Publica
Hier finden Sie wissenschaftliche Publikationen aus den FraunhoferInstituten. Block Newton method and block rayleigh quotient iteration for computing invariant subspaces of general complex matrices
 Linear algebra and its applications 526 (2017), S.6094 ISSN: 00243795 

 Englisch 
 Zeitschriftenaufsatz 
 Fraunhofer IVI () 
 eigenvalue problem; Block Newton method; block RQI 
Abstract
We consider the Block Newton Method and a modification of it, the Block Rayleigh Quotient Iteration, for approximating a simple pdimensional invariant subspace X=im(X) and the corresponding eigenvalues collected in the projection L=(XHX)−1XHAX of an arbitrary complex matrix A∈Cn×n. Both methods generate a sequence of bases {Uk} such that the subspaces im(Uk) approximate the target subspace X. The Block Newton Method is Newton's method applied to the extended system AX−XL=0, WHX=I with respect to (X,L) where W is a normalization matrix. We give a direct convergence analysis which, in contrast to the approach by Beyn/Kless/Thümmler [2001], explicitly exploits the product form of the second derivative and, therefore, leads to new and essentially sharper error recursions for each of the sequences {Uk} and {Θk} where Uk and Θk are the kth X and L iterates, resp. Then, following Lösche/Schwetlick/Timmermann [1998] where the case of real symmetric A is considered, we modify the Block Newton Method as follows: In step k we set W=Uk and use only the Xupdate from the Newton method to improve Uk and then, with the improved basis, perform a Rayleigh–Schur process and take the orthonormal basis and the generalized Rayleigh quotient matrix obtained from it as new iterates. So the method can be considered as a Block Rayleigh Quotient Iteration. We show that for a sufficiently good initial approximation U0 the method converges in that the sequence {sinξk} with ξk=∡(im(Uk),X) converges Qquadratically to zero provided that the invariant subspace X is simple, i.e., that the eigenvalues corresponding to X which are the eigenvalues of the projection L=(XHX)−1(XHAX) of A onto X are separated from those corresponding to X⊥ which are given by λ(A)−λ(L). The convergence results are directly proven – without exploiting standard Newton convergence results – by using the angles between the corresponding subspaces instead of working with the norm differences of the bases as done by Beyn/Kless/Thümmler [2001]. As a byproduct the method is shown to converge cubically if A=AH as in the case p=1.