Padé Approximant -- from Wolfram MathWorld

Calculus and Analysis > Series > Series Expansions >

Interactive Entries > Interactive Demonstrations >

Padé Approximant

Approximants derived by expanding a function as a ratio of two power series and determining both the numerator and denominator coefficients. Padé approximations are usually superior to Taylor expansions when functions contain poles, because the use of rational functions allows them to be well-represented.

The Padé approximant corresponds to the Maclaurin series. When it exists, the Padé approximant to any power series

(1)

is unique. If is a transcendental function, then the terms are given by the Taylor series about

(2)

The coefficients are found by setting

(3)

and equating coefficients. can be multiplied by an arbitrary constant which will rescale the other coefficients, so an additional constraint can be applied. The conventional normalization is

(4)

Expanding (3) gives

			(5)
			(6)

These give the set of equations

	(7)
	(8)
	(9)
	(10)
	(11)
	(12)
	(13)
	(14)

where for and for . Solving these directly gives

[L/M]=(|a_(L-m+1) a_(L-m+2) ... a_(L+1); | | ... |; a_L a_(L+1) ... a_(L+M); sum_(j=M)^(L)a_(j-M)x^j sum_(j=M-1)^(L)a_(j-M+1)x^j ... sum_(j=0)^(L)a_jx^j|)/(|a_(L-M+1) a_(L-M+2) ... a_(L+1); | | ... |; a_L a_(L+1) ... a_(L+M); x^M x^(M-1) ... 1|),

(15)

where sums are replaced by a zero if the lower index exceeds the upper. Alternate forms are

[L/M]=sum_(j=0)^(L-M)a_jx^j+x^(L-M+1)w_(L/M)^(T)W_(L/M)^(-1)w_(L/M)
=sum_(j=0)^(L+n)a_jx^j+x^(L+n+1)w_((L+M)/M)^TW_(L/M)^(-1)w_((L+n)/M)

(16)

for

			(17)
			(18)

and .

For example, the first few Padé approximants for are

			(19)
			(20)
			(21)
			(22)
			(23)
			(24)
			(25)
			(26)
			(27)
			(28)
			(29)
			(30)
			(31)
			(32)
			(33)
			(34)

Two-term identities include

			(35)
			(36)
			(37)
			(38)
			(39)
			(40)

where is the C-determinant. Three-term identities can be derived using the Frobenius triangle identities (Baker 1975, p. 32).

A five-term identity is

(41)

Cross ratio identities include

			(42)
			(43)
			(44)
			(45)
			(46)

REFERENCES:

Baker, G. A. Jr. "The Theory and Application of The Pade Approximant Method." In Advances in Theoretical Physics, Vol. 1 (Ed. K. A. Brueckner). New York: Academic Press, pp. 1-58, 1965.

Baker, G. A. Jr. Essentials of Padé Approximants in Theoretical Physics. New York: Academic Press, pp. 27-38, 1975.

Baker, G. A. Jr. and Graves-Morris, P. Padé Approximants. New York: Cambridge University Press, 1996.

Brent, R. P.; Gustavson, F. G.; and Yun, D. Y. Y. "Fast Solution of Toeplitz Systems of Equations and Computation of Padé Approximants." J. Algorithms 1, 259-295, 1980.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Padé Approximants." §5.12 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 194-197, 1992.

Weisstein, E. W. "Books about Padé Approximants." http://www.ericweisstein.com/encyclopedias/books/PadeApproximants.html.

CITE THIS AS:

Weisstein, Eric W. "Padé Approximant." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PadeApproximant.html