Introduction to ARPACK

  ARPACK is a collection of Fortran77  subroutines designed to solve large scale eigenvalue problems. ARPACK   stands for ARnoldi PACKage. ARPACK software is capable of solving large scale Hermitian, non-Hermitian, standard or generalized eigenvalue problems from significant application areas. The software is designed to compute a few, say k, eigenvalues with user specified features such as those of largest real part or largest magnitude using $n \cdot {\cal O}(k) + {\cal O}(k^2)$ storage. No auxiliary storage is required. A set of Schur basis vectors for the desired k dimensional eigen-space is computed which is numerically orthogonal to working precision. Eigenvectors are also available upon request.

The Arnoldi process is a technique for approximating a few eigenvalues and corresponding eigenvectors of a general $n \times n$ matrix. It is most appropriate for large structured matrices ${\bf A}$ where structured means that a matrix-vector product ${\bf w}\leftarrow{\bf A}{\bf v}$ requires ${\cal O}(n)$rather than the usual ${\cal O}(n^2)$ floating point operations (Flops). This software is based upon an algorithmic variant of the Arnoldi process called the Implicitly Restarted Arnoldi Method (IRAM).   When the matrix ${\bf A}$is symmetric it reduces to a variant of the Lanczos  process called the Implicitly Restarted Lanczos Method (IRLM).   These variants may be viewed as a synthesis of the Arnoldi/Lanczos process with the Implicitly Shifted QR scheme that is suitable for large scale problems. For many standard problems, a matrix factorization is not required. Only the action of the matrix on a vector is needed.

In this chapter, we give an overview of the package. Chapter 2 explains how the user can quickly start using ARPACK while Chapter 3 gives a comprehensive description of how to utilize the full capabilities of ARPACK. An overview of the theory of Krylov subspace projection methods and the underlying algorithms implemented in ARPACK is the subject of Chapter 4. The final chapter discusses the implementation details of the main computational routines in ARPACK. Appendix A is a guide on how to use the example driver routines as templates. Experienced users who are already familiar with large scale eigenvalue computations may find it most productive to go directly to this appendix, locate the suitable driver and modify that for the particular application. Appendix B describes the trace debugging capability that is easily turned on in order to monitor progress and output important intermediate computed quantities. Checkpointing to guard against loss of intermediate computational results due to system or hardware failure is possible. A description of how to recover and restart in the event of a fault is provided in Appendix B.