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The main aim of this paper is to establish iterative algorithms for computing the Takagi factorization of complex symmetric matrices. Similar to the classical iterative algorithms of computing the eigenpairs of real symmetric matrices, we derive power-like iterations for computing the Takagi values and associated Takagi vectors of complex symmetric matrices, i.e., the power-like method, the orthogonal-like iteration and the complex symmetric QR-like iteration. We analyze the convergence of these algorithms under some mild conditions. We also investigate the Jacobi-like methods for computing the Takagi factorization of complex symmetric matrices like Jacobi's methods for real symmetric eigenvalue problems. We illustrate our algorithms via numerical examples. [ABSTRACT FROM AUTHOR]

The Shortley–Weller method is a standard finite difference method for solving the Poisson equation with Dirichlet boundary condition. Unless the domain is rectangular, the method meets an inevitable problem that some of the neighboring nodes may be outside the domain. In this case, an usual treatment is to extrapolate the function values at outside nodes by quadratic polynomial. The extrapolation may become unstable in the sense that some of the extrapolation coefficients increase rapidly when the grid nodes are getting closer to the boundary. A practical remedy, which we call artificial perturbation, is to treat grid nodes very near the boundary as boundary points. The aim of this paper is to reveal the adverse effects of the artificial perturbation on solving the linear system and the convergence of the solution. We show that the matrix is nearly symmetric so that the ratio of its minimum and maximum eigenvalues is an important factor in solving the linear system. Our analysis shows that the artificial perturbation results in a small enhancement of the eigenvalue ratio from O ( 1 / ( h ⋅ h m i n ) to O ( h − 3 ) and triggers an oscillatory order of convergence. Instead, we suggest using Jacobi or ILU-type preconditioner on the matrix without applying the artificial perturbation. According to our analysis, the preconditioning not only reduces the eigenvalue ratio from O ( 1 / ( h ⋅ h m i n ) to O ( h − 2 ) , but also keeps the sharp second order convergence. [ABSTRACT FROM AUTHOR]

In this study, we give the necessary and sufficient conditions for the existence of a solution to the Hochstadt inverse eigenvalue problem (HIEP), i.e., the problem of recovering n unknown parameters of a Jacobi matrix from all n of its eigenvalues and n − 1 known parameters. Effective algorithms are proposed for solving the HIEP. [ABSTRACT FROM AUTHOR]

For the eigenvalue decomposition in angle symmetric matrices, a new sorted Jacobi algorithm (S-Jacobi) is proposed. This algorithm sorts the eigenvalues automatically by exploiting both inner and outer angles in each Jacobi rotation. With the condition of convergence that can be easily satisfied in practice, the convergence speed of S-Jacobi is faster than conventional Jacobi algorithms that do not involve eigenvalue sorting. Furthermore, the rotation angle computing circuit proposed for the parallel implementation of S-Jacobi needs only small additional hardware with respect to the case of conventional Jacobi algorithms. [ABSTRACT FROM AUTHOR]

Numerical simulation of open structures requires the truncation of the computational domain and the definition of appropriate boundary conditions. However, any artificial boundary at finite distances causes additional undesired modes in the eigenanalysis of waveguides which must be excluded from the computed spectrum. We propose an extension of the Jacobi-Davidson method by introducing a special weighting function that enables a reliable suppression of undesired eigenmodes already during the solution process. Moreover, the number of iteration steps as well as the computation time can be drastically reduced by this process. We show the improvement of efficiency by means of an eigenmode computation in a photonic crystal fiber discretized by the finite integration technique. [ABSTRACT FROM AUTHOR]

Let the system matrix of a linear system be p-cyclic and consistently ordered. Under the assumption that the pth power of the associated Jacobi matrix has only non-positive eigenvalues, it is known that the optimal spectral radius of the SOR-k iteration matrix is strictly increasing as k increases from 2 to p. In this paper, we first show that the optimal parameter of the SOR-k method as a function of k is strictly increasing. The behaviour of the spectral radius of the SOR-k method (for fixed parameter) is then studied. [ABSTRACT FROM AUTHOR]