Mathematics 371 -- Numerical Analysis

Adaptive Quadrature

November 7, 2001

An example: f( x ) = . This is a good candidate for adaptive

quadrature over an interval like [1,3] since the graph of the function is slowly varying

over portions of the interval and rapidly varying over others:

> f:=x->100*x^2*exp(-4*x)*sin(10/x);

> plot(f(x),x=1..3);

>

Here is a listing of the Maple code for Adaptive Quadrature

> interface(verboseproc=2):

> read `/home/fac/little/public_html/Num01/Adapt.map`;

> print(Adapt);

Using adaptive quadrature:

> count := 0;

> approx1:=Adapt(f,1.0,3.0,.00005,true);

"starting integration on [", 1.0, 3.0, "]"

"starting integration on [", 1.0, 2.000000000, "]"

"starting integration on [", 1.0, 1.500000000, "]"

"starting integration on [", 1.0, 1.250000000, "]"

"starting integration on [", 1.0, 1.125000000, "]"

"starting integration on [", 1.0, 1.062500000, "]"

"tolerance met"

"starting integration on [", 1.062500000, 1.125000000, "]"

"tolerance met"

"starting integration on [", 1.125000000, 1.250000000, "]"

"tolerance met"

"starting integration on [", 1.250000000, 1.500000000, "]"

"starting integration on [", 1.250000000, 1.375000000, "]"

"tolerance met"

"starting integration on [", 1.375000000, 1.500000000, "]"

"tolerance met"

"starting integration on [", 1.500000000, 2.000000000, "]"

"starting integration on [", 1.500000000, 1.750000000, "]"

"tolerance met"

"starting integration on [", 1.750000000, 2.000000000, "]"

"tolerance met"

"starting integration on [", 2.000000000, 3.0, "]"

"starting integration on [", 2.000000000, 2.500000000, "]"

"tolerance met"

"starting integration on [", 2.500000000, 3.0, "]"

"tolerance met"

Maple's computed value:

> exact:=evalf(Int(f(x),x=1..3));

> err:=abs(exact1-approx1);

The number of times the Adapt procedure was called recursively:

> count;

CAUTION: It is not the case that the error from the adaptive

quadrature procedure will always be less than or equal to the

error tolerance value (even ignoring round-off error phenomena).

(The derivation of the error estimate uses the assumption that two

values of the fourth derivative of f are equal, which may be far

from true.) For example:

> approx:=Adapt(f,1.0,3.0,.0001,false);

> abs(approx-exact);

Note that the actual error is larger than the error tolerance in this case.