This is a tutorial made solely for the purpose of education and it was designed for students taking Applied Math 0330. It is primarily for students who have very little experience or have never used Mathematica before and would like to learn more of the basics for this computer algebra system. As a friendly reminder, don't forget to clear variables in use and/or the kernel.

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Return to Part IV of the course APMA0330

Padé Approximation

Henri Eugène Padé.

A Padé approximant is the "best" approximation of a function by a rational function of given order -- under this technique, the approximant's power series agrees with the power series of the function it is approximating. The technique was developed around 1890 by the French mathematician Henri Padé (1863--1953), but goes back to the German mathematician Georg Frobenius (1849--1917) who introduced the idea and investigated the features of rational approximations of power series. Henri Eugène Padé, while preparing his doctorate under Charles Hermite, he introduced what is now known as the Padé approximant.

Given a function f and two integers \( m \ge 0 \quad\mbox{and}\quad n \ge 1, \) the Padé approximant of order [m/n] is the rational function

\[ R(x) = \frac{\sum_{j=0}^m a_j x^j}{1 + \sum_{k=1}^n b_k x^k} = \frac{a_0 + a_1 x + a_2 x^2 + \cdots + a_m x^m}{1 + b_1 x + b_2 x^2 + \cdots + b_n x^n} , \]
which agrees with f(x) to the highest possible order, which amounts to
\begin{eqnarray*} f(0) &=& R(0) , \\ f' (0) &=& R' (0) , \\ f'' (0) &=& R'' (0) , \\ &\vdots & \\ f^{(n+m)} &=& R^{(n+m)} (0) . \end{eqnarray*}
Equivalently, if R(x) is expanded in a Maclaurin (or Taylor) series its first m + n terms would cancel the first m + n terms of f(x), and as such:
\[ f(x) - R(x) = c_1 x^{m+n+1} + c_2 x^{m+n+2} + \cdots = O\left( x^{n+m+1} \right) . \]
The Padé approximant is unique for given m and n, that is, the coefficients \( a_0 , \ldots , a_m , b_1 , \ldots b_n \) can be uniquely determined.

Example: Consider Padé approximants for cosine functions

\begin{align*} r_2 (x) &= \dfrac{1 - \frac{5}{12}\, x^2}{1 + \frac{1}{12}\, x^2} , \\ r_4 (x) &= \dfrac{1 - \frac{115}{252}\,x^2 + \frac{313}{15120}\,x^4}{1+ \frac{11}{252}\,x^2 + \frac{13}{15120}\,x^4} , \\ r_8 (x) &= \dfrac{1-\frac{260735}{545628}\,x^2 + \frac{4375409}{141863280}\,x^4 - \frac{7696415}{13108167072}\,x^6 + \frac{80737373}{23594700729600}\, x^8}{1 + \frac{12079}{545628}\, x^2 + \frac{34709}{141863280}\,x^4 + \frac{109247}{65540835360}\, x^6 + \frac{11321}{1814976979200}\, x^8} . \end{align*}




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