Wave Propagation in Linear Media | re-examined

7.4 Auxiliary functions
While[iterations < it && Accuracy[intlim] < ac && Precision[intlim] < pc,
parttab = Join[Last[parttab],
PartitionPoints[fzero,a,il,offs,ne+nt+1,opts]];
seqtab = Join[seqtab,
Table[NIntegrate[f[x],x,parttab[[i]],parttab[[i+1]],
Method->GaussKronrod],
i,1,Length[parttab]-1]];
ne = ne + il;
iterations++;
intlim = SequenceLimit[Take[FoldList[Plus,firstval,seqtab],
-ne],WynnDegree->wd]
];
Example 7.3.2 We want to compute R 1
sin x
x
dx to the precision of 10 signi cant digits. Note that
0
the integrand as well as the argument are given as pure functions. If we choose the working precision
too low, we hit the iteration limit.
In[5]:= OscIntControlled[Sin[#]/#&,#&,0,
WorkingPrecision->17,PrecisionControl->10]
OscInt::accfail:
OscInt failed to achieve the desired accuracy or precision after 6
iterations. Increase MaxIterations or WorkingPrecision (currently 17).
Out[5]= 1.570798
If we increase the working precision, we obtain the desired result.
In[6]:= OscIntControlled[Sin[#]/#&,#&,0,
WorkingPrecision->20,PrecisionControl->10]
Out[6]= 1.570796327
7.4 Auxiliary functions
In addition to the functions that are needed for pure quadrature, a number of supporting
routines have been added to the package. These functions facilitate the use of the quadrature
programs in that they provide information required by the quadrature algorithm.
A set of such functions returns the zeros fx0;x1;x2;:::g of sin f(x) for quadratic and hyperbolic
arguments f(x). In order to be suitable as partition points for the quadrature, they
are de ned such that the rst zero x0 always lies to the right of the extremum of f(x). It is
thus necessary to nd solutions for the equation f(x) =(k+o) , where k 0 and the o set
o = f(xe)= is chosen so that f(x) is monotonic for x>xe.
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