%0 Journal Article %1 longsine80:LAA-34-195 %A Longsine, D. E. %A McCormick, S. F. %D 1980 %J Linear Algebra and its Applications %K usyd 65f15-numerical-eigenvalues-eigenvectors %P 195--234 %R 10.1016/0024-3795(80)90166-4 %T Simultaneous Rayleigh-Quotient Minimization Methods for \$Ax=Bx\$ %U http://dx.doi.org/10.1016/0024-3795(80)90166-4 %V 34 %X New simultaneous iteration techniques are developed for solving the generalized eigenproblem Ax=λBx, where A and B are real symmetric matrices and B is positive definite. The approach is to minimize the generalized Rayleigh quotient in some sense over several independent vectors simultaneously. In particular, each new vector iterate is formed from a linear combination of current iterates and correction vectors that are derived from either gradient or conjugate-gradient techniques. A Ritz projection or simultaneous iteration process is used to accelerate convergence. For one of the gradient versions, convergence and asymptotic rates of convergence are established. Also, some numerical experiments are reported that demonstrate the convergence behavior of these methods.

@article{longsine80:LAA-34-195, abstract = {{New simultaneous iteration techniques are developed for solving the generalized eigenproblem Ax=λBx, where A and B are real symmetric matrices and B is positive definite. The approach is to minimize the generalized Rayleigh quotient in some sense over several independent vectors simultaneously. In particular, each new vector iterate is formed from a linear combination of current iterates and correction vectors that are derived from either gradient or conjugate-gradient techniques. A Ritz projection or simultaneous iteration process is used to accelerate convergence. For one of the gradient versions, convergence and asymptotic rates of convergence are established. Also, some numerical experiments are reported that demonstrate the convergence behavior of these methods.}}, added-at = {2017-06-29T07:13:07.000+0200}, author = {Longsine, D. E. and McCormick, S. F.}, biburl = {https://www.bibsonomy.org/bibtex/2d36aff7c2961810978916d4d5c55254e/gdmcbain}, citeulike-article-id = {2441996}, citeulike-linkout-0 = {http://dx.doi.org/10.1016/0024-3795(80)90166-4}, doi = {10.1016/0024-3795(80)90166-4}, interhash = {33b5ab2c90ddf15b2fec0936f791f95a}, intrahash = {d36aff7c2961810978916d4d5c55254e}, journal = {Linear Algebra and its Applications}, keywords = {usyd 65f15-numerical-eigenvalues-eigenvectors}, pages = {195--234}, posted-at = {2008-02-28 10:10:49}, priority = {2}, timestamp = {2019-02-28T23:45:33.000+0100}, title = {Simultaneous {Rayleigh}-Quotient Minimization Methods for \$Ax=\lambda Bx\$}, url = {http://dx.doi.org/10.1016/0024-3795(80)90166-4}, volume = 34, year = 1980 }