The bounds for the eigenvalues of the stiffness matrices in the finite element discretization corresponding to $Lu := - u'' $ with zero boundary conditions by quadratic hierarchical basis are shown explicitly. The condition number of the resulting system behaves like $O(\frac{1}{h})$ where $h$ is the mesh size. We also analyze a main diagonal preconditioner of the stiffness matrix which reduces the condition number of the preconditioned system to $O(1)$.

AD (Alternating direction) Galerkin schemes for $d$-dimensional nonlinear pseudo-hyperbolic equations are studied. By using patch approximation technique, AD procedure is realized, and calculation work is simplified. By using Galerkin approach, highly computational accuracy is kept. By using various priori estimate techniques for differential equations, difficulty coming from non-linearity is treated, and optimal $H^1$ and $L^2$ convergence properties are demonstrated. Moreover, although all the existed AD Galerkin schemes using patch approximation are limited to have only one order accuracy in time increment, yet the schemes formulated in this paper have second order accuracy in it. This implies an essential advancement in AD Galerkin analysis.

We analyze three one parameter families of approximations and show that they are symplectic in Lagrangian sense and can be related to symplectic schemes in Hamiltonian sense by different symplectic mappings. We also give a direct generalization of Veselov variational principle for construction of scheme of higher order differential equations. At last, we present numerical experiments.

In this paper the uniform convergence of Hermite-Fejér interpolation and Grünwald type theorem of higher order on an arbitrary system of nodes are presented.

 A new kind of matrix-valued rational interpolants is recursively established by means of generalized Samelson inverse for matrices, with scalar numerator and matrix-valued denominator. In this respect, it is essentially different from that of the previous works [7, 9], where the matrix-valued rational interpolants is in Thiele-type continued fraction form with matrix-valued numerator and scalar denominator. For both univariate and bivariate cases, sufficient conditions for existence, characterisation and uniqueness in some sense are proved respectively, and an error formula for the univariate interpolating function is also given. The results obtained in this paper are illustrated with some numerical examples.

 This paper is mainly concerned with solving the following two problems:
Problem Ⅰ. Given $X\in R^{n\times m},B\in R^{m\times m}$. Find $A\in P_n$ such that $$\|X^TAX-B\|_F=\min,$$ where $P_n=\{A\in R^{n\times n}| x^TAx\geq 0, \forall\,x\in R^n\}$.
Problem Ⅱ. Given $\widetilde{A}\in R^{n\times n}.$ Find $\widetilde{A}\in S_E$ such that $$\|\widetilde{A}-\hat{A}\|_F=\min_{A\in S_E}\|\widetilde{A}-A\|_F,$$ where $\|\cdot\|_F$ is Frobenius norm, and $S_E$ denotes the solution set of Problem I.
The general solution of problem I has been given. It is proved that there exists a unique solution for Problem II. The expression of this solution for corresponding Problem II for some special case will be derived.

Least-squares solution of $AXB = D$ with respect to symmetric positive semidefinite matrix $X$ is considered. By making use of the generalized singular value decomposition, we derive general analytic formulas, and present necessary and sufficient conditions for guaranteeing the existence of the solution. By applying MATLAB 5.2, we give some numerical examples to show the feasibility and accuracy of this construction technique in the finite precision arithmetic.

 In this paper, the concept of generalized Nekrasov matrices is introduced, some properties of these matrices are discussed, obtained equivalent representation of generalized diagonally dominant matrices.

 The convergence problem of the family of Euler-Halley methods is considered under the Lipschitz condition with the $L$-average, and a united convergence theory with its applications is presented.

 In this note, we consider the backward errors for more general inverse eigenvalue problems by extending Sun's approach. The optimal backward errors are defined for diagonalization matrix inverse eigenvalue problem with respect to an approximate solution, and the upper and lower bounds are derived for the optimal backward errors. The results may be useful for testing the stability of practical algorithms.

This paper presents a new trust-region algorithm for $n$-dimension nonlinear optimization subject to $m$ nonlinear inequality constraints. Equivalent KKT conditions are derived, which is the basis for constructing the new algorithm. Global convergence of the trial steps, local quadratic convergence theorem is proved for nondegenerate minimizer point. Numerical experiment is presented to show the effectiveness of our approach.

Trust region (TR) algorithms are a class of recently developed algorithms for nonlinear optimization. A new family of TR algorithms for unconstrained optimization, which is the extension of the usual TR method, is presented in this paper. When the objective function is bounded below and continuously differentiable, and the norm of the Hesse approximations increases at most linearly with the iteration number, we prove the global convergence of the algorithms. Limited numerical results are reported, which indicate that our new TR algorithm is competitive.

In this paper we propose a self-adaptive trust region algorithm. The trust region radius is updated at a variable rate according to the ratio between the actual reduction and the predicted reduction of the objective function, rather than by simply enlarging or reducing the original trust region radius at a constant rate. We show that this new algorithm preserves the strong convergence property of traditional trust region methods. Numerical results are also presented.

In this paper we present a nonmonotone trust region algorithm for general nonlinear constrained optimization problems. The main idea of this paper is to combine Yuan's technique [1] with a nonmonotone method similar to Ke and Han [2]. This new algorithm may not only keep the robust properties of the algorithm given by Yuan, but also have some advantages led by the nonmonotone technique. Under very mild conditions, global convergence for the algorithm is given. Numerical experiments demonstrate tre efficiency of the algorithm.

A new algorithm for inequality constrained optimization is presented, which solves a linear programming subproblem and a quadratic subproblem at each iteration. The algorithm can circumvent the difficulties associated with the possible inconsistency of $QP$ subproblem of the original $SQP$ method. Moreover, the algorithm can converge to a point which satisfies a certain first-order necessary condition even if the original problem is itself infeasible. Under certain condition, some global convergence results are proved and local superlinear convergence results are also obtained. Preliminary numerical results are reported.