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Padé Approximant

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A Padé approximant is a rational function whose power series agrees with that of a given function to the highest possible order. Padé approximations are usually superior to Taylor series when functions contain poles, because the use of rational functions allows them to be well-represented.

The Padé approximant R_(L/0) corresponds to the Maclaurin series. When it exists, the R_(L/M)=[L/M] Padé approximant to any power series

A(x)=sum_(j=0)^inftya_jx^j
(1)

is unique. If A(x) is a transcendental function, then the terms are given by the Taylor series about x_0

a_n=1/(n!)A^((n))(x_0).
(2)

The coefficients are found by setting

A(x)-(P_L(x))/(Q_M(x))=0
(3)

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

Q_M(0)=1.
(4)

Expanding (3) gives

P_L(x)=p_0+p_1x+...+p_Lx^L
(5)
Q_M(x)=1+q_1x+...+q_Mx^M.
(6)

These give the set of equations

a_0=p_0
(7)
a_1+a_0q_1=p_1
(8)
a_2+a_1q_1+a_0q_2=p_2
(9)
|
a_L+a_(L-1)q_1+...+a_0q_L=p_L
(10)
a_(L+1)+a_Lq_1+...+a_(L-M+1)q_M=0
(11)
|
a_(L+M)+a_(L+M-1)q_1+...+a_Lq_M=0,
(12)

where a_n=0 for n<0 and q_j=0 for j>M. 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|),
(13)

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)^TW_(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)
(14)

for

W_(L/M)=[a_(L-M+1)-xa_(L-M+2) ... a_L-xa_(L+1); | ... |; a_L-xa_(L+1) ... a_(L+M-1)-xa_(L+M)]
(15)
w_(L/M)=[a_(L-M+1); a_(L-M+2); |; a_L],
(16)

and 0<=n<=M.

For example, the first few Padé approximants for e^x are

exp_(0/0)(x)=1
(17)
exp_(0/1)(x)=1/(1-x)
(18)
exp_(0/2)(x)=2/(2-2x+x^2)
(19)
exp_(0/3)(x)=6/(6-6x+3x^2-x^3)
(20)
exp_(1/0)(x)=1+x
(21)
exp_(1/1)(x)=(2+x)/(2-x)
(22)
exp_(1/2)(x)=(6+2x)/(6-4x+x^2)
(23)
exp_(1/3)(x)=(24+6x)/(24-18x+6x^2-x^3)
(24)
exp_(2/0)(x)=(2+2x+x^2)/2
(25)
exp_(2/1)(x)=(6+4x+x^2)/(6-2x)
(26)
exp_(2/2)(x)=(12+6x+x^2)/(12-6x+x^2)
(27)
exp_(2/3)(x)=(60+24x+3x^2)/(60-36x+9x^2-x^3)
(28)
exp_(3/0)(x)=(6+6x+3x^2+x^3)/6
(29)
exp_(3/1)(x)=(24+18x+6x^2+x^3)/(24-6x)
(30)
exp_(3/2)(x)=(60+36x+9x^2+x^3)/(60-24x+3x^2)
(31)
exp_(3/3)(x)=(120+60x+12x^2+x^3)/(120-60x+12x^2-x^3).
(32)

Two-term identities include

(P_(L+1)(x))/(Q_(M+1)(x))-(P_L^'(x))/(Q_M^'(x))=(C_((L+1)/(M+1))^2x^(L+M+1))/(Q_(M+1)(x)Q_M^'(x))
(33)
(P_(L+1)(x))/(Q_M(x))-(P_L^'(x))/(Q_M^'(x))=(C_((L+1)/M)C_((L+1)/(M+1))x^(L+M+1))/(Q_M(x)Q_M^'(x))
(34)
(P_L(x))/(Q_(M+1)(x))-(P_L^'(x))/(Q_M^'(x))=(C_(L/(M+1))C_((L+1)/(M+1))x^(L+M+1))/(Q_M(x)Q_M^'(x))
(35)
(P_L(x))/(Q_(M+1)(x))-(P_(L+1)^'(x))/(Q_M^')=(C_((L+1)/(M+1))^2x^(L+M+2))/(Q_(M+1)Q_M^')
(36)
(P_(L+1))/(Q_M(x))-(P_(L-1)^'(x))/(Q_M^'(x))=(C_(L/(M+1))C_((L+1)/M)x^(L+M)+C_(L/M)C_((L+1)/(M+1))x^(L+M+1))/(Q_M(x)Q_M^'(x))
(37)
(P_L(x))/(Q_(M+1)(x))-(P_L^'(x))/(Q_(M-1)^'(x))=(C_(L/(M+1))C_((L+1)/M)x^(L+M)-C_(L/M)C_((L+1)/(M+1))x^(L+M+1))/(Q_(M+1)(x)Q_(M-1)^'(x)),
(38)

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

A five-term identity is

S_((L+1)/M)S_((L-1)/M)-S_(L/(M+1))S_(L/(M-1))=S_(L/M)^2.
(39)

Cross ratio identities include

((R_(L/M)-R_(L/(M+1)))(R_((L+1)/M)-R_((L+1)/(M+1))))/((R_(L/M)-R_((L+1)/M))(R_(L/(M+1))-R_((L+1)/(M+1))))=(C_(L/(M+1))C_((L+2)/(M+1)))/(C_((L+1)/M)C_((L+1)/(M+2)))
(40)
((R_(L/M)-R_((L+1)/(M+1)))(R_((L+1)/M)-R_(L/(M+1))))/((R_(L/M)-R_(L/(M+1)))(R_((L+1)/M)-R_((L+1)/(M+1))))=(C_((L+1)/(M+1))^2x)/(C_(L/(M+1))C_((L+2)/(M+1)))
(41)
((R_(L/M)-R_((L+1)/(M+1)))(R_((L+1)/M)-R_(L/(M+1))))/((R_(L/M)-R_((L+1)/M))(R_(L/(M+1))-R_((L+1)/(M+1))))=(C_((L+1)/(M+1))^2x)/(C_((L+1)/M)C_((L+1)/(M+2)))
(42)
((R_(L/M)-R_((L+1)/(M-1)))(R_(L/(M+1))-R_((L+1)/M)))/((R_(L/M)-R_(L/(M+1)))(R_((L+1)/(M+1))-R_((L+1)/M)))=(C_((L+1)/M)C_((L+1)/(M+1))x)/(C_(L/(M+1))C_((L+2)/M))
(43)
((R_(L/M)-R_((L-1)/(M+1)))(R_((L+1)/M)-R_(L/(M+1))))/((R_(L/M)-R_((L+1)/M))(R_((L-1)/(M+1))-R_(L/(M+1))))=(C_(L/(M+1))C_((L+1)/(M+1))x)/(C_((L+1)/M)C_(L/(M+2))).
(44)

See also

C-Determinant, Economized Rational Approximation, Frobenius Triangle Identities

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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.

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Padé Approximant

Cite this as:

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

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