Hi folks,

There's an error in Problem 1 on Homework 2.  The problem 
asks you to compute M as an approximate right inverse for A, 
by minimizing norm(A*M-I,'fro') over matrices M with the 
same nonzero pattern as A.  However, Matlab's GMRES uses its 
preconditioner as an approximate *left* inverse for A, which
of course is not the same in general for nonsymmetric A.

So your "spai" routine needs to compute M to minimize 
norm(M*A-I,'fro').  You could do this by building up M one
row at a time instead of one column at a time.  But that's
an inefficient way to build a sparse matrix in Matlab 
(since Matlab sparse matrices are kept by columns).  A
better way is to compute M as the transpose of an
approximate right inverse of A'.  

Got that?  The Frobenius norm of a matrix is the same as
the Frobenius norm of its transpose, so if we compute N
to minimize

     norm( (A'*N - I),  'fro')

   = norm( (A'*N - I)', 'fro')

   = norm( (N'*A - I),  'fro')

and then take M = N', we have the approximate left
inverse that we want, which minimizes

     norm( (M*A - I),   'fro')

You might try both preconditioners for WATT1.  Why is their
performance so similar?

Many thanks to Stephanie Taylor for noticing the error.

Second remark:  WEST0479 is a hard matrix to precondition.
Don't knock yourself out trying to get good results on it
with SPAI -- just report the results you do get!

Cheers,

- John