Gauss–Seidel method for multi-valued inclusions with Z mappings

We consider a problem of solution of a multi-valued inclusion on a cone segment. In the case where the underlying mapping possesses Z type properties we suggest an extension of Gauss–Seidel algorithms from nonlinear equations. We prove convergence of a modified double iteration process under rather mild additional assumptions. Some results of numerical experiments are also presented.     

Performance Optimization for the Parallel Gauss–Seidel Smoother

Simulation, e.g., in the field of computational fluid dynamics, accounts for a major part of the computing time on highperformance systems. Many simulation packages still rely on Gauss–Seidel iteration, either as the main linear solver or as a smoother for multigrid schemes. Straight-forward implementations of this solver have efficiency problems on today's most common high-performance computers, i.e., multiprocessor clusters with pronounced memory hierarchies. In this work we present two simple techniques for improving the performance of the parallel Gauss–Seidel method for the 3D Poisson equation by optimizing cache usage as well as reducing the number of communication steps. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The convergence of the modified Gauss–Seidel methods for consistent linear systems

In this paper we present a convergence analysis for the modified Gauss–Seidel methods given in Gunawardena et al. (Linear Algebra Appl. 154–156 (1991) 125) and Kohno et al. (Linear Algebra Appl. 267 (1997) 113) for consistent linear systems. We prove that the modified Gauss–Seidel method converges for some values of the parameters in the preconditioned matrix.     

On the Gauss–Newton method

We provide a new semilocal convergence analysis of the Gauss–Newton method (GNM) for solving nonlinear equation in the Euclidean space. Using a combination of center-Lipschitz, Lipschitz conditions, and our new idea of recurrent functions, we provide under the same or weaker hypotheses than before (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982), a finer convergence analysis. The results can be extended in case outer or generalized inverses are used. Numerical examples are also provided to show that our results apply, where others fail (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982).     

The geometry of the Gauss–Picard modular group

We give a construction of a fundamental domain for PU(2,1,mathbbZ [i]){{rm PU}(2,1,mathbb{Z} [i])}, that is the group of holomorphic isometries of complex hyperbolic space with coefficients in the Gaussian ring of integers mathbbZ [i]{mathbb{Z} [i]}. We obtain from that construction a presentation of that lattice and relate it, in particular, to lattices constructed by Mostow.     

Correction of eigenvalues estimated by the Legendre–Gauss Tau method

It is known that the accuracy in estimating a large number of eigenvalues deteriorates when the standard numerical methods are applied, because of the sharp oscillatory behavior of the corresponding eigenfunctions. One method which has proved to be efficient in treating such problems is the Legendre–Gauss Tau method. In this paper we present an exponentially fitted version of this method and we develop practical formulae to correct the estimated eigenvalues.     

Extending the applicability of the Gauss–Newton method under average Lipschitz–type conditions

We extend the applicability of the Gauss–Newton method for solving singular systems of equations under the notions of average Lipschitz–type conditions introduced recently in Li et al. (J Complex 26(3):268–295, 2010). Using our idea of recurrent functions, we provide a tighter local as well as semilocal convergence analysis for the Gauss–Newton method than in Li et al. (J Complex 26(3):268–295, 2010) who recently extended and improved earlier results (Hu et al. J Comput Appl Math 219:110–122, 2008; Li et al. Comput Math Appl 47:1057–1067, 2004; Wang Math Comput 68(255):169–186, 1999). We also note that our results are obtained under weaker or the same hypotheses as in Li et al. (J Complex 26(3):268–295, 2010). Applications to some special cases of Kantorovich–type conditions are also provided in this study.     

The factorization method for the Askey–Wilson polynomials

A special Infeld–Hull factorization is given for the Askey–Wilson second order q-difference operator. It is then shown how to deduce a generalization of the corresponding Askey–Wilson polynomials.     

The Gauss–Wilson theorem for quarter-intervals

We define a Gauss factorial N _n ! to be the product of all positive integers up to N that are relatively prime to n. It is the purpose of this paper to study the multiplicative orders of the Gauss factorials $\left\lfloor\frac{n-1}{4}\right\rfloor_{n}!$ for odd positive integers n. The case where n has exactly one prime factor of the form p≡1(mod4) is of particular interest, as will be explained in the introduction. A fundamental role is played by p with the property that the order of  $\frac{p-1}{4}!$ modulo p is a power of 2; because of their connection to two different results of Gauss we call them Gauss primes. Our main result is a complete characterization in terms of Gauss primes of those n of the above form that satisfy $\left\lfloor\frac{n-1}{4}\right\rfloor_{n}!\equiv 1\pmod{n}$ . We also report on computations that were required in the process.     

