The most commonly used approximation techniques for large sparse nonsymmetric eigenvalue problems are based on the Arnoldi process. In 1992, Sorensen developed the Implicitly Restarted Arnoldi (IRA) method. This resulted in the FORTRAN software package called ARPACK (1996). In 1997, we created a block-version, the Implicitly Restarted Block Lanczos (IRBL) method for solving symmetric eigenvalue problems. When there are multiple or very close eigenvalues our method outperforms ARPACK. In 2002, we developed a MATLAB code irbleigs.m that implements the IRBL method. A block code has many advantages, in particular, it allows the use of Level 3 BLAS matrix-matrix products to increase performance and can locate multiple or clustered eigenvalues without the need of any deflation techniques. Although, Lehoucq and Maschhoff created a block Arnoldi method in 1997 which is a straightforward generalization of the (IRA) method there are implementation issues involved that warrant further investigation before a viable computer code can be created. We propose to develop the implicitly restarted block Arnoldi (IRBA) method to solve the nonsymmetric eigenvalue problem and create a public domain computer code. Our method will use Householder transformations instead of the Gram-Schimdt process to maintain strong orthogonality and will be based on the equivalent Wu and Simon's (2001) approach.
Solving eigenvalue problems are a common part of our life whether we realize it or not. The software package that I will develop will be very helpful in solving numerous applications, including; noise reduction in airplanes, vibrational analysis in buildings and bridges, quantum mechanics, image deblurring, and face recognition.
