cc} \Delta _ {n,m} &f _ {n+ 1} \\ {} &\cdot \\ {} &\cdot \\ {} &\cdot \\ {} &f _ {n+ m} \\ z ^ {m} \dots z & 1 \\ \end{array} \right | $$

(the normalization is $ q _ {n,m} ( 0) = 1 $; an explicit formula can also be written down for the numerator of the function $ \pi _ {n,m} $). Moreover,

$$ ( f - \pi _ {n,m} )( z) = A _ {n,m} z ^ {n+ m+ 1} + \dots . $$

The latter relation is sometimes taken as the definition of a Padé approximant; in this case a Padé approximant need not exist for certain $ ( n, m) $. The Padé approximant of type $ ( n, m) $ of the given power series $ f $ is often denoted by the symbol

$$ [ n/m ] = [ n/m ] _ {f} . $$

For an effective calculation of Padé approximants, it is more convenient to make use not of explicit formulas but of recurrence relations existing in the Padé table. A large number of algorithms have been constructed for the automatic calculation of a Padé approximant; these problems are of specific importance in connection with applications (see [17], [18]).

The first general problem concerning the interpolation of given values of a function at $ n+ m+ 1 $ different points by means of rational functions of class $ R _ {n,m} $ was considered by A.L. Cauchy [1]; C.G.J. Jacobi [2] extended Cauchy's results to the case of multiple-point interpolation. The case of one $(n+m+1)$-multiple point corresponds to Padé approximation. The idea of Padé approximation was formulated at the end of the 19th century within the classical theory of continued fractions (G. Frobenius [3], H. Padé [4]). Fundamental results on diagonal Padé approximants were obtained by P.L. Chebyshev, A.A. Markov and T.J. Stieltjes in terms of continued fractions. They discovered and studied the relations of diagonal Padé approximants with orthogonal polynomials, quadrature formulas, moment problems, and other problems of classical analysis (see [7]–). The origin of the study of the rows of the Padé table lays in the work on the radius of meromorphy of a function defined by a power series and on the convergence of the row of the Padé table in the discs of meromorphy (see [5], [6]).

From the beginning of the 20th century onwards, Padé approximation has become an independent object of analysis and constitutes an important chapter in the theory of rational approximation of analytic functions. Using local data (coefficients of a power series) for their construction, they allow one to study global properties of the corresponding analytic function (analytic continuation, the character and distribution of singularities, etc.) and to compute the value of a function outside the disc of convergence of the power series.

Along with classical Padé approximation, various generalizations have been considered: general interpolation processes by means of rational functions with free poles (multiple-point Padé approximation); rational approximation of series with respect to given systems of polynomials (e.g. with respect to orthogonal polynomials); joint Padé approximation (Padé–Hermite approximation); rational (Padé type) approximation of power series of several variables, and other topics.

#### References

[1] | A.L. Cauchy, "Cours d'analyse de l'école royale polytechnique" , 1 , Paris (1821) |

[2] | C.G.J. Jacobi, "Ueber die Darstellung einer Reihe gegebener Werthe durch eine gebrochene rationale Funktion" J. Reine Angew. Math. , 30 (1846) pp. 127–156 |

[3] | G. Frobenius, "Ueber Relationen zwischen den Näherungsbrüchen von Potenzen" J. Reine Angew. Math. , 90 (1881) pp. 1–17 |

[4] | H. Padé, "Sur la représentation approchée d'une fonction par des fractions rationnelles" Ann. Sci. Ecole Normale Sup. , 9 (Suppl.) (1892) pp. 1–93 |

[5] | J. Hadamard, "Essai sur l'étude des fonctions données par leur développement de Taylor" J. Math. Pures Appl. , 8 (1892) pp. 101–186 |

[6] | R.F.B. Montessus de Ballore, "Sur les fractions continues algébriques" Bull. Soc. Math. France , 30 (1902) pp. 26–36 |

[7] | P.L. Chebyshev, "Collected works" , 3 , Moscow-Leningrad (1948) (In Russian) |

[8] | A.A. Markov, "Selected work on the theory of continued fractions and the theory of functions deviating least from zero" , Moscow-Leningrad (1948) (In Russian) |

[9a] | T.J. Stieltjes, "Recherches sur les fractions continues" Ann. Fac. Sci. Univ. Toulouse , 8 (1894) pp. 1–122 |

[9b] | T.J. Stieltjes, "Recherches sur les fractions continues" Ann. Fac. Sci. Univ. Toulouse , 9 (1895) pp. 1–47 |

[10] | H.S. Wall, "Analytic theory of continued fractions" , Chelsea (1973) MR0025596 MR0008102 Zbl 0035.03601 Zbl 0060.16502 |

[11] | O. Perron, "Die Lehre von den Kettenbrüchen" , 2 , Teubner (1977) MR0085349 MR0064172 MR0037384 Zbl 0077.06602 Zbl 0056.05901 Zbl 0041.18206 Zbl 55.0262.09 Zbl 43.0283.04 |

[12] | G.A. Baker jr. (ed.) J.L. Gamel (ed.) , The Padé approximant in theoretical physics , Acad. Press (1970) |

