Hua Dai

Nanjing University of Aeronautics & Astronautics · Department of Mathematics

This paper deals with the problem of updating simultaneously mass and stiffness matrices. The desired matrix properties, including satisfaction of the characteristic equation, symmetry, positive semidefiniteness, and boundedness, are imposed as side constraints to form the matrix pencil minimization problem. This problem is related to finite elemen...

We propose a novel projected gradient-like method to solve a minimization problem with simple constraints. The new search direction is consistent with the first-order optimality condition for the simple constrained optimization problem, and makes the search step-size easier to determine. Convergence properties of the method are analyzed. Numerical...

The global Krylov subspace iterative methods are an attractive class of iterative solvers for solving linear systems with several right-hand sides. In this paper, the global version of the GMRES method is applied to solve linear discrete ill-posed problems that arise from the discretization of linear ill-posed problems, pattern classification and d...

Locality preserving projection (LPP), as a well-known technique for dimensionality reduction, is designed to preserve the local structure of the original samples which usually lie on a low-dimensional manifold in the real world. However, it suffers from the undersampled or small-sample-size problem, when the dimension of the features is larger than...

Sparsity preserving projection (SPP), as a widely used linear unsupervised dimensionality reduction (DR) method, is designed to preserve the sparse reconstructive relationship of the raw data. SPP constructs an affinity weight matrix by solving a sparse representation model which does not need any parameters. Moreover, the obtained projection may c...

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)...

Using the equivalent block two-by-two real linear systems and relaxing technique, we establish a new block preconditioner for a class of complex symmetric indefinite linear systems. The new preconditioner is much closer to the original block two-by-two coefficient matrix than the Hermitian and skew-Hermitian splitting (HSS) preconditioner. We analy...

In this paper we consider the computation of some eigenpairs with smallest eigenvalues in modulus of large-scale polynomial eigenvalue problem. Recently, a partially orthogonal projection method and its refinement scheme were presented for solving the polynomial eigenvalue problem. The methods preserve the structures and properties of the original...

Based on the splitting-based block preconditioners presented by Yan and Huang (2014), a class of inexact splitting-based block preconditioners are constructed. Then, computing exactly inversion of the matrix αW+T can be avoided for solving the linear subsystem during the preconditioning process. Spectral properties of the preconditioned matrices ar...

A quadratic inverse eigenvalue problem arising from damped structural model updating is presented in this paper. The existence of solution to the problem is analyzed. A new model updating method for damped structural systems is proposed. The new approach can update directly the damped model using the spatially incomplete modes without using model r...

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.

In this paper, we consider preconditioned simplified Hermitian normal splitting (PSHNS) iteration method for solving complex symmetric indefinite linear systems, analyze the convergence of the PSHNS iteration method and discuss the spectral properties of the PSHNS preconditioned matrix. Using discrete Sine transform (DST), we apply a fast algorithm...

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,...

In this paper, we propose a successive mth (m≥ 2) approximation method for the nonlinear eigenvalue problem (NEP) and analyze its local convergence. Applying the partially orthogonal projection method to the successive mth approximation problem, we present the partially orthogonal projection method with the successive mth approximation for solving...

This paper considers the problem of finding the least change adjustment to a given matrix pencil. The desired matrix properties, including satisfaction of the characteristic equation, symmetry, positive semidefiniteness, and sparsity, are imposed as side constraints to form the optimal matrix pencil approximation problem. This problem is related to...

We consider computation of the derivatives of the semisimple eigenvalues and corresponding eigenvectors of a symmetric quadratic eigenvalue problem. Using the normalization condition, we can compute the derivatives of the differentiable eigenvalues of the quadratic eigenvalue problem. Using the constrained generalized inverse, we present an efficie...

