Adaptive eigenvalue computations using Newton's method on the Grassmann manifold
2002 (English)In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, Vol. 23, no 3, 819-839 p.Article in journal (Refereed) Published
We consider the problem of updating an invariant subspace of a large and structured Hermitian matrix when the matrix is modified slightly. The problem can be formulated as that of computing stationary values of a certain function with orthogonality constraints. The constraint is formulated as the requirement that the solution must be on the Grassmann manifold, and Newton's method on the manifold is used. In each Newton iteration a Sylvester equation is to be solved. We discuss the properties of the Sylvester equation and conclude that for large problems preconditioned iterative methods can be used. Preconditioning techniques are discussed. Numerical examples from signal subspace computations are given in which the matrix is Toeplitz and we compute a partial singular value decomposition corresponding to the largest singular values. Further we solve numerically the problem of computing the smallest eigenvalues and corresponding eigenvectors of a large sparse matrix that has been slightly modified.
Place, publisher, year, edition, pages
2002. Vol. 23, no 3, 819-839 p.
Conjugate gradient method, Differential geometry, Eigenvalue, Eigenvector, Grassmann manifold, Newton's method, Preconditioner, Signal subspace problem, Singular values and vectors, Sparse matrix, Toeplitz matrix
Engineering and Technology
IdentifiersURN: urn:nbn:se:liu:diva-47142DOI: 10.1137/S0895479899354688OAI: oai:DiVA.org:liu-47142DiVA: diva2:268038