We consider the solution of ill-conditioned symmetric and positive definite large sparse linear systems of equations. These arise, for instance, when using some symmetrizable preconditioning technique for solving a general (possibly unsymmetric) ill-conditioned linear system, or in domain decomposition of a numerically difficult elliptic problem. We are also concerned with the consecutive solut...

In this paper we introduce a parameter dependent class of Krylov-based methods, namely CD, for the solution of symmetric linear systems. We give evidence that in our proposal we generate sequences of conjugate directions, extending some properties of the standard Conjugate Gradient (CG) method, in order to preserve the conjugacy. For specific values of the parameters in our framework we obtain ...

In this article, conditions for the preservation of quadratic and Hamiltonian invariants by numerical methods which can be written as B-series are derived in a purely algebraical way. The existence of a modified invariant is also investigated and turns out to be equivalent, up to a conjugation, to the preservation of the exact invariant. A striking corollary is that a symplectic method is forma...

Many scientific applications require one to solve successively linear systems Ax = b with different right-hand sides b and a symmetric positive definite matrix A. The conjugate gradient method applied to the first system generates a Krylov subspace which can be efficiently recycled thanks to orthogonal projections in subsequent systems. A modified conjugate gradient method is then applied with ...

The effects of gain and loss on the band structures a bulk topological dielectric photonic crystal (PC) with ${C}_{6v}$ symmetry PC-air-PC interface are studied based first-principle calculation. To illustrate importance parity-time (PT) symmetry, three systems considered, namely PT-symmetric, PT-asymmetric, lossy systems. We find that system distributed in PT symmetric manner exhibits phase tr...

SIAM J. Matrix Anal Appl. 26 (2005) 748–757 Abstract. Let R ∈ C and S ∈ C be nontrivial involutions; i.e., R = R 6= ±Im and S = S 6= ±In. We say that A ∈ C is (R,S)-symmetric ((R,S)-skew symmetric) if RAS = A (RAS = −A). We give an explicit representation of an arbitrary (R,S)-symmetric matrix A in terms of matrices P andQ associatedwith R and U and V associatedwith S. If R = R, then the least ...

In the present article, using subordination principle, authors employed certain generalized multiplier transform to define two new subclasses of analytic functions with respect symmetric and conjugate points. particular, bi-univalent conditions for function f(z) belonging these their relevant connections famous Fekete-Szegö inequality |a3−va22| were investigated a succinct mathematical approach.

We combine linear algebra techniques with finite element techniques to obtain a reliable stopping cri-terion for Krylov method based algorithms. The Conjugate Gradient method has for a long time beensuccessfully used in the solution of the symmetric and positive definite systems obtained from thefinite-element approximation of self-adjoint elliptic partial differential equations...

A new proof is presented for the Motzkin theorem saying that if a set consists of d− 1 complex points and is symmetric relative to the real axis, then there exists a monic, irreducible, and integral polynomial of degree d whose roots are as close to each of these d−1 points as we wish. Unlike the earlier proofs, the new proof is efficient, i.e., it gives both an explicit construction of the pol...

It is very important to achieve high performance in the SC-FDMA (single carrier frequency division multiple access) system that is used for LTE (long term evolution) and LTE-advanced system. To simultaneously resolve ICI(inter-subcarrier interference) and PAPR(peak to average power ratio) problem, focused on the two transmit(Tx) / one receive(Rx) antenna configuration, phase noise influence on ...