• Journal of the Institute of Electronics and Information Engineers
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• v.51 no.7
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• pp.142-148
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• 2014

Krylov-Schur iteration method has been applied to the 3-Dim. resonant cavity of a cylindrical form. The vector Helmholtz equation has been analysed for the resonant field strength in homogeneous media by FEM. An eigen-equation has been constructed from element equations basing on tangential edges of the tetrahedra element. This equation made up of two square matrices associated with the curl-curl form of the Helmholtz operator. By performing Krylov-Schur iteration loops on them, Eigen-values and their modes have been determined from the diagonal components of the Schur matrices and its transforming matrices. Eigen-pairs as a result have been revealed visually in the schematic representations. The spectra have been compared with each other to identify the effect of boundary conditions.

• Journal of the Institute of Electronics and Information Engineers
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• v.51 no.2
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• pp.10-14
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• 2014

Krylov-Schur iteration method has been applied to the 2-Dim. waveguides of the varied geometrical structure. The eigen-equations for them have been constructed from FEM based on the tangential edge vectors of triangular elements. The eigen-values and their modes have been determined from the diagonal components of the Schur matrices and its transforming matrices. The eigen-pairs as the results have been revealed visually in the schematic representations.

• Journal of the Institute of Electronics and Information Engineers
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• v.50 no.11
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• pp.28-35
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• 2013

The Krylov-Schur algorithm has been applied to reveal the eigen-properties of the wave guide having the square cross section. The eigen-matrix equation has been constructed from FEM with the basis function of the tangential edge-vectors of the triangular element. This equation has been treated firstly with Arnoldi decomposition to obtain a upper Hessenberg matrix. The QR algorithm has been carried out to transform it into Schur form. The several eigen values satisfying the convergent condition have appeared in the diagonal components. The eigen-modes for them have been calculated from the inverse iteration method. The wanted eigen-pairs have been reordered in the leading principle sub-matrix of the Schur matrix. This sub-matrix has been deflated from the eigen-matrix equation for the subsequent search of other eigen-pairs. These processes have been conducted several times repeatedly. As a result, a few primary eigen-pairs of TE and TM modes have been obtained with sufficient reliability.

• Journal of applied mathematics & informatics
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• v.8 no.2
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• pp.305-326
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• 2001

We introduce a class of multilevel recursive incomplete LU preconditioning techniques (RILUM) for solving general sparse matrices. This techniques is based on a recursive two by two block incomplete LU factorization on the coefficient martix. The coarse level system is constructed as an (approximate) Schur complement. A dynamic preconditioner is obtained by solving the Schur complement matrix approximately. The novelty of the proposed techniques is to solve the Schur complement matrix by a preconditioned Krylov subspace method. Such a reduction process is repeated to yield a multilevel recursive preconditioner.