Re: Definite integral with huge variations in magnitude

In article <ebbcn2$n42$1@xxxxxxxxxxxxxx>, axlq <axlq@xxxxxxxxxxx> wrote:
What numerical integration technique would be best for a definite
integral, evaluated over a finite interval.

* Definite integral over a finite interval, z = z1 to z2.

* The integrand is continuous over the interval, as are its
derivatives.

* The integrand is well-behaved in magnitude over MOST of the
interval.

* Over a TINY part of the interval, it makes some oscillations
many orders of magnitude above the values along the rest of the
interval. For example, over 99% of the interval the integrand
varies smoothly with values between +/- 10e5. In a tiny piece of
the interval it makes a sudden oscillation between +/- 10e11.

The integrand is below. It appears to have no analytical solution,
so I have to solve it numerically. (Believe it or not, I have to
implement this in an Excel Visual Basic macro).

integrand = (d2*Cos[k*R] - d1*Sin[k*R]) dz

where
d1 = k*R * (3*a^2 - 2*R^2)
d2 = 2*R^2 + a^2 * (k^2 * R^2 - 3)
R = Sqrt[a^2 + (q-z)^2]

k = 400 * Pi / 3
a = 3.75E-6
q = 0.0000875

interval: z1 = -0.0025, z2 = 0.0025

The wild oscillations occur between 0.00086 and 0.00089. I have
confirmed that the function IS continuous here although at first
glance it looks like a blow-up to infinite values.

There are no such wild oscillations for the function you gave us.
It seems extremely smooth there. Certainly the obvious bounds show
that it can't get large: at most about 0.00002622 on the
interval -0.0025 to 0.0025 according to Maple. Perhaps you had
in mind something different from what you wrote.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.