Wave Propagation in Linear Media | re-examined

6Towards a quadrature routine
Example 6.3.1 The function Precision returns the number of digits a oating-point number
posesses. For exact numbers, such as integers or fractions of integers, it returns Infinity. For
ordinary machine precision numbers, the result is $MachinePrecision, even if the number has less
digits [146].
In[12]:= Precision[1234.5678901234567890]
Out[12]= 19
In[13]:= Precision[1234.56]
Out[13]= 16
The function Accuracy, on the other hand, refers to the digits to the right of the decimal point. Like
before, it returns Infinity if the number is exact, and assumes machine precision numbers if the
argument has less than | in our case | 16 digits.
In[14]:= Accuracy[1234.5678901234567890]
Out[14]= 16
In[15]:= Accuracy[1234.56]
Out[15]= 13
In fact, both functions signify the error bounds of a oating-point number in Mathematica. The
accuracy of a number x then is the absolute error bound log j j while the precision means the
relative error bound log jx= j [144]. The maximum error clearly depends on the working precision.
Both values are rounded to the nearest integer and thus look like numbers of signi cant digits although
they can be incorrect by one digit due to the rounding.
All other parameters of NIntegrate are not of primary importance for our purpose here and
will be explained as needed in the following chapters.
We will now explore the two di erent quadrature algorithms as they are applied to the rst
partial integral over the interval [a; x0] between the left integration limit and the rst partition
point. Let us return to the example (6.8) of the previous section. If we set xm = 40, we obtain
an index o set for the step in the sequence of partial sums o = 509 such that we select our rst
partition point to be the 510-th zero. We then want to know how long it takes to compute
this integral in one.
In[16]:= Timing[
NIntegrate[f[x],fx,zero[0],zero[510]g,MaxRecursion->10,
Method->GaussKronrod]
]
142