for the subspace spanned by the eigenvectors. Thus, the matrix A
is used only to compute the matrix-vector product As, and implementations
of these algorithms need not make any assumptions
about the structure/storage format of A. In one sense,
the efficiency of these methods is determined
primarily by the performance of the matrix-vector product
and the storage scheme used for the matrix.
The Lanczos method solves the eigenvalue problem 
through partial tridiagonalizations of the matrix C using
Krylov bases. Unlike
direct factorization methods (e.g., QR factorization),
no intermediate dense submatrices (resulting from fill-in) are generated. Also,
information about C's extremal eigenvalues tend to emerge long
before tridiagonalization is complete. Hence, the Lanczos algorithm is
particularly useful in situations where only a few of C's largest or
smallest eigenvalues are desired.
An alternative method (referred to as CSI-MSVD) for producing tridiagonal matrices whose eigenvalues approximate those of the original matrix C was presented in [10]. This method, based on the extraction of modified moments from the Chebyshev semi-iterative method [23], is an attractive alternative given its scope for parallelism and reduced memory requirements.