As an exercise in regularization, calculate the singular values for the
matrix **A**, where **A** is . Look at how the values are
distributed and notice that there is a large jump between the first few
values and the other values. Recalculate the solution of the system, this
time using the SVD program, and zero out the entry that correspond to the
smallest singular value.

You will find, perhaps to your surprise, that the solution now gives you
reasonable results (at least more reasonable than your first, unregularized
solution). What you have just done is a form of regularization known as
truncated singular value decomposition (TSVD). In effect, you have applied
a filter to the system, which filters out the high frequency * noise*.
At the same time, you are also filtering out useful information.
Basically, you are giving up some information for an added degree of
smoothness of the solution. Unfortunately, this method doesn't work well
for all ill-conditioned matrices (you might try to figure out for which
cases the truncated SVD method would work well and for which cases it would
not). Other methods have been developed which try to regularize the
system, not by completely removing the singular values, but by modifying
them so that they can contribute to the information content. Such methods
were developed by the Russian mathematician Tikhonov [16].