Linear convergence of the row cyclic Jacobi and Kogbetliantz methods

Abstract

Summary. Linear convergence of the row cyclic Jacobi and Kogbetliantz methods can be guaranteed if certain constraints concerning the angles of rotations are implemented. Unlike the results of Forsythe and Henrici, convergence can be achieved without under-rotations.

One-Bit Null Space Learning for MIMO underlay cognitive radio

“…However, this result did not determine the convergence rate. The convergence rate problem was addressed in [7], which proved that the CJT for real symmetric matrices has a global linear convergence rate 2 if θ k ∈ [−π/4, π/4] for every k. This result was extended to complex Hermitian matrices in [8]. It was later shown in [9,10] that for matrix with well separated eigenvalues, the CJT has a quadratic convergence rate 3 .…”

“…Theorem 3: Let G be a finite dimensional n t × n t complex Hermitian matrix and P k denotes the Frobenius norm of the off diagonal upper triangular (or lower triangular) part of A k = W * k−1 GW k−1 where W k is defined in (8) and let m = n t (n t − 1)/2. Let η be the accuracy of the binary search (see (16)), then the OBNSLA satisfies…”

“…Let P k be the Frobenius norm of the off diagonal upper triangular part of (8) and let m = (n 2 t − n t )/2. Assume that the modified O/BNSLA has reached a stage k, such that P…”

Implicit Cholesky algorithms for singular values and vectors of triangular matrices

“…The answer is that the classical Jacobi method in which the largest offdiagonal element is chosen as the doomed element is no longer the preferred version for computation of eigenvalues. Instead the row (or column) cyclic annihilations are used in Jacobi algorithms and this strategy does not have good initial convergence properties; see Fernando [8], although asymptotically the convergence is quadratic. Hence the use of a preconditioner for cyclic Jacobi is not a futile effort.…”

Accuracy of the Kogbetliantz method for scaled diagonally dominant triangular matrices

“…Its algorithm [6][7][8][9][10][11][12][13][14], the global [8,9,15,11,16] and the asymptotic [10,13,[17][18][19][20][21] convergence are well understood. Its implementation details and adaptation for different computer architectures can be found in [12,[22][23][24] and the references cited therein.…”