A Preconditioned COCG Method for Solving Complex Symmetric Linear Systems Arising from Scattering Problems

Quasi-Minimal Residual Variants of the COCG and COCR Methods for Complex Symmetric Linear Systems in Electromagnetic Simulations

“…From the complex symmetric Lanczos process, it follows that (2) where and is defined as the scaling matrix . In fact, we often take for convenience.…”

“…In addition, a three-element buffer is placed between the equivalent source surface and the PML interface. Therefore, we obtain a system of linear equations of size 51611 with a complex coefficient matrix containing 428 771 nonzero elements in the upper triangular (see [2] for details for this model issue). Here we set the maximal iteration number .…”

“…Such complex-value systems of linear equations, which are strongly required to solve efficiently, often come from many valuable electromagnetic applications such as modeling waveguide discontinuities [1] and simulating the radar cross section (RCS) [2]. In order to solve such systems efficiently, van der Vorst and Mellissen [3] exploited and extended the conjugate gradient (CG) method [4] to establish the well-known conjugate orthogonal conjugate gradient (COCG) method.…”

“…In order to solve such systems efficiently, van der Vorst and Mellissen [3] exploited and extended the conjugate gradient (CG) method [4] to establish the well-known conjugate orthogonal conjugate gradient (COCG) method. Meanwhile, Li et al [2], [5] capitalized upon incomplete Cholesky factorization preconditioning strategies combined with the COCG method to solve 3-D electromagnetic scattering problems. Besides, other relatively complicated, but robust solvers such as quasi-minimal residual (QMR) [6], complex symmetric method (CSYM) [7], conjugate -orthogonal conjugate residual (COCR) [8], [9], symmetric complex bi-conjugate gradient (SCBiCG ) method [10], [11] and bi-conjugate gradient conjugate residual-type (BiCGCR) method [11], [12] are also favorable.…”

“…54-55]. CSYM is derived from the combination of QMR and tridiagonalization of by Householder reflections and it does not belong to the Krylov subspace methods (refer to [2] and [7] for this discussion). Moreover, BiCGCR and SCBiCG can be obtained from the special cases in the bi-conjugate gradient (BiCG) method [13], [14], which has been proposed for solving non-Hermitian linear systems.…”