[13] | P.R. Grave-Morris (ed.) , Padé approximants and their application (Canterbury, 1972) , Acad. Press (1973) |

[14] | H. Cabannes (ed.) , Padé approximants method and its applications to mechanics , Springer (1973) |

[15] | G.A. Baker, "Essentials of Padé approximants" , Acad. Press (1975) |

[16] | E.B. Saff (ed.) R.S. Varga (ed.) , Padé and Rational Approximation (Tampa, 1976) , Acad. Press (1977) MR458010 |

[17] | J. Gilewicz, "Approximants de Padé" , Springer (1978) |

[18] | L. Wuytack (ed.) , Padé approximation and its applications (Antwerp, 1979) , Lect. notes in math. , 888 , Springer (1979) MR561441 |

#### Comments

Over the last ten years, there has been an enormous increase in the number of new results concerning Padé and Hermite–Padé approximation. One of the main influences has been the interplay between this field and the fields of continued fractions, orthogonal polynomials, moment problems, potential theory, and functional analysis.

Also, the generalizations in the directions of two-point Padé approximation, simultaneous approximation (already indicated in the work by Ch. Hermite and Padé) and multivariate approximation have led to new results.

Most of the relevant publications can be found in the proceedings of the many conferences on the subject and in recent volumes of the "Journal of Approximation Theory" or the "Journal of Constructive Approximation".

Reference [a2] is a valuable and interesting historical survey of the subject. The article by C. Brezinski in [a3] contains a rather extensive bibliography; a much more complete one has been compiled by Brezinski and is available on request.

#### References

[a1] | G.A. Baker jr., P. Graves-Morris, "Padé approximants and its applications" , 2 , Addison-Wesley (1981) |

[a2] | M.G. de Bruin (ed.) H. van Rossum (ed.) , Padé approximation and its applications (Amsterdam, 1980) , Lect. notes in math. , 888 , Springer (1981) |

[a3] | C. Brezinski, "Padé-type approximation and general orthogonal polynomials" , Birkhäuser (1980) MR0561106 Zbl 0418.41012 |

[a4] | A. Cuyt, "Padé approximation for operators: theory and applications" , Springer (1984) |

[a5] | P.P. Petrushev, V.A. Popov, "Rational approximation of real functions" , Chapt. 12 , Cambridge Univ. Press (1987) |

[a6] | Ch. Hermite, "Sur la fonction exponentielle" C.R. Acad. Sci. , LXXVII (1873) pp. 18–24; 74–79; 226–233; 285–293 |

[a7] | K. Mahler, "Perfect systems" Compos. Math. , 19 (1968) pp. 95–166 (Writting during his stay in Holland in 1934–1935) |

[a8] | W.B. Jones, W.J. Thron, "Continued fractions and their applications" , Addison-Wesley (1980) |

[a9] | , First French–Polish Meeting on Padé Approximation and Convergence Acceleration Techniques (Warsaw, 1981) , CPT-81/PE 1354 , CNRS (1982) |

[a10] | H. Werner (ed.) H.-J. Bünger (ed.) , Padé approximation and its application (Bad Honnef, 1983) , Lect. notes in math. , 1071 , Springer (1984) |

[a11] | P.R. Graves-Morris (ed.) E.B. Saff (ed.) R.S. Varga (ed.) , Rational Approximation and Interpolation (Tampa, 1983) , Lect. notes in math. , 1105 , Springer (1984) |

[a12] | C. Brezinski (ed.) A. Draux (ed.) A.P. Magnus (ed.) P. Maroni (ed.) A. Ronveaux (ed.) , Polynômes Orthogonaux et Applications (Bar-le-Duc, 1984) , Lect. notes in math. , 1171 , Springer (1985) |

[a13] | "Internat. Conf. Computational and Applied Mathematics (Leuven, 1984)" J. Comput. Applied Math. , 12–13 (1985) |

[a14] | J. Gilewicz (ed.) M. Pindor (ed.) W. Siemasko (ed.) , Rational Approximation and its Application in Mathematics and Physics (Lańcut, 1985) , Lect. notes in math. , 1237 , Springer (1987) |

[a15] | A. Cuyt (ed.) , Nonlinear numerical methods and rational approximation (Antwerp, 1987) , Reidel (1988) MR1005348 |

[a16] | M. Alfaro (ed.) J.S. Dehesa (ed.) F.J. Marcellna (ed.) J.L. Rubio de Francia (ed.) , Orthogonal Polynomials and Their Applications , Lect. notes in math. , 1329 , Springer (1988) |

[a17] | "Conf. Constructive Function Theory (Edmonton, 1986)" Rocky Mt. J. of Math. , 19 (1989) |

[a18] | H. Werner (ed.) et al. (ed.) , Computational aspects of complex analysis , Reidel (1983) |

[a19] | "Volume dedicated to H. Werner" J. Comput. Appl. Math. , 19 : 1 (1987) |

[a20] | F. Broeckx (ed.) et al. (ed.) , Proc. Internat. Conf. Comput. and Appl. Math. (Leuven, July 1983) , North-Holland (1985) |

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Padé approximation.

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