Implicity Restarted Block Arnoldi (IRBA) Method

大型稀疏非对称特征值问题最常用的逼近技术是基于Arnoldi过程的。1992年，Sorensen发展了隐式重启Arnoldi(IRA)方法。这导致了FORTRAN软件包被称为ARPACK(1996)。1997年，我们创建了求解对称特征值问题的块形式--隐式重启块Lanczos(IRBL)方法。当存在多个或非常接近的特征值时，我们的方法优于ARPACK。在2002年，我们开发了一个实现IRBL方法的MATLAB代码irbleigs.m。分组码具有许多优点，尤其是，它允许使用第3级BLAS矩阵-矩阵乘积来提高性能，并且可以在不需要任何收缩技术的情况下定位多个或聚集的特征值。虽然Lehoucq和Maschhoff在1997年创建了块Arnoldi方法，这是(IRA)方法的直接推广，但在创建可行的计算机代码之前，涉及的实现问题需要进一步研究。我们建议发展隐式重启块Arnoldi(IRBA)方法来求解非对称特征值问题，并创建公共域计算机代码。我们的方法将使用Household变换而不是Gram-Schimdt过程来保持强正交性，并且将基于等价的Wu和Simon(2001)方法。无论我们是否意识到，解决本征值问题都是我们生活中常见的一部分。我将开发的软件包将非常有助于解决许多应用，包括飞机降噪、建筑物和桥梁的振动分析、量子力学、图像去模糊和人脸识别。

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.