Mathematica Solutions to the ISSAC '97 Systems
Wolfram Research, Inc.
Find the largest eigenvalue lambda to 13 significant digits for the
following integral equation.
Method 1: Solve a corresponding discrete version.
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.
We iterate the eigenfunction expansion with an increasing number of
subdivisions until we have the desired precision (actually
is already enough).
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 ). With increasing iterations we increase
the precision too.
This is the largest eigenvalue.
Here are the iterated eigenfunctions; only the first few are visibly
Here are the local logarithmic differences between the iterated
eigenvectors and the last eigenvector. The dip in the middle comes from
the normalization condition.
This shows how the eigenvalues converge.
This shows the convergence of the differences between the iterated
eigenvalues and the last iteration.