Eigenvalues/eigenvectors for a matrix of the form (A^T)A

I have an m-dimensional real-valued symmetric matrix M which arises
from the product (A^T)A where A is an nxm real-valued matrix which is
not generally sparse. 'n' is much greater that 'm'. I would like to
determine the eigenvalues and eigenvectors of M.

The straightforward method would be to calculate M, then solve the
eigenproblem. My problem is that M is strongly ill-conditioned, so
that there are quantization problems in determining the eigenvalues.
In the past, I have used the fact that the EVD (ie, eigenvalue
decomposition) of 'M' is easily derived from the SVD of 'A' (but with
the singular values of A being the square root of the eigenvalues of M
-- goodbye quantization problems). Since n >> m, however, this results
in an intermediate nxn matrix that I never really use anyway.

It seems to me that this structure should lend itself to an efficient
determination of the EVD of M without calculating M itself. Is there a
good way to go about this without producing large, unused intermediate
matrices?

Thank you in advance.

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