Wave Propagation in Linear Media | re-examined

7 Mathematica implementation of a quadrature function
If[N[zeros[[1]]] < N[a],
zeros = Select[zeros,(N[#] >= N[a])&];
Message[OscInt::lowfirst,Length[zeros]]];
If[N[zeros[[1]]] == N[a], 0,
NIntegrate[f[x],fx,a,zeros[[1]]g,
Method->DoubleExponential]] +
If[N[zeros[[1]]] == Infinity, 0,
seqtab = Table[NIntegrate[f[x],fx,zeros[[i]],zeros[[i+1]]g,
Method->GaussKronrod],
fi,1,Length[zeros]-1g];
SequenceLimit[Take[FoldList[Plus,0,seqtab],-ne],WynnDegree->wd]]
7.3.6 OscIntControlled
This function is probably the most intricate of the package. It is basically a copy ofPartInt
extended by a control loop. The user interface is exactly the same as that of OscInt, it is
invoked by
OscIntControlled[f, fzero, a, opts].
The only di erence in the behaviour towards the user is that the setting of WorkingPrecision
is not touched in any way, which means that the user himself must choose a reasonable value
(in particular one greater than the machine precision). The meta algorithm that is used is
shown in g. 7.3. This algorithm is of course applied only if error control is enabled, otherwise
the function works much like PartInt.
compute initial set of partition points fx0;x1;::: ;xng
compute rst partial integral R x0
a f(x) dx
compute partial integrals R xi+1
f(x) dx
xi
compute sequence limit
iterations := 1
while not reached precision or accuracy or maximum iterations do
compute il additional partition points
compute additional partial integrals
iterations := iterations + 1
compute sequence limit
check error conditions
Figure 7.3: The meta algorithm for OscIntControlled
The portion of the function body that implements the control loop is given below. Note
that the table of partition points parttab is not augmented in each pass, but only the last
element is kept before the new points are added. For the sake of simplicity, the variable ne
that initially holds the value of NSumExtraTerms is increased by the number il of additional
sequence members directly.
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