A fast convergent iterative solver for approximate inverse of matrices

In this paper, a rapid iterative algorithm is proposed to find robust approximations for the inverse of nonsingular matrices. The analysis of convergence reveals that this high‐order method possesses eighth‐order convergence. The interesting point is that, this rate is attained using less number of matrix‐by‐matrix multiplications in contrast to the existing methods of the same type in the literature. The extension of the method for finding Moore–Penrose inverse of singular or rectangular matrices is also presented. Numerical comparisons will be given to show the applicability, stability and consistency of the new scheme by paying special attention on the computational time. Copyright © 2013 John Wiley & Sons, Ltd.     

Staggered discontinuous Galerkin methods for the incompressible Navier–Stokes equations: Spectral analysis and computational results

The goal of this paper is to create a fruitful bridge between the numerical methods for approximating PDEs in fluid dynamics and the (iterative) numerical methods for dealing with the resulting large linear systems. Among the main objectives are the design of new, efficient iterative solvers and a rigorous analysis of their convergence speed. The link we have in mind is either the structure or the hidden structure that the involved coefficient matrices inherit, both from the continuous PDE and from the approximation scheme; in turn, the resulting structure is used for deducing spectral information, crucial for the conditioning and convergence analysis and for the design of more efficient solvers. As a specific problem, we consider the incompressible Navier–Stokes equations; as a numerical technique, we consider a novel family of high‐order, accurate discontinuous Galerkin methods on staggered meshes, and as tools, we use the theory of Toeplitz matrices generated by a function (in the most general block, the multilevel form) and the more recent theory of generalized locally Toeplitz matrix sequences. We arrive at a somehow complete picture of the spectral features of the underlying matrices, and this information is employed for giving a forecast of the convergence history of the conjugate gradient method, together with a discussion on new and more advanced techniques (involving preconditioning, multigrid, multi‐iterative solvers). Several numerical tests are provided and critically illustrated in order to show the validity and the potential of our analysis.     

High‐order numerical solution of second‐order one‐dimensional hyperbolic telegraph equation using a shifted Gegenbauer pseudospectral method

We present a high‐order shifted Gegenbauer pseudospectral method (SGPM) to solve numerically the second‐order one‐dimensional hyperbolic telegraph equation provided with some initial and Dirichlet boundary conditions. The framework of the numerical scheme involves the recast of the problem into its integral formulation followed by its discretization into a system of well‐conditioned linear algebraic equations. The integral operators are numerically approximated using some novel shifted Gegenbauer operational matrices of integration. We derive the error formula of the associated numerical quadratures. We also present a method to optimize the constructed operational matrix of integration by minimizing the associated quadrature error in some optimality sense. We study the error bounds and convergence of the optimal shifted Gegenbauer operational matrix of integration. Moreover, we construct the relation between the operational matrices of integration of the shifted Gegenbauer polynomials and standard Gegenbauer polynomials. We derive the global collocation matrix of the SGPM, and construct an efficient computational algorithm for the solution of the collocation equations. We present a study on the computational cost of the developed computational algorithm, and a rigorous convergence and error analysis of the introduced method. Four numerical test examples have been carried out to verify the effectiveness, the accuracy, and the exponential convergence of the method. The SGPM is a robust technique, which can be extended to solve a wide range of problems arising in numerous applications. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 307–349, 2016     

Iterative solutions to the extended Sylvester-conjugate matrix equations

This paper is concerned with iterative solutions to a class of complex matrix equations. By applying the hierarchical identification principle, an iterative algorithm is constructed to solve this class of complex matrix equations. The range of the convergence factor is given to guarantee that the proposed algorithm is convergent for arbitrary initial matrix by applying a real representation of a complex matrix as a tool. By using some properties of the real representation, a sufficient convergence condition that is easier to compute is also given by original coefficient matrices. Two numerical examples are given to illustrate the effectiveness of the proposed methods.     

On the convergence of basic iterative methods for convection–diffusion equations

In this paper we analyze convergence of basic iterative Jacobi and Gauss–Seidel type methods for solving linear systems which result from finite element or finite volume discretization of convection–diffusion equations on unstructured meshes. In general the resulting stiffness matrices are neither M‐matrices nor satisfy a diagonal dominance criterion. We introduce two newmatrix classes and analyse the convergence of the Jacobi and Gauss–Seidel methods for matrices from these classes. A new convergence result for the Jacobi method is proved and negative results for the Gauss–Seidel method are obtained. For a few well‐known discretization methods it is shown that the resulting stiffness matrices fall into the new matrix classes. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical analysis for the quadratic matrix equations from a modification of fixed‐point type

In this paper, we study the quadratic matrix equations. To improve the application of iterative schemes, we use a transform of the quadratic matrix equation into an equivalent fixed‐point equation. Then, we consider an iterative process of Chebyshev‐type to solve this equation. We prove that this iterative scheme is more efficient than Newton's method. Moreover, we obtain a local convergence result for this iterative scheme. We finish showing, by an application to noisy Wiener‐Hopf problems, that the iterative process considered is computationally more efficient than Newton's method.     

Some extensions of the improved modified Gauss–Seidel iterative method for H‐matrices

In this paper, first we present a convergence theorem of the improved modified Gauss–Seidel iterative method, referred to as the IMGS method, for H‐matrices and compare the range of parameters α_i with that of the parameter ω of the SOR iterative method. Then with a more general splitting, the convergence analysis of this method for an H‐matrix and its comparison matrix is given. The spectral radii of them are also compared. Copyright © 2006 John Wiley & Sons, Ltd.     

Iterative algorithms for solving a class of complex conjugate and transpose matrix equations

This paper is concerned with iterative solutions to a class of complex matrix equations, which include some previously investigated matrix equations as special cases. By applying the hierarchical identification principle, an iterative algorithm is constructed to solve this class of matrix equations. A sufficient condition is presented to guarantee that the proposed algorithm is convergent for an arbitrary initial matrix with a real representation of a complex matrix as tools. By using some properties of the real representation, a convergence condition that is easier to compute is also given in terms of original coefficient matrices. A numerical example is employed to illustrate the effectiveness of the proposed methods.     

Weighted matrix means and symmetrization procedures

Here we prove the convergence of the Ando–Li–Mathias and Bini–Meini–Poloni procedures for matrix means. Actually it is proved here that for a two-variable function which maps pairs of positive definite matrices to a positive definite matrix and is not greater than the square mean of two positive definite matrices, the Ando–Li–Mathias and Bini–Meini–Poloni procedure converges. In order to be able to set up the Bini–Meini–Poloni procedure, a weighted two-variable matrix mean is also needed. Therefore a definition of a two-variable weighted matrix mean corresponding to every symmetric matrix mean is also given. It is also shown here that most of the properties considered by Ando, Li and Mathias for the n-variable geometric mean hold for all of these n-variable maps that we obtain by this two limiting process for all two-variable matrix means. As a consequence it also follows that the Bini–Meini–Poloni procedure converges cubically for every matrix mean.     

Block filtering decomposition

This paper introduces a new preconditioning technique that is suitable for matrices arising from the discretization of a system of PDEs on unstructured grids. The preconditioner satisfies a so‐called filtering property, which ensures that the input matrix is identical with the preconditioner on a given filtering vector. This vector is chosen to alleviate the effect of low‐frequency modes on convergence and so decrease or eliminate the plateau that is often observed in the convergence of iterative methods. In particular, the paper presents a general approach that allows to ensure that the filtering condition is satisfied in a matrix decomposition. The input matrix can have an arbitrary sparse structure. Hence, it can be reordered using nested dissection, to allow a parallel computation of the preconditioner and of the iterative process. We show the efficiency of our preconditioner through a set of numerical experiments on symmetric and nonsymmetric matrices. Copyright © 2014 John Wiley & Sons, Ltd.     

On a new family of high‐order iterative methods for the matrix pth root

The main goal of this paper is to approximate the principal pth root of a matrix by using a family of high‐order iterative methods. We analyse the semi‐local convergence and the speed of convergence of these methods. Concerning stability, it is well known that even the simplified Newton method is unstable. Despite it, we present stable versions of our family of algorithms. We test numerically the methods: we check the numerical robustness and stability by considering matrices that are close to be singular and badly conditioned. We find algorithms of the family with better numerical behavior than the Newton and the Halley methods. These two algorithms are basically the iterative methods proposed in the literature to solve this problem. Copyright © 2015 John Wiley & Sons, Ltd.     

Strong Convergence Theorems for Mixed Typ e Asymptotically Nonexpansive Mappings

The purpose of this paper is to study a new two-step iterative scheme with mean errors of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappin...     

Solution of Non-linear Eigenvalue Problems by the Continuation Method

The Problem of finding the roots (eigenvalues) of the equationdet A()=0, where A in an nxn matrix, is studied. There existseveral efficient local iterative methods for this problem.However, no efficient global method is available. We describethe application of the continuation method to this problem andsolve two examples by it. We conclude that the continuationmethod is a practical global strategy for locating eigenvaluesof non-linear matrices. This method is even more effective whenit is combined with an appropriate iterative scheme.     

Tangential frequency filtering decompositions for symmetric matrices

Summary. The tangential frequency filtering decomposition (TFFD) is introduced. The convergence theory of an iterative scheme based on the TFFD for symmetric matrices is the focus of this paper. The existence of the TFFD and the convergence of the induced iterative algorithm is shown for symmetric and positive definite matrices. Convergence rates independent of the number of unknowns are proven for a smaller class of matrices. Using this framework, the convergence independent of the number of unknowns is shown for Wittum's frequency filtering decomposition. Some characteristic properties of the TFFD are illustrated and results of several numerical experiments are presented. Received April 1, 1996 / Revised version July 4, 1996     

Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods

Use of the stochastic Galerkin finite element methods leads to large systems of linear equations obtained by the discretization of tensor product solution spaces along their spatial and stochastic dimensions. These systems are typically solved iteratively by a Krylov subspace method. We propose a preconditioner, which takes an advantage of the recursive hierarchy in the structure of the global matrices. In particular, the matrices posses a recursive hierarchical two‐by‐two structure, with one of the submatrices block diagonal. Each of the diagonal blocks in this submatrix is closely related to the deterministic mean‐value problem, and the action of its inverse is in the implementation approximated by inner loops of Krylov iterations. Thus, our hierarchical Schur complement preconditioner combines, on each level in the approximation of the hierarchical structure of the global matrix, the idea of Schur complement with loops for a number of mutually independent inner Krylov iterations, and several matrix–vector multiplications for the off‐diagonal blocks. Neither the global matrix nor the matrix of the preconditioner need to be formed explicitly. The ingredients include only the number of stiffness matrices from the truncated Karhunen–Loève expansion and a good preconditioned for the mean‐value deterministic problem. We provide a condition number bound for a model elliptic problem, and the performance of the method is illustrated by numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

On the convergence of right transforming iterations for the numerical solution of PDE‐constrained optimization problems

We present an iterative solver, called right transforming iterations (or right transformations), for linear systems with a certain structure in the system matrix, such as they typically arise in the framework of Karush–Kuhn–Tucker (KKT) conditions for optimization problems under PDE constraints. The construction of the right transforming scheme depends on an inner approximate solver for the underlying PDE subproblems. We give a rigorous convergence proof for the right transforming iterative scheme in dependence on the convergence properties of the inner solver. Provided that a fast subsolver is available, this iterative scheme represents an efficient way of solving first‐order optimality conditions. Numerical examples endorse the theoretically predicted contraction rates. Copyright © 2011 John Wiley & Sons, Ltd.     

The equivalent inclusion technique and its convergence

This paper addresses the convergence characteristics of an iterative solution scheme of the Neumann‐type useful for obtaining homogenized mechanical material properties within an RVE. The analysis is based on the idea of “equivalent inclusions” and, within the context of stress/strain analysis, allows modeling of elastically highly heterogeneous bodies with the aid of discrete Fourier transforms. Within the iterative scheme the proof of convergence depends critically upon the choice of an appropriate, auxiliary stiffness matrix, which also determines the speed of convergence. Mathematically speaking it is based on Banach's fixpoint theorem and only results in necessary convergence conditions. However, for all cases of elastic heterogeneity that are of practical importance convergence can be demonstrated.     

Diagonally compensated reduction and related preconditioning methods

When solving linear algebraic equations with large and sparse coefficient matrices, arising, for instance, from the discretization of partial differential equations, it is quite common to use preconditioning to accelerate the convergence of a basic iterative scheme. Incomplete factorizations and sparse approximate inverses can provide efficient preconditioning methods but their existence and convergence theory is based mostly on M-matrices (H-matrices). In some application areas, however, the arising coefficient matrices are not H-matrices. This is the case, for instance, when higher-order finite element approximations are used, which is typical for structural mechanics problems. We show that modification of a symmetric, positive definite matrix by reduction of positive offdiagonal entries and diagonal compensation of them leads to an M-matrix. This diagonally compensated reduction can take place in the whole matrix or only at the current pivot block in a recursive incomplete factorization method. Applications for constructing preconditioning matrices for finite element matrices are described.     

Solutions of symmetry‐constrained least‐squares problems

In this paper, two new matrix‐form iterative methods are presented to solve the least‐squares problem: and matrix nearness problem: where matrices and are given; ??₁ and ??₂ are the set of constraint matrices, such as symmetric, skew symmetric, bisymmetric and centrosymmetric matrices sets and S_XY is the solution pair set of the minimum residual problem. These new matrix‐form iterative methods have also faster convergence rate and higher accuracy than the matrix‐form iterative methods proposed by Peng and Peng (Numer. Linear Algebra Appl. 2006; 13 : 473–485) for solving the linear matrix equation AXB+CYD=E. Paige's algorithms, which are based on the bidiagonalization procedure of Golub and Kahan, are used as the framework for deriving these new matrix‐form iterative methods. Some numerical examples illustrate the efficiency of the new matrix‐form iterative methods. Copyright © 2008 John Wiley & Sons, Ltd.     

An experiment on the consistency of aggregated comparison matrices in AHP

The analytic hierarchy process can be used for group decision making by aggregating individual judgments or individual priorities. The most commonly used aggregation methods are the geometric mean method and the weighted arithmetic mean method. While it is known that the weighted geometric mean comparison matrix is of acceptable consistency if all individual comparison matrices are of acceptable consistency, this paper addresses the following question: Under what conditions would an aggregated geometric mean comparison matrix be of acceptable consistency if some (or all) of the individual comparison matrices are not of acceptable consistency? Using Monte Carlo simulation, results indicate that given a sufficiently large group size, consistency of the aggregate comparison matrix is guaranteed, regardless of the consistency measures of the individual comparison matrices, if the geometric mean is used to aggregate. This result implies that consistency at the aggregate level is a non-issue in group decision making when group size exceeds a threshold value and the geometric mean is used to aggregate individual judgments. This paper determines threshold values for various dimensions of the aggregated comparison matrix.