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 $R_{L/0}$ corresponds to the Maclaurin Series. When it exists, the $R_{L/M}\equiv [L/M]$ Padé approximant to any Power Series

$$A(x)=\sum_{j=0}^\infty a_j x^j$$ (1)

is unique. If $A(x)$ is a Transcendental Function, then the terms are given by the Taylor Series about $x_0$

$$a_n={1\over n!} A^{(n)}(x_0).$$ (2)

The Coefficients are found by setting

$$A(x)-{P_L(x)\over Q_M(x)}=0$$ (3)

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

$$Q_M(0)=1.$$ (4)

Expanding (3) gives

$$P_L(x) = p_0+p_1 x+\ldots+p_L x^L$$ (5)

$$Q_M(x) = 1+q_1x+\ldots+q_M x^M.$$ (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)

$$\vdots$$

$$a_L+a_{L-1}q_1+\ldots+a_0q_L = p_L$$ (10)

$$a_{L+1}+a_Lq_1+\ldots+a_{L-M+1}q_M = 0$$ (11)

$$\vdots$$

$$q_{L+M}+a_{L+M-1}q_1+\ldots+a_Lq_M = 0,$$ (12)

where $a_n=0$ for $n<0$ and $q_j=0$ for $j>M$. Solving these directly gives

$$[L/M]={\left\vert\matrix{ a_{L-m+1} & a_{L-m+2} & \cdots & a... ...dots & a_{L+M}\cr x^M & x^{M-1} & \cdots & 1\cr}\right\vert},$$ (13)

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

$[L/M]=\sum_{j=0}^{L-M} a_jx^j+x^{L-M+1} {\bf w}_{L/M}^{\rm T}{\hbox{\sf W}}^{-1}_{L/M}{\bf w}_{L/M}$
$=\sum_{j=0}^{L+n} a_jx^j+x^{L+n+1}{\bf w}^{\rm T}_{(L+M)/M}{\hbox{\sf W}}^{-1}_{L/M}{\bf w}_{(L+n)/M}$

for

$${\hbox{\sf W}}_{L/M} = \left[\begin{array}{ccc} a_{L-M+1}-xa_{L-M+2} & \cdots & a_L-xa_{... ...\ddots & \vdots\\ a_L-xa_{L+1} & \cdots & a_{L+M-1}-xa_{L+M}\end{array}\right]$$ (14)

$${\bf w}_{L/M} = \left[\begin{array}{c}a_{L-M+1}\\ a_{L-M+2}\\ \vdots\\ a_L\end{array}\right],$$ (15)

and $0\leq n\leq M$.

The first few Padé approximants for $e^x$ are

$$\begin{eqnarray*} \mathop{\rm exp}\nolimits _{0/0}(x)&=&1\\ \mathop{\rm exp}\... ...nolimits _{3/3}(x)&=&{120+60x+12x^2+x^3\over 120-60x+12x^2-x^3}. \end{eqnarray*}$$

Two-term identities include

$${P_{L+1}(x)\over Q_{M+1}(x)}-{P'_L(x)\over Q'_M(x)} = {{C_{(L+1)/(M+1)}}^2 x^{L+M+1}\over Q_{M+1}(x)Q'_M(x)}$$ (16)

$${P_{L+1}(x)\over Q_M(x)}-{P'_L(x)\over Q'_M(x)} = {C_{(L+1)/M}C_{(L+1)/(M+1)}x^{L+M+1}\over Q_M(x)Q'_M(x)}$$ (17)

$${P_L(x)\over Q_{M+1}(x)}-{P'_L(x)\over Q'_M(x)} = {C_{L/(M+1)}C_{(L+1)/(M+1)}x^{L+M+1}\over Q_M(x)Q'_M(x)}$$ (18)

$${P_L(x)\over Q_{M+1}(x)}-{P'_{L+1}(x)\over Q'_M} = {{C_{(L+1)/(M+1)}}^2x^{L+M+2}\over Q_{M+1}Q'_M}$$ (19)

$${P_{L+1}\over Q_M(x)}-{P'_{L-1}(x)\over Q'_M(x)} = {C_{L/(M+... .../M}x^{L+M}+C_{L/M}C_{(L+1)/(M+1)}x^{L+M+1}\over Q_M(x)Q'_M(x)}$$ (20)

$${P_L(x)\over Q_{M+1}(x)}-{P'_L(x)\over Q'_{M-1}(x)}= {C_{L/(... ...}-C_{L/M}C_{(L+1)/(M+1)}x^{L+M+1}\over Q_{M+1}(x)Q'_{M-1}(x)},$$ (21)

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

$$S_{(L+1)/M}S_{(L-1)/M}-S_{L/(M+1)}S_{L/(M-1)}={S_{L/M}}^2.$$ (22)

Cross ratio identities include

$${(R_{L/M}-R_{L/(M+1)})(R_{(L+1)/M}-R_{(L+1)/(M+1)})\over (R_... ...= {C_{L/(M+1)}C_{(L+2)/(M+1)}\over C_{(L+1)/M}C_{(L+1)/(M+2)}}$$ (23)

$${(R_{L/M}-R_{(L+1)/(M+1)})(R_{(L+1)/M}-R_{L/(M+1)})\over (R_... ...1)})} = {{C_{(L+1)/(M+1)}}^2x\over C_{L/(M+1)}C_{(L+2)/(M+1)}}$$ (24)

$${(R_{L/M}-R_{(L+1)/(M+1)})(R_{(L+1)/M}-R_{L/(M+1)})\over (R_... ...1)})} = {{C_{(L+1)/(M+1)}}^2x\over C_{(L+1)/M}C_{(L+1)/(M+2)}}$$ (25)

$${(R_{L/M}-R_{(L+1)/(M-1)})(R_{L/(M+1)}-R_{(L+1)/M})\over (R_... ...)} = {C_{(L+1)/M}C_{(L+1)/(M+1)}x\over C_{L/(M+1)}C_{(L+2)/M}}$$ (26)

$${(R_{L/M}-R_{(L-1)/(M+1)})(R_{(L+1)/M}-R_{L/(M+1)})\over (R_... ...} = {C_{L/(M+1)}C_{(L+1)/(M+1)}x\over C_{(L+1)/M}C_{L/(M+2)}}.$$ (27)

See also C-Determinant, Economized Rational Approximation, Frobenius Triangle Identities

References

Padé Approximants

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.

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.