Hua Dai

Generalized global conjugate gradient squared algorithm

In this paper, a generalized global conjugate gradient squared method for solving nonsymmetric linear systems with multiple right-hand sides is presented. The method can be derived by using products of two nearby global BiCG polynomials and formal orthogonal polynomials, of which global CGS and global BiCGSTAB are just particular cases. We also show to apply the method for solving the Sylvester matrix equation. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.

Stability analysis of time-delay systems using a contour integral method

In this paper we propose a contour integral method for computing the rightmost characteristic roots of systems of linear time-delay differential equations (DDEs). These roots are very important in the context of stability analysis of the time-delay systems. The effectiveness of the proposed method is illustrated by some numerical experiments.

Inverse problems for nonsymmetric matrices with a submatrix constraint

Abstract In this paper the following problems are considered: Problem I(a): Given matrices X ∈ R n × p with full column rank, B ∈ R p × p and A 0 ∈ R r × r , find a matrix A ∈ R n × n such that X T AX = B , A ( [ 1 , r ] ) = A 0 , where A ( [ 1 , r ] ) is the r × r leading principal submatrix of the matrix A. Problem I(b): Given matrices X ∈ R n × p , B ∈ R p × p and A 0 ∈ R r × r , find a matrix A ∈ R n × n such that ‖ X T AX - B ‖ = min s.t. A ( [ 1 , r ] ) = A 0 . Problem II: Given a matrix A ∼ ∈ R n × n with A ∼ ( [ 1 , r ] ) = A 0 , find A ^ ∈ S E such that ‖ A ∼ - A ^ ‖ = inf A ∈ S E ‖ A ∼ - A ‖ , where S E is the solution set of Problem I(a). By applying the generalized singular value decomposition (GSVD) and the canonical correlation decomposition (CCD) of a matrix pair, the solvability conditions for Problem I(a) and the general solution of Problem I are derived. The expression of the solution of Problem II is presented. A numerical algorithm for solving the problems is provided.

On the construction of a real symmetric five-diagonal matrix from its three eigenpairs☆

In this paper, an inverse eigenvalue problem of constructing a real symmetric five-diagonal matrix from its three eigenpairs is considered. The necessary and sufficient conditions for the existence and uniqueness of the solutions are derived. Three numerical algorithms and three numerical experiments are given.

An inverse eigenvalue problem for the finite element model of a vibrating rod

An inverse eigenvalue problem for the finite element model of a longitudinally vibrating rod whose one end is fixed and the other end is supported on a spring is considered. It is known that the mass and stiffness matrices are both tridiagonal for the finite element model of the rod based on linear shape functions. It is shown that the cross section areas can be determined from the spectrum of the rod. The inverse vibration problem can be recast into an inverse eigenvalue problem of a special Jacobi matrix. The necessary and sufficient conditions for the construction of a physically realizable rod with positive cross section areas are established. A numerical method is presented and an illustrative example is given.

An inverse eigenvalue problem for Jacobi matrix

The paper considers an inverse eigenvalue problem of Jacobi matrix which is obtained from reconstruction of a fixed-free mass-spring system of size 2 n from its spectrum and from existing physical parameters of the first half of the particles. The necessary and sufficient conditions for the solvability of the problem are derived. Two numerical algorithms and some numerical examples are given.

Inverse mode problems for the finite element model of a vibrating rod

The inverse mode problems for the finite element model of an axially vibrating rod are formulated and solved. It is known that for the finite element model, based on linear shape functions, of the rod, the mass and stiffness matrices are both tridiagonal. It is shown that the finite element model of the rod can be constructed from two eigenvalues, their corresponding eigenvectors and the total mass of the rod. The necessary and sufficient conditions for the construction of a physically realizable rod with positive mass and stiffness elements from two eigenpairs and the total mass of the rod are established. If these conditions are satisfied, then the construction of the model is unique.

A novel numerical method to determine the algebraic multiplicity of nonlinear eigenvalues

We generalize the algebraic multiplicity of the eigenvalues of nonlinear eigenvalue problems (NEPs) to the rational form and give the extension of the argument principle. In addition, we propose a novel numerical method to determine the algebraic multiplicity of the eigenvalues of the NEPs in a given region by the contour integral method. Finally, some numerical experiments are reported to illustrate the effectiveness of our method.

A new iterative method for solving complex symmetric linear systems

Based on implementation of the quasi-minimal residual (QMR) and biconjugate A-orthogonal residual (BiCOR) method, a new Krylov subspace method is presented for solving complex symmetric linear systems. The new method can be combined with arbitrary symmetric preconditioners. The preconditioned modified Hermitian and Skew-Hermitian splitting (PMHSS) preconditioner is used to accelerate the convergence rate of this method. Numerical experiments indicate that the proposed method and its preconditioned version are efficient and robust, in comparison with other Krylov subspace methods.