Software Availability

As mentioned before, all of the software from Chapter 4 for the Hermitian eigenvalue problem can be applied to one of $A^* A$, $AA^*$, or $H(A)$. However, some of this software and other software is specialized for the SVD, as we now describe.

Implicitly restarted Lanczos from §4.5 is available in ARPACK, as documented in the ARPACK Users' Guide [295]. Any of its variations may be applied to $AA^*$, $A^* A$, or $H(A)$. Both ARPACK and a parallel version, PARPACK, are available electronically. ARPACK also has a driver for the SVD which applies its algorithm directly to $A^* A$, i.e., assuming only the ability to multiply by $A$ and $A^*$. This is most suitable for finding a few of the largest singular values and their singular vectors.

PLANSO [462] is a parallel Lanczos algorithm with ``thick restart'' that compares favorably to PARPACK in speed and memory.

SVDPACK [49] is designed specifically for finding a few of the largest singular values of $A$. It includes many methods, of which the fastest is Lanczos (§4.4), which applies to $A^* A$. Other methods in SVDPACK are subspace iteration (§4.3.4), a variation of block-Lanczos that is a simpler version of the one in §4.6, and trace minimization. Any of these methods may be applied either to $A^* A$ or $H(A)$. When applying block Lanczos to $H(A)$, a variation on the algorithm in §6.3.3 is used. Trace minimization is a case of the methods for nonlinear eigenproblems discussed in §9.4.