On the convergence of complex Jacobi methods

Abstract

In this paper we prove the global convergence of the complex Jacobi method for Hermitian matrices for a large class of generalized serial pivot strategies. For a given Hermitian matrix A of order n we find a constant γ < 1 depending on n, such that S(A ′ ) ≤ γS(A), where A ′ is obtained from A by applying one or more cycles of the Jacobi method and S(·) stands for the off-norm. Using the theory of complex Jacobi operators, the result is generalized so it can be used for proving convergence of more general Jacobi-type processes. In particular, we use it to prove the global convergence of Cholesky-Jacobi method for solving the positive definite generalized eigenvalue problem.

Vectorization of the Jacobi-type singular value decomposition method

“…It also handles the 2 × 2 terminal cases in the QR-based [18,19] and the MRRRbased [12,13] algorithms for the eigendecomposition of Hermitian/symmetric matrices. Its direct application, the two-sided Jacobi-type EVD method for symmetric [24] and Hermitian [20] matrices, has not been included in LAPACK but is widely known.…”

“…Vectorization of eigendecompositions of order two. Let A be a symmetric (Hermitian) matrix of order two, U an orthogonal (unitary) matrix of its eigenvectors, i.e., its diagonalizing Jacobi rotation, real [24] or complex [20], and Λ a real diagonal matrix of its eigenvalues. In the complex case, (2.1)…”

“…Assume that A, represented by its lower triangle elements a 11 , a 21 , and a 22 , has already been scaled. From the annihilation condition U * AU = Λ, as shown in Appendix A.1 of the supplementary material similarly to [20], but using the fused multiply-add operation (fma(a, b, c) = RN(a • b + c)) with a single rounding of its result [21], and assuming arg 0 = 0 for determinacy, it follows In the real case (2.4) leads to e iα = sign a 21 = ±1 (note, sign ±0 = ±1). Also, (2.5)…”

Approximate matrix and tensor diagonalization by unitary transformations: convergence of Jacobi-type algorithms

“…Most of the results for the matrix case are on the convergence of f (U k ) to A 2 (or the off-norm (2.4) to zero). The rate is linear and asymptotically quadratic, for the cyclic strategies of choice of pairs and a class of other strategies, see [24, §8.4.3] and [26,27] for an overview. Moreover, the result [40] guarantees that in this case U H k AU k converges to a diagonal matrix.…”

“…The algorithm[32] was proposed for a particular problem of Tucker approximation, but the convergences result of[32] are valid for arbitrary smooth functions, see discussion in[38].3 i.e., for any starting point, the iterations converge to a single limit point. Note that global convergence does not imply convergence to a global minimum; also, the notion of "global convergence" often has a different meaning in the numerical linear algebra community[27].…”