A DIFFERENTIAL-GEOMETRIC LOOK AT THE JACOBI–DAVIDSON FRAMEWORK
Pierre-Antoine Absil, Michiel E. Hochstenbach, K. Hüper, Trumpf, J. · TU/e Research Portal · 2012
Abstract. The problem of computing a p-dimensional invariant subspace of a symmetric positive-definite matrix pencil of dimension n is interpreted as computing a zero of a tangent vector field on the Grassmann manifold of p-planes in Rn. The theory of Newton’s method on manifolds is applied to this problem, and the resulting Newton equations are interpreted as block versions of the Jacobi–Davidson correction equation for the generalized eigenvalue problem. 1. Introduction. The Jacobi–Davidson method (JD) [32] is a method to com-pute certain eigenpairs of standard or generalized eigenvalue problems. JD has been particularly successful for standard eigenproblems where interior eigenvalues are re-quired, and for generalized types of eigenproblems. JD belongs to the the class of subspace methods, where low-dimensional subspaces are exploited to find approxima-