Company
Mathematica Solutions to the ISSAC '97 Systems Challenge

Wolfram Research, Inc.


Problem 9

Find the largest eigenvalue lambda to 13 significant digits for the following integral equation.

[Graphics:ISSACChallengegr158.gif]

Result
lambda=37.5291455603353...


Method 1: Solve a corresponding discrete version.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr159.gif]

We get the abscissas and weights for the Gaussian quadrature of order 32 in the interval (0,1) to 20 digits. Any order between, say, 10 and 40 would also work. Going to 20 digits makes sure that arbitrary-precision arithmetic is used; this controls the errors in the calculations.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr160.gif]

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr161.gif]

We iterate the eigenfunction expansion with an increasing number of subdivisions until we have the desired precision (actually [Graphics:ISSACChallengegr162.gif] is already enough).

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr163.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr164.gif]


Method 2: Iterate toward the largest eigenvalue.

For this kernel we can iterate the application of the integral operator starting with some vector that has a nonvanishing component in the direction of the largest eigenvalue (see [6]). With increasing iterations we increase the precision too.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr165.gif]

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr166.gif]

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr167.gif]

This is the largest eigenvalue.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr168.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr169.gif]

Here are the iterated eigenfunctions; only the first few are visibly different.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr170.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr171.gif]

Here are the local logarithmic differences between the iterated eigenvectors and the last eigenvector. The dip in the middle comes from the normalization condition.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr172.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr173.gif]

This shows how the eigenvalues converge.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr174.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr175.gif]

This shows the convergence of the differences between the iterated eigenvalues and the last iteration.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr176.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr177.gif]