TOPICS
Search

Padé Approximant

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

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 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)
|
(10)
a_L+a_(L-1)q_1+...+a_0q_L=p_L
(11)
a_(L+1)+a_Lq_1+...+a_(L-M+1)q_M=0
(12)
|
(13)
a_(L+M)+a_(L+M-1)q_1+...+a_Lq_M=0,
(14)

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|),
(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

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)]
(17)
w_(L/M)=[a_(L-M+1); a_(L-M+2); |; a_L],
(18)

and 0<=n<=M.

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

exp_(0/0)(x)=1
(19)
exp_(0/1)(x)=1/(1-x)
(20)
exp_(0/2)(x)=2/(2-2x+x^2)
(21)
exp_(0/3)(x)=6/(6-6x+3x^2-x^3)
(22)
exp_(1/0)(x)=1+x
(23)
exp_(1/1)(x)=(2+x)/(2-x)
(24)
exp_(1/2)(x)=(6+2x)/(6-4x+x^2)
(25)
exp_(1/3)(x)=(24+6x)/(24-18x+6x^2-x^3)
(26)
exp_(2/0)(x)=(2+2x+x^2)/2
(27)
exp_(2/1)(x)=(6+4x+x^2)/(6-2x)
(28)
exp_(2/2)(x)=(12+6x+x^2)/(12-6x+x^2)
(29)
exp_(2/3)(x)=(60+24x+3x^2)/(60-36x+9x^2-x^3)
(30)
exp_(3/0)(x)=(6+6x+3x^2+x^3)/6
(31)
exp_(3/1)(x)=(24+18x+6x^2+x^3)/(24-6x)
(32)
exp_(3/2)(x)=(60+36x+9x^2+x^3)/(60-24x+3x^2)
(33)
exp_(3/3)(x)=(120+60x+12x^2+x^3)/(120-60x+12x^2-x^3).
(34)

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))
(35)
(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))
(36)
(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))
(37)
(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^')
(38)
(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))
(39)
(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)),
(40)

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.
(41)

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)))
(42)
((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)))
(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/(M+1))-R_((L+1)/(M+1))))=(C_((L+1)/(M+1))^2x)/(C_((L+1)/M)C_((L+1)/(M+2)))
(44)
((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))
(45)
((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))).
(46)

See also

C-Determinant, Economized Rational Approximation, Frobenius Triangle Identities

Explore with Wolfram|Alpha

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.

Referenced on Wolfram|Alpha

Padé Approximant

Cite this as:

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

Subject classifications