In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive-definite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the ...

Abstract: The conjugate gradient method is an algorithm for the numerical solution of systems of linear equations whose matrices are positive-definite and symmetric. It is an iterative method that can even be applied to solve a sparse system of equations. In this paper, we applied the conjugate gradient method of higher order to wave propagation control problems. The algorithm of this method wa...

UltraWideband (UWB) systems use Multi-Band OFDM (MB-OFDM) techniques for transmission. UWB based systems are power limited by the regulation of Federal Communications Commission (FCC). CP introduces correlation in the transmitted data sequence and hence introduces ripples in the power spectral density (PSD) of the transmitted data. This in turn reduces the range of data transmission. The overla...

This paper is concerned with proving theoretical results related to the convergence of the Conjugate Gradient method for solving positive definite symmetric linear systems. New relations for ratios of the A-norm of the error and the norm of the residual are provided starting from some earlier results of Sadok [13]. These results use the well-known correspondence between the Conjugate Gradient m...

In this paper we define the Evans function for Sturm-Liouville problems. We show that the Evans function is analytic in the spectral parameter, has zeros in one-to-one correspondence with the eigenvalues, and is under certain conditions what we call conjugate symmetric. We conclude by showing that the Evans function can be used to track the movement of the eigenvalues as the coefficients in the...

The notion of Newton–like inequalities is extended and an inductive approach is utilized to show that the generalized Newton–like inequalities hold on elementary symmetric functions with self–conjugate variables in the right half–plane.

Let R be an n by n nontrivial real symmetric involution matrix, that is, R R−1 R / In. An n × n complex matrix A is termed R-conjugate if A RAR, where A denotes the conjugate of A. We give necessary and sufficient conditions for the existence of the Hermitian R-conjugate solution to the system of complex matrix equations AX C andXB D and present an expression of the Hermitian R-conjugate soluti...

We present a residual smoothing algorithm for the complex symmetric conjugate gradients method. As in the real case, the residual smoothing method reproduces the (quasi-minimising) QMR method, and is considerably simpler to compute.

The pure symmetric automorphism group of a finitely generated free group consists of those automorphisms which send each standard generator to a conjugate of itself. We prove that these groups are duality groups.