Wave Propagation in Linear Media | re-examined

7.4.3 Approximation error control
7 Mathematica implementation of a quadrature function
Another auxilliary function implements the criterion of equation (6.13),
0 (x)
0 (x) , 0 (x) = k;
which has been given as a quality measure for asymptotic partition. It relates the argument
function and its polynomial approximation by means of their respective oscillation frequencies.
The di erence between these two cause the actual partition points to deviate from the ideal
phase invariant positions, which in turn produces wavepacket-like pictures in the sequence of
partial sums. Hence the above formula hence gives the length of such awave packet depending
on the abscissa value. We would like, however, to prescribe k and get the corresponding x as
a result in order to know where to start the extrapolation. The functions needed to this end
exploit the symbolical calculation capabilities of Mathematica.
To nd the polynomial approximation of a given function f(x) for x ! 1, we de ne two
small routines that determine the polynomial degree of f(x) and then compute the asymptotic
expansion to the correct order.
PolynomialDegree[f_,x_Symbol] :=
Length[CoefficientList[Series[f,fx,Infinity,1g],x]]-1;
AsymptoticExpand[f_,x_Symbol] :=
Normal[Series[f,fx,Infinity,PolynomialDegree[f,x]g]];
Remark (Series expansion) The function Series[f, x, x0, n] generates a power series
expansion for a given function to the order n about the point x0. In PolynomialDegree
we use it just to determine the highest order terms, that is why we set n = 1. With
CoefficientList we then obtain a list of the coe cients of the resulting polynomial
and, by regarding the length of this list, the degree of the polynomial.
In AsymptoticExpand, we again calculate the power series, but now to the correct order.
The command Normal nally converts the series expansion to a normal expression by
cutting o the error term that is also returned by Series.
Example 7.4.1 Consider the function f(x) =x+ p x 4 +x 3 ,x+1. The series expansion calculated
to nd the degree of the polynomial includes only the highest order terms and an error term.
In[7]:= Series[x+Sqrt[x^4+x^3-x+1],fx,Infinity,1g]
Out[7]= 2 3 x 1 0
x + --- + O[-]
2 x
The length of the list of coe cients (starting with x 0 ) gives the desired value.
In[8]:= CoefficientList[%,x]
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