We discuss the parameterized generalized inverse eigenvalue problem (PGIEP): For given matrices Ai, Bi ∈ Cn×n (i = 0, 1,⋯, n), find complex numbers ci ∈ C (i = 1, 2,⋯,n) such that the generalized eigenvalue problem (A0 + Σni=1 ciAi)x = λ(B0 + Σni=1 ciBi)x has the prescribed eigenvalues λ1, λ2,⋯, λn. We show that this problem is equivalent to a mult...

Recently, the conjugate A-orthogonal residual squared (CORS) method is often competitive and superior to other Krylov subspace methods for solving complex non-Hermitian linear systems from many realistic problems. However, like the conjugate gradient squared (CGS) method, the CORS method often exhibits irregular convergence behavior with wild oscil...

Recently, some novel variants of Lanczos-type methods were explored that are based on the biconjugate A-orthonormalization process. Numerical experiments coming from some physical problems indicate that these new methods are competitive with or superior to other popular Krylov subspace methods. Among them, the biconjugate A-orthogonal residual (BiC...

An algorithm is derived for the computation of eigenpair derivatives of asymmetric quadratic eigenvalue problem with distinct and repeated eigenvalues. In the proposed method, the eigenvector derivatives of the damped systems are divided into a particular solution and a homogeneous solution. By introducing an additional normalization condition, we...

The paper considers an inverse eigenvalue problem of Jacobi matrix which is obtained from reconstruction of a fixed-free mass-spring system of size 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 algorith...

An efficient algorithm is derived for computation of eigenvalue and eigenvector derivatives of symmetric nonviscously damped systems with repeated eigenvalues. In the proposed method, the mode shape derivatives of the nonviscously damped systems are divided into a particular solution and a homogeneous solution. A simplified method is given to calcu...

The global BiCGSTAB (Gl-BiCGSTAB) method has been found to perform very well in many practical cases. However, like the BiCGSTAB method, it often converges slowly or stagnates for some non-symmetric linear systems with complex spectrum. In this paper, we propose a new global generalized product-type method based on the global BiCG (Gl-BCG) method t...

This paper concerns the eigenvalue embedding problem of undamped gyroscopic systems. Based on a low-rank correction form, the approach moves the unwanted eigenvalues to desired values and the remaining large number eigenvalues and eigenvectors of the original system do not change. In addition, the symmetric structure of mass and stiffness matrices...

In this paper, we consider computing the derivatives of the semisimple eigenvalues and corresponding eigenvectors of symmetric quadratic eigenvalue problem. In the proposed method, the eigenvector derivatives of the symmetric quadratic eigenvalue problem are divided into a particular solution and a homogeneous solution; a simplified method is given...

Recently numerous numerical experiments on realistic calculation have shown that the conjugate A-orthogonal residual squared (CORS) method is often competitive with other popular methods. However, the CORS method, like the CGS method, shows irregular convergence, especially appears large intermediate residual norm, which may lead to worse approxima...

Updating an existing but inaccurate structural dynamics model with measured data can be mathematically reduced to the problem of the best approximation to a given matrix pencil in the Frobenius norm under a given spectral constraint and a submatrix pencil constraint. In this paper, a direct method and the associated mathematical theories for solvin...

The partial eigenvalue assignment problem concerns reassigning a few of undesired eigenvalues of a linear system to suitably chosen locations and keeping the other large number of eigenvalues and eigenvectors unchanged (no spill-over). This paper considers the partial eigenvalue assignment problem with time delay robustness. A time delay robustness...

In this paper, we consider the partial eigenvalue assignment problem (PEAP) of high order linear system by state feedback. To prevent spill-over phenomenon, we establish the new orthogonality relations for the eigenvectors of matrix polynomial such that the unwanted eigenvalues are moved to prescribed values and the wanted eigenvalues remain unchan...

This paper considers the condition numbers of a nondefective multiple eigenvalue of a nonsymmetric matrix pencil. Based on the directional derivatives of a nondefective multiple eigenvalue of a nonsymmetric matrix pencil analytically dependent on several parameters, different condition numbers of a nondefective multiