A band Lanczos method for Hermitian matrices was first proposed in . The Hermitian band Lanczos Algorithm 4.12 described here, however, is somewhat different in that it uses both the first Lanczos vectors (4.28) and the candidate vectors (4.30) for the following Lanczos vectors. This makes it possible to build only the matrix . The algorithm in  constructs the first Lanczos vectors, and as a result, it needs to build a somewhat larger matrix, instead of . The version of the band Lanczos method described here was first derived in  as a special case of the general nonsymmetric band Lanczos method proposed in .
The version of the band Lanczos method stated as
performs all multiplications with as matrix-vector products
with single vectors.
However, at any stage of the algorithm, it is also possible to
precompute the next matrix-vector products, which will
be needed in the next iterations, as a single matrix-matrix
product with a block of vectors.
The procedure is as follows.
Instead of performing step (7) in Algorithm 4.12, we
compute the matrix-matrix product
For Hermitian positive definite matrices ,
yet another variant of Algorithm 4.12, based on coupled
recurrences, was proposed in .
The motivation for this variant was the observation that for
Hermitian positive definite matrices , the smallest eigenvalue
obtained from the Lanczos matrix
may become negative;
see [34,35] for examples.
In exact arithmetic, this would be impossible, since the positive
definiteness of implies that
However, in finite precision arithmetic, roundoff may cause the
generated by Algorithm 4.12 to be
This can be avoided easily by using coupled recurrences to generate
a Cholesky factor of
, instead of
More precisely, the recurrences summarized in (4.32)
are replaced by coupled recurrences of the form
Finally, we note that the block Lanczos method is the more traditional way of extending the standard Lanczos algorithm for a single starting vector to multiple starting vectors. When the block Lanczos method is combined with deflation (as described, e.g., in ), then the resulting algorithm is mathematically equivalent to the band Lanczos method described here. However, the band Lanczos method has two distinct advantages over the block Lanczos method. First, deflation is extremely simple, since it requires only to check the norm of a single vector. In contrast, in the block Lanczos method, one needs to check if a whole block of vectors has full rank, and if not, find the linearly dependent columns. Typically, this requires the computation of a QR factorization of each block. Second, the band Lanczos method actually performs slightly fewer explicit orthogonalizations, resulting in a Lanczos matrix that is slightly sparser than the one produced by the block Lanczos method.