A real algorithm for the Hermitian eigenvalue decomposition

Modifying complex plane rotations, we derive a new Jacobi-type algorithm for the Hermitian eigendecomposition, which uses only real arithmetic. When the fast-scaled rotations are incorporated, the new algorithm brings a substantial reduction in computational costs. The new method has the same convergence properties and parallelism as the symmetric Jacobi algorithm. Computational test results show that it produces accurate eigenvalues and eigenvectors and achieves great reduction in computational time.The work of this author was supported in part by the National Science Foundation grant CCR-8813493 and by the University of Minnesota Army High Performance Computing Research Center contract DAAL 03-89-C-0038.The work of this author was supported in part by the University of Minnesota Army High Performance Computing Research Center contract DAAL 03-89-C-0038.     

On the cubic convergence of the Paardekooper method

We show a simple way how asymptotic convergence results can be conveyed from a simple Jacobi method to a block Jacobi method. Our pilot methods are the well known symmetric Jacobi method and the Paardekooper method for reducing a skew-symmetric matrix to the real Schur form. We show resemblance in the quadratic and cubic convergence estimates, but also discrepances in the asymptotic assumptions. By numerical tests we confirm that our asymptotic assumptions for the Paardekooper method are most general.     

On quadratic convergence bounds for theJ-symmetric Jacobi method

Summary This paper deals with quadratic convergence estimates for the serialJ-symmetric Jacobi method recently proposed by Veseli. The method is characterized by the use of orthogonal and hyperbolic plane rotations. Using a new technique recently introduced by Hari we prove sharp quadratic convergence bounds in the general case of multiple eigenvalues.     

On the global and cubic convergence of a quasi-cyclic Jacobi method

Summary In this paper we consider the global and the cubic convergence of a quasi-cyclic Jacobi method for the symmetric eigenvalue, problem. The method belongs to a class of quasi-cyclic methods recently proposed by W. Mascarenhas. Mascarenhas showed that the methods from his class asymptotically converge cubically per quasi-sweep (one quasi-sweep is equivalent to 1.25 cyclic sweeps) provided the eigenvalues are simple. Here we prove the global convergence of our method and derive very sharp asymptotic convergence bounds in the general case of multiple eigenvalues. We discuss the ultimate cubic convergence of the method and present several numerical examples which all well comply with the theory.This work was supported in part by the University of Minnesota Army High Performance Computing Research Center and the U.S Army Contract DAAL03-89-C-0038. The paper was partly written while this author was a visiting faculty in the Department of Mathematics, University of Kansas, Lawrence, Kansas. The first version of this paper was made in July 1990 while this author was visiting AHPCRC.     

Globally convergent Jacobi methods for positive definite matrix pairs

The paper derives and investigates the Jacobi methods for the generalized eigenvalue problem A x = λ B x, where A is a symmetric and B is a symmetric positive definite matrix. The methods first “normalize” B to have the unit diagonal and then maintain that property during the iterative process. The global convergence is proved for all such methods. That result is obtained for the large class of generalized serial strategies from Hari and Begovi? Kova? (Trans. Numer. Anal. (ETNA) 47, 107–147, 2017). Preliminary numerical tests confirm a high relative accuracy of some of those methods, provided that both matrices are positive definite and the spectral condition numbers of Δ _A AΔ _A and Δ _B BΔ _B are small, for some nonsingular diagonal matrices Δ _A and Δ _B .     

EPR study of a copper impurity center in a single crystal of 2-thiothymine

EPR spectroscopy was used to study the complex formed in crystals of 5-methyl-2-thiouracil (2-thiothymine) containing traces of copper. The copper impurities, originally present as Cu(I)-complex of 2-thiothymine in the lattice of 2-thiothymine, are transformed into paramagnetic Cu(II)-complex by ionizing radiation. It was found that the complex is planar, the plane being defined by two pairs of S and N atoms, from two adjacent 2-thiothymine molecules. The structure of the complex suggests that a pair of hydrogen bonds between two neighboring molecules is replaced by a stronger Cu-coordination bonding, with two sulfur and two nitrogen atoms as ligands. The spectroscopic parameters (g-tensor, A(63Cu) and A(14N) tensors) are essentially similar to those earlier observed for copper planar centers in other systems.     

Radicals formed in cytosine–hydrochloride(thiocytosine) single crystals: Insights from a DFT study

Specific properties of chlorine interaction with thiocytosine in comparison to chlorine interaction with cytosine in the system formed by the incorporation of small amount of thiocytosine in a cytosine–hydrochloride crystal lattice have been investigated by DFT calculations of the g-tensor of sulfur centered radical. Taking account of the crystal environment of the radical site it has been shown that the main reason for difficulties in interpretation of spectroscopic as well as theoretical results in this model system is the considerable spin density spread on both chlorine and sulfur atoms. From comparison of the results for the proposed model structure with accessible spectroscopic data, the specific direction in which charge transfer in the investigated system may take place has been pointed out.     

On sharp quadratic convergence bounds for the serial Jacobi methods

Summary Using a new technique we derive sharp quadratic convergence bounds for the serial symmetric and SVD Jacobi methods. For the symmetric Jacobi method we consider the cases of well and poorely separated eigenvalues. Our result implies the result proposed, but not correctly proved, by Van Kempen. It also extends the well-known result of Wilkinson to the case of multiple eigenvalues.This work was supported in part by the U.S. Army. Contract DAAL03-89-C-0038     

On the complex Falk–Langemeyer method

Numerical Algorithms - A new algorithm for the simultaneous diagonalization of two complex Hermitian matrices is derived. It is a proper generalization of the known Falk–Langemeyer algorithm...