engleski

Convergence to diagonal form of general Jacobi-type processes

The standard fast SVD solvers for a general matrix first reduce it to bidiagonal form. This initial reduction can deteriorate the relative accuracy of the smallest singular values even if the initial matrix is well-behaved for accurate SVD computation. For such matrices, the one-sided Jacobi method has proved to be very accurate and fast, especially on parallel computers. Similar arguments holds for the solvers of other matrix problems: hyperbolic SVD, eigenvalue problem for non-Hermitian matrices, generalized Hermitian eigenvalue problem and generalized singular value problem. The global convergence of one-sided Jacobitype processes reduces to the convergence of their two-sided counterparts. This report considers convergence to diagonal form of a general two-sided Jacobi-type process A(k+1) = [P(k)]∗ A(k)Q(k), k ≥ 0, where P(k) and Q(k) are regular elementary matrices, which differ from the identity matrix in one principal submatrix. The modulus pivot strategy is assumed since it is weakly equivalent to the most common cyclic strategies for sequential and parallel processing. The technique uses the theory of Jacobi annihilators which is due to Henrici and Zimmermann. The main result provides sufficient conditions for the convergence of such a process to diagonal form. Recent research includes the block Jacobi type processes.

Jacobi-type process; convergence

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