Improved local convergence analysis of the Gauss–Newton method under a majorant condition

We present a local convergence analysis of Gauss-Newton method for solving nonlinear least square problems. Using more precise majorant conditions than in earlier studies such as Chen (Comput Optim Appl 40:97–118, 2008), Chen and Li (Appl Math Comput 170:686–705, 2005), Chen and Li (Appl Math Comput 324:1381–1394, 2006), Ferreira (J Comput Appl Math 235:1515–1522, 2011), Ferreira and Gonçalves (Comput Optim Appl 48:1–21, 2011), Ferreira and Gonçalves (J Complex 27(1):111–125, 2011), Li et al. (J Complex 26:268–295, 2010), Li et al. (Comput Optim Appl 47:1057–1067, 2004), Proinov (J Complex 25:38–62, 2009), Ewing, Gross, Martin (eds.) (The merging of disciplines: new directions in pure, applied and computational mathematics 185–196, 1986), Traup (Iterative methods for the solution of equations, 1964), Wang (J Numer Anal 20:123–134, 2000), we provide a larger radius of convergence; tighter error estimates on the distances involved and a clearer relationship between the majorant function and the associated least squares problem. Moreover, these advantages are obtained under the same computational cost.     

The Sinc-collocation method for solving the Thomas–Fermi equation

A numerical technique for solving nonlinear ordinary differential equations on a semi-infinite interval is presented. We solve the Thomas–Fermi equation by the Sinc-Collocation method that converges to the solution at an exponential rate. This method is utilized to reduce the nonlinear ordinary differential equation to some algebraic equations. This method is easy to implement and yields very accurate results.     

An iterative method for Bayesian Gauss–Markov image restoration

The purpose of this paper is to introduce a new method for the restoration of images that have been degraded by a blur and an additive white Gaussian noise. The model adopted here is assumed to be Bayesian Gauss–Markov linear model. By exploiting the structure of the blurring matrix and by using Kronecker product approximations, the image restoration problem is formulated as matrix equations which will be solved iteratively by projection methods onto Krylov subspaces. We give some theoretical and experimental results with applications to image restoration.     

The Gauss–Bonnet–Chern mass for graphic manifolds

In this paper, we prove a positive mass theorem and Penrose-type inequality of the Gauss–Bonnet–Chern mass $m_2$ for the graphic manifold with flat normal bundle.     

Integration of the Gauss–Codazzi Equations

The Gauss–Codazzi equations imposed on the elements of the first and the second quadratic forms of a surface embedded in are integrable by the dressing method. This method allows constructing classes of Combescure-equivalent surfaces with the same rotation coefficients. Each equivalence class is defined by a function of two variables (master function of a surface). Each class of Combescure-equivalent surfaces includes the sphere. Different classes of surfaces define different systems of orthogonal coordinates of the sphere. The simplest class (with the master function zero) corresponds to the standard spherical coordinates.     

The Farrell–Hsiang method revisited

We present a sufficient condition for groups to satisfy the Farrell–Jones Conjecture in algebraic K-theory and L-theory. The condition is formulated in terms of finite quotients of the group in question and is motivated by work of Farrell–Hsiang.     

A variable fixing version of the two-block nonlinear constrained Gauss–Seidel algorithm for \ell _1-regularized least-squares

The problem of finding sparse solutions to underdetermined systems of linear equations is very common in many fields as e.g. signal/image processing and statistics. A standard tool for dealing with sparse recovery is the \(\ell _1\) -regularized least-squares approach that has recently attracted the attention of many researchers. In this paper, we describe a new version of the two-block nonlinear constrained Gauss–Seidel algorithm for solving \(\ell _1\) -regularized least-squares that at each step of the iteration process fixes some variables to zero according to a simple active-set strategy. We prove the global convergence of the new algorithm and we show its efficiency reporting the results of some preliminary numerical experiments.