Eigenproblems appear ubiquitously all across applied science and engineering, and their efficient computations have become an integral part of related research and teaching. The research component of this proposal is concerned with numerical linear algebra problems that have significant impacts on related applications. Although the existing general purposed software libraries such as LINPACK, EISPACK, and LAPACK solve many common eigenvalue and singular value problems quite efficiently, many problems from applied science and engineering such as primary eigenvector computations for object recognition in image processing can be solved much more efficiently if their own intrinsic characteristics that are mostly ignored by general purpose libraries are incorporated properly. This project demonstrates that point of view with a preliminary and promising study that combines numerical linear algebra techniques and characteristics of image data to attack a huge eigenvalue (singular value) problem for a large set of high resolution images. The research objective is to exploit in depth the structural properties of applied problems from their application background and, consequently, will develop accurate and efficient algorithms. The research into relative perturbation theory and highly relative accurate methods for matrix computational problems has been extremely active in the last ten years, and exciting advances are being made. The results could have a significant impact on research in other disciplines that previously used conventional algorithms to carry out eigenproblem computations. The education component of this project is to implement a set of ideas and concrete projects, including the exploitation of new information technologies for more effective teaching such as avoiding too much time for students' note taking, keeping constantly in touch with struggling students, and bringing math material alive in the classroom. The project is also concerned with modifying the curriculum of the existing two-semester numerical linear algebra courses to include applied problem lectures that ultimately will lead to a new course on solving applied computational problems.
