《150 Years odl and still alive eigenproblems》.pdf

Years old and still alive eigenproblems Henk A van der Vorst Utrech t University Gene H Golub Stanford University Dedicated to JH Wilkinson th on the A nniversary of his death Intro duction Linear eigenproblems continue to b e an imp ortant and highly relevant area of research in numerical linear algebra Therefore it should b e no surprise that nu merical algorithms for eigenproblems are among the oldest known in the mo dern literature In exactly years ago Jacobi wrote his famous pap er on T the computation of solutions for the problem Ax x with A A Strictly sp eaking this is not correct since matrix notation was unknown at that time So Jacobi formulated the problem in terms of systems of equations written out elementwise His pap er contains computational elements that are still in use like the plane rotations for making a system more diagonally dominant and the GaussJacobi iteration metho d for diagonally dominant systems These elements were combined in his approach for the computation of eigenvalues to gether with a clever way of setting up a correction equation for eigenvalue and eigenvector approximations Later on other metho ds b ecame p opular such as the Power iteration metho d and when to ols b ecame more sophisticated eigenproblems other than the stan dard Ax x problem required solution for instance the generalized eigen problems Ax B x higher order p olynomial eigenproblems n A 0 A 1 An x and of course the singular value decomp osition Through pioneering work of