where we use tick marks to denote the updated versions of D(t) and s(t) after receiving the new training data, and D(t) 1 2 is the matrix square-root of D(t). The updates to A consist of adding or subtracting an outer-product of a matrix of size (d × k)-by-d, which implies that each update has rank at most d. If we have already computed the eigenvalue decomposition of the old A, we can compute ...

We describe a method for the calculation of guaranteed bounds for the K lowest eigenvalues of second-order problems with Neumann boundary conditions. Using P2 approximations for the eigenfunctions and RT1 approximations for the gradients of the eigenfunctions in H(div; ), an error bound for the eigenfunctions is established for weak approximations in H1( ). In addition, the rest of the spectrum...

Ren-Cang Li University of Texas at Arlington 15.1 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-1 15.2 Singular Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-6 15.3 Polar Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-7 15.4 Generalized Eigenvalue Problems . . . . . . . . . . . . . . . . . . . 15-...

Given an undamped gyroscopic system GðλÞ 1⁄4 Mλ þ CλþK with M , K symmetric and C skew-symmetric, this paper presents a real-valued spectral decomposition of GðλÞ by a real standard pair ðX;TÞ and a skew-symmetric parameter matrix S . When T is assumed to be a block diagonal matrix, the parameter matrix S has a special structure. This spectral decomposition is applied to solve the quadratic inv...

A structure preserving Sort-Jacobi algorithm for computing eigenvalues or singular values is presented. The proposed method applies to an arbitrary semisimple Lie algebra on its (−1)-eigenspace of the Cartan involution. Local quadratic convergence for arbitrary cyclic schemes is shown for the regular case. The proposed method is independent of the representation of the underlying Lie algebra an...