Important Features

The important features of ARPACK are:

• A reverse communication interface.  
• Ability to return k eigenvalues which satisfy a user specified criterion such as largest real part, largest absolute value, largest algebraic value (symmetric case), etc. For many standard problems, the action of the matrix on a vector ${\bf w}\leftarrow{\bf A}{\bf v}$ is all that is needed.
• A fixed pre-determined storage requirement suffices throughout the computation. Usually this is $n \cdot {\cal O}(k) + {\cal O}(k^2)$ where k is the number of eigenvalues to be computed and n is the order of the matrix. No auxiliary storage or interaction with such devices is required during the course of the computation.
• Sample driver routines are included that may be used as templates to implement various spectral transformations to enhance convergence and to solve the generalized eigenvalue problem.
• Special consideration is given to the generalized problem ${\bf A}{\bf x}={\bf M}{\bf x}\lambda$ for singular or ill-conditioned symmetric positive semi-definite ${\bf M}$.
• Eigenvectors and/or Schur vectors may be computed on request. A Schur basis of dimension k is always computed. The Schur basis consists of vectors which are numerically orthogonal to working accuracy. Computed eigenvectors of symmetric matrices are also numerically orthogonal.
• The numerical accuracy of the computed eigenvalues and vectors is user specified. Residual tolerances may be set to the level of working precision. At working precision, the accuracy of the computed eigenvalues and vectors is consistent with the accuracy expected of a dense method such as the implicitly shifted QR iteration.
• Multiple eigenvalues offer no theoretical difficulty. This is possible through deflation techniques similar to those used with the implicitly shifted QR algorithm for dense problems. With the current deflation rules, a fairly tight convergence tolerance and sufficiently large subspace will be required to capture all multiple instances. However, since a block method is not used, there is no need to ``guess" the correct block size that would be needed to capture multiple eigenvalues.