Tim Davis

This function attempts to do the right thing when computing selected entries of the inverse, by solving a sequence of linear equations after doing an LU factorization. However, the implementation is flawed.

One suggestion: the statement [L,U,P]=lu(A) does not allow LU to permute A to improve sparsity. It would be better to use the 4-output LU, [L,U,P,Q]=lu(A), except in your case you'd need to use another name for Q since you have a Q variable already.

You would then need to use the column permutation Q in the forward/backsolve steps.

Including a "waitbar" in a function like this is not a good idea.

There seems to be a bug. The documentation says that s=smartinv(A) returns the diagonal of the inverse, but in fact I get the whole inverse back.

If I do s=smartinv(A,1), I get a diagonal matrix, but not the diagonal entries of the inverse, which is what the documentatioon said I would get. (I tried with this matrix, which is included in MATLAB):

load west0479
A=west0479;

With smartinv(A,1) or smartinv(A,1,'mono'), I should get the diagonal of inv(A) as the result, but I don't. SOmething else is returned.

Finally, the method is very slow. Computing the entire inverse takes 0.13 seconds using S=inv(A). Using S=smartinv(A) I get the same result, but it takes 15 seconds.

In the profiler, all the time is spent in this one line of code:

 Q(q0+1:q0+w,q0+1:q0+w) = insblock;

that's a very bad idea. You should never use sparse submatrix assignment, if at all possible. A better approach is to build the matrix all at once, from the blocks, using the "sparse" command.

So there appears to be at least three serious bugs in the code:
(1) does not perform according to the documentation (smartinv(A) returns inv(A), not diag(inv(A))).
(2) very slow as compared with just "inv"
(3) cannot compute the diagonal of the inverse, in spite of the fact that the documentation says it can do so. Instead it returns something else altogether.

And 2 serious drawbacks:
(1) Not really a bug, but a serious performance issue: doesn't allow for pivoting to maintain sparsity, and is thus unsuitable for large matrices.
(2) Remove the waitbar. If your function is used within other codes, the end user isn't interested in the "Inverting matrix..." message.

You should clarify in your help information that the matrix produced has its columns listed in the opposite order of the vander.m function in MATLAB.

Also, providing the determinant is nice, but it would be better to return the log or log10 of the determinant. That way, you can deal with determinants that are very large and very small (it's quite easy for det(A) to underflow or overflow).

Otherwise, nice work, and nicely commented. (You should have a "see also vander" line, though).

Nice use of cumprod. I'm surpised that this function is slower than vander, though.

Thanks Tim!

This is very popular in the current ML contest (flooding a matrix with colour, Fall 2009).

What mathworks are waiting to incorporate this package into Matlab?