Company
Mathematica Solutions to the ISSAC '97 Systems Challenge

Wolfram Research, Inc.


Problem 2

What is the value of [Graphics:ISSACChallengegr30.gif] to 7 significant digits?

Result
[Graphics:ISSACChallengegr31.gif]


Method 1: Use a built-in function.

The most natural approach gives the correct result but with a warning message.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr32.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr33.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr34.gif]

To guarantee the required digits and to avoid the message, we use arbitrary-precision arithmetic, allow more recursive subdivisions at the right endpoint, and use a more appropriate integration method.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr35.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr36.gif]


Method 2: Substitute to get a nicer integrand.

We make the substitution [Graphics:ISSACChallengegr37.gif]. Using [Graphics:ISSACChallengegr38.gif], this is equivalent to [Graphics:ISSACChallengegr39.gif][Graphics:ISSACChallengegr40.gif]. This makes the integrand much better behaved, getting rid of any subdivision problems.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr41.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr42.gif]