•Recall from Fourier
analysis, any function (in this case, the error in our approximate solution) can be considered as
a linear combination of
basis vectors - here, sine functions of different frequencies.
•These sine
functions are in fact the eigenvectors of the T(i) matrix. The following figure shows some of these sine-curves when i=5, so T(5) is 31-by-31. The top plot is a
plot of the eigenvalues
(frequencies) of 4^5*T(5), from lowest to highest. The subsequent plots are the eigenvectors
(sine-curves), starting with
the lowest 4 frequencies, and then a few
more frequencies up to the highest (number 31).