Existence of fixed point in orthogonal metric spaces has been initiated recently by Eshaghi and et al. [On orthogonal sets and Banach fixed Point theorem, Fixed Point Theory, in press]. In this paper, we introduce the notion of the strongly orthogonal sets and prove a genuine generalization of Banach' fixed point theorem and Walter's theorem. Also, we give an example showing that our main theor...

4. 4.1. 4.1.1. 4.1.2. 4.1.3. 4.2. 4.3. Gaussian quadrature with preassigned nodes Christoffel's work and related developments Kronrod's extension of quadrature rules Gaussian quadrature with multiple nodes The quadrature formula of Turan Arbitrary multiplicities and preassigned nodes Power-orthogonal polynomials Constructive aspects and applications Further miscellaneous extensions Product-type...

We consider the non-de nite linear functionals Ln[f] = ∫ R w(x)f (x) dx and prove the nonexistence of orthogonal polynomials, with all zeros real, in several cases. The proofs are based on the connection with moment preserving spline approximation. c © 1998 Elsevier Science B.V. All rights reserved.

We consider the non-definite linear functionals Ln[f ] = ∫ IR w(x)f (n)(x) dx and prove the nonexistence of orthogonal polynomials, with all zeros real, in several cases. The proofs are based on the connection with moment preserving spline approximation.

lanczos-type algorithms are well known for their inherent instability. they typically breakdown occurs when relevant orthogonal polynomials do not exist. current approaches to curing breakdown rely on jumping over the non-existent polynomials to resume computation. this may have to be used many times during the solution process. we suggest an alternative to jumping, which consists of restarting...

In this paper we explore two sets of polynomials, the orthogonal polynomi-als and the residual polynomials, associated with a preconditioned conjugate gradient iteration for the solution of the linear system Ax = b. In the context of preconditioning by the matrix C, we show that the roots of the orthogonal polynomials, also known as generalized Ritz values, are the eigenvalues of an orthogonal ...

In this study a numerical method is developed to solve the Hammerstein integral equations. To this end the kernel has been approximated using the leastsquares approximation schemes based on Legender-Bernstein basis. The Legender polynomials are orthogonal and these properties improve the accuracy of the approximations. Also the nonlinear unknown function has been approximated by using the Berns...