# Preface

This section gives an introduction to approximations of functions by a rational function of given order.

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---a rational functions that provides a smallest overall error on the interval [𝑎, b].

Actually, the first rational approximation was obtained in seventh century by the Indian mathematician Bhaskara (c.600 – c.680). He has given the rule for a remarkable approximation of the sine function:

$\sin \left( \theta^\circ \right) = \frac{4\theta \left( 180-\theta \right)}{40500 - \theta \left( 180 - \theta \right)} , \qquad 0 \le \theta \le 180 ,$
and cosine function
$\cos \left( \theta^\circ \right) = \frac{32400-4\,\theta^2}{32400 + \theta^2} \qquad\mbox{in degrees }\theta .$
Now we know much better Padé approximation of sine function (in radians):
$\sin \left( x \right) = \frac{\left( 12671/4363920 \right) x^5 - \left( 2363/18183 \right) x^3 + x}{1 + \left( 445/12122 \right) x^2 + \left( 601/872784 \right) x^4 + \left( 121/16662240 \right) x^6} .$
num[x_] = 12671/4363920*x^5 - 2363/18183*x^3 + x;
den[x_] = 1 + 445/12122*x^2 + 601/872784*x^4 + 121/16662240*x^6;
si[x_] = num[x]/den[x]
Then we check the differences of our approximation and the standard buildin sine function at some points:
Sin[0.487] - si[0.487]
-2.58127*10^-14
Sin[2.1457] - si[2.1457]
-4.62412*10^-6

When the power series representation of a function diverges, it indicates the inability of the power series to approximate the function in a certain region. A theorem from complex analysis states that if the Taylor series of a function diverges, then that function has singularities in the complex plane. A Padé approximant is a ratio of polynomials that contains the same information that a truncated power series does. Because the polynomial in the denominator may have roots in the region of interest, the Padé approximant may accurately indicate the presence of singularities.

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 quotient of two polynomials Pm(x) and Qn(x) of degrees m and n, respectively:

$R(x) = \frac{P_m (x)}{Q_n (x)} = \frac{\sum_{j=0}^m a_j x^j}{1 + \sum_{k=1}^n b_k x^k} = \frac{a_0 + a_1 x + a_2 x^2 + \cdots + a_m x^m}{1 + b_1 x + b_2 x^2 + \cdots + b_n x^n} ,$
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, denoted by Pmn(), 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. There is nothing special with the chosen point x = 0. A change of variable can be used to shift the calculations over to an interval that contains zero.

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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Example: Suppose we wish to approximate the solution of the ordinary differential equation

$y'=y^2,\qquad y(0)=1.$
Since the equation is separable its solution is y(x) = 1/(1-x). If we tried to find the Taylor series of y(x) directly from equation, we would obtain
$y(x) = 1+x+x^2+x^3+x^4+\cdots.$
This geometric series is convergent, of course, only for |x|<1. The solution has a singularity at x = 1, but this fact is not readily apparent from the expansion in the given equation.

The diagonal sequence of Padé approximants corresponding to the differential equation is

$\begin{split} P_1^1(x)&= \frac{1}{1-x},\\ P_2^2(x)&= \frac{1}{1-x},\\ P_3^3(x)&= \frac{1}{1-x}. \end{split}$
Therefore, the diagonal sequence of Padé approximants recovers the exact solution to the differential equation from only a few terms in the Taylor series. Of course, this is an exceptional example.    ■

Example: Suppose we wish to approximate the solution of the ordinary differential equation

$y'=1+y^2, \qquad y(0)=0.$
The given differential quation is separable and its solution is y(x) = tan(x). If we tried to find the Taylor series of y(x) directly from the equation, we would find
$y(x) = x+\frac{x^3}{3}+\frac{2x^5}{15}+\frac{17x^7}{315}+\cdots .$
Note that the exact solution has singularities at x = ±(2n+1)π/2, whereas the Taylor series approximation does not appear to show this behavior. Using Maclaurin expansion for the solution, we can compute the first few elements of the Padé diagonal sequence
$\begin{split} P_2^2(x) &= \frac{3x}{3-x^2}, \\ P_3^3(x) &= \frac{x(x^2-15)}{3(2x^2-5)}, \\ P_4^4(x) &= \frac{5x(21-2x^2)}{x^4-45x^2+105}. \end{split}$
Note that these Padé approximants have singularities where the denominator vanishes:
• For P22(x), these singularities are at x ≈ ±1.7.
• For P3(x), these singularities are at x ≈ ±1.58.
• For P44(x), these singularities are at x ≈ ±1.5712, and x ≈ ±6.5.
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Example: Suppose we wish to approximate the solution of the ordinary differential equation

$y'= x^2 - y^2,\qquad y(0)=1.$
The Maclaurin expansion of the solution can be obtained with the Mathematica comamnd
AsymptoticDSolveValue[{y'[x]==x^2 - (y[x])^2, y==1}, y[x],{x,0,10}]
1 + x + x^2 + (4 x^3)/3 + (7 x^4)/6 + (6 x^5)/5 + (37 x^6)/30 + ( 404 x^7)/315 + (369 x^8)/280 + (428 x^9)/315 + (1961 x^10)/1400
$y(x) = 1 - x + x^2 - \frac{2}{3}\,x^3 + \frac{5}{6}\,x^4 - \frac{4}{5}\,x^5 + \frac{23}{30}\,x^6 - \frac{236}{315}\,x^7 + \frac{201}{280}\,x^8 - \frac{218}{315}\,x^9 + \cdots .$
Fortunately, Mathematica has a dedicated command to determine the Padé approximant:
(1 - (429180 x)/13651769 + (362890 x^2)/13651769 + ( 25896601 x^3)/81910614 + (81656941 x^4)/382249532 + ( 104675 x^5)/81910614)/(1 + (13222589 x)/13651769 - ( 66290 x^2)/13651769 + (256801 x^3)/27303538 + ( 46165979 x^4)/1146748596 + (41585323 x^5)/1911247660)
So we define the corresponding {5,5}-Padé approximant
$P_5^5 (x) = \frac{1 -0.03\,x - 0.0265\, x^2 + 0.32\,x^3 + 0.21\,x^4 -0.0012779\,x^5}{1+ 0.968\,x - 0.0048\,x^2 +0.0094\,x^3 +0.04\,x^4 +0.021\,x^5} .$ Finally, we plot the solution along with its {5,5}-Padé approximation: pade[x_] = N[%]; s = NDSolve[{y'[x]==x^2 -(y[x])^2, y==1}, y, {x,0,2}]; Plot[{pade[x],Evaluate[y[x]/.s]}, {x,0,2}, PlotRange->All, PlotLabels->{"Pade","true"}] Padé approximation along with the rtue solution to the Riccati equation. Mathematica code

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