Modified Newton-MDPMHSS method for solving nonlinear systems with block two-by-two complex symmetric Jacobian matrices

Numerical Algorithms - In this study, an efficient iterative method is given to solve large sparse nonlinear systems with block two-by-two complex symmetric Jacobian matrices. Based on the...     

Modified Newton-DSS method for solving a class of systems of nonlinear equations with complex symmetric Jacobian matrices

Numerical Algorithms - Double-step scale splitting (DSS) iteration method is proved to be an unconditionally convergent iteration method, which is also efficient and robust for solving a class of...     

Alternating-directional PMHSS iteration method for a class of two-by-two block linear systems

In Bai et al. (2013), a preconditioned modified HSS (PMHSS) method was proposed for a class of two-by-two block systems of linear equations. In this paper, the PMHSS method is modified by adding one more parameter in the iteration. Convergence of the modified PMHSS method is guaranteed. Theoretic analysis and numerical experiment show that the modification improves the PMHSS method.     

On SSOR iteration method for a class of block two-by-two linear systems

In this paper, the optimal iteration parameters of the symmetric successive overrelaxation (SSOR) method for a class of block two-by-two linear systems are obtained, which result in optimal convergence factor. An accelerated variant of the SSOR (ASSOR) method is presented, which significantly improves the convergence rate of the SSOR method. Furthermore, a more practical way to choose iteration parameters for the ASSOR method has also been proposed. Numerical experiments demonstrate the efficiency of the SSOR and ASSOR methods for solving a class of block two-by-two linear systems with the optimal parameters.     

Additive block diagonal preconditioning for block two-by-two linear systems of skew-Hamiltonian coefficient matrices

For a class of block two-by-two systems of linear equations with certain skew-Hamiltonian coefficient matrices, we construct additive block diagonal preconditioning matrices and discuss the eigen-properties of the corresponding preconditioned matrices. The additive block diagonal preconditioners can be employed to accelerate the convergence rates of Krylov subspace iteration methods such as MINRES and GMRES. Numerical experiments show that MINRES preconditioned by the exact and the inexact additive block diagonal preconditioners are effective, robust and scalable solvers for the block two-by-two linear systems arising from the Galerkin finite-element discretizations of a class of distributed control problems.     

Modified Newton’s method for systems of nonlinear equations with singular Jacobian

It is well known that Newton’s method for a nonlinear system has quadratic convergence when the Jacobian is a nonsingular matrix in a neighborhood of the solution. Here we present a modification of this method for nonlinear systems whose Jacobian matrix is singular. We prove, under certain conditions, that this modified Newton’s method has quadratic convergence. Moreover, different numerical tests confirm the theoretical results and allow us to compare this variant with the classical Newton’s method.     

Generalized shift-splitting iteration method for a class of two-by-two linear systems

In this paper, we propose a generalized-shift splitting (GSS) iteration method for solving a broad class of two-by-two linear systems. The convergence theory of the GSS iteration method is established and the spectral properties of the corresponding preconditioned matrix of some special choices for the parameter matrices are analyzed. In addition, the optimal choice of the iterative parameter is discussed. Numerical experiments are used to verify the validity of the theoretical results and the effectiveness of the new method.     

On nonsingularity of block two-by-two matrices

We derive necessary and sufficient conditions for guaranteeing the nonsingularity of a block two-by-two matrix by making use of the singular value decompositions and the Moore–Penrose pseudoinverses of the matrix blocks. These conditions are complete, and much weaker and simpler than those given by Decker and Keller [D.W. Decker, H.B. Keller, Multiple limit point bifurcation, J. Math. Anal. Appl. 75 (1980) 417–430], and may be more easily examined than those given by Bai [Z.-Z. Bai, Eigenvalue estimates for saddle point matrices of Hermitian and indefinite leading blocks, J. Comput. Appl. Math. 237 (2013) 295–306] from the computational viewpoint. We also derive general formulas for the rank of the block two-by-two matrix by utilizing either the unitarily compressed or the orthogonally projected sub-matrices.     

Jacobian elliptic function method for nonlinear differential-difference equations

An algorithm is devised to derive exact travelling wave solutions of differential-difference equations by means of Jacobian elliptic function. For illustration, we apply this method to solve the discrete nonlinear Schrödinger equation, the discretized mKdV lattice equation and the Hybrid lattice equation. Some explicit and exact travelling wave solutions such as Jacobian doubly periodic solutions, kink-type solitary wave solutions are constructed.     

Structured preconditioners for nonsingular matrices of block two-by-two structures

For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to efficient and high-quality preconditioning matrices for some typical matrices from the real-world applications.

     

A method of solving systems of nonlinear equations of large dimension with banded Jacobian matrix

We consider the concept of bandedness in a matrix. We study iterative methods of solving systems of linear and nonlinear equations of large dimension with a banded Jacobian matrix. We propose a method of computing the row submatrix of a certain full Jacobian matrix that is efficient in a certain sense for numerical differentiation of a vector-valued function of several arguments.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 21–24.     

An efficient multistep iteration scheme for systems of nonlinear algebraic equations associated with integral equations

The aim of this research is to present a new iterative procedure in approximating nonlinear system of algebraic equations with applications in integral equations as well as partial differential equations (PDEs). The presented scheme consists of several steps to reach a high rate of convergence and also an improved index of efficiency. The theoretical parts are furnished, and several computational tests mainly arising from practical problems are given to manifest its applicability.     

Robust additive block triangular preconditioners for block two-by-two linear systems

Numerical Algorithms - In this paper, a class of additive block triangular preconditioners are constructed for solving block two-by-two linear systems with symmetric positive (semi-)definite...     

Block by block method for the systems of nonlinear Volterra integral equations

The approach given in this paper leads to numerical methods for solving system of Volterra integral equations which avoid the need for special starting procedures. The method has also the advantages of simplicity of application and at least four order of convergence which is easy to achieve. Also, at each step we get four unknowns simultaneously. A convergence theorem is proved for the described method. Finally numerical examples presented to certify convergence and accuracy of the method.     

On symmetric block triangular splitting iteration method for a class of complex symmetric system of linear equations

For solving a class of complex symmetric linear system, we first transform the system into a block two-by-two real formulation and construct a symmetric block triangular splitting (SBTS) iteration method based on two splittings. Then, eigenvalues of iterative matrix are calculated, convergence conditions with relaxation parameter are derived, and two optimal parameters are obtained. Besides, we present the optimal convergence factor and test two numerical examples to confirm theoretical results and to verify the high performances of SBTS iteration method compared with two classical methods.     

A stable Richardson iteration method for complex linear systems

Summary The Richardson iteration method is conceptually simple, as well as easy to program and parallelize. This makes the method attractive for the solution of large linear systems of algebraic equations with matrices with complex eigenvalues. We change the ordering of the relaxation parameters of a Richardson iteration method proposed by Eiermann, Niethammer and Varga for the solution of such problems. The new method obtained is shown to be stable and to have better convergence properties.Research supported by the National Science Foundation under Grant DMS-8704196     

Some geometrical iteration methods for nonlinear equations

This paper describes geometrical essentials of some iteration methods （e.g. Newton iteration, secant line method, etc.） for solving nonlinear equations and advances some geometrical methods of iteration that are flexible and efficient.     

Variational iteration technique for solving nonlinear equations

In this paper, we use the variational iteration technique to suggest some new iterative methods for solving nonlinear equations f(x)=0. We also discuss the convergence criteria of these new iterative methods. Comparison with other similar methods is also given. These new methods can be considered as an alternative to the Newton method. We also give several examples to illustrate the efficiency of these methods. This technique can be used to suggest a wide class of new iterative methods for solving system of nonlinear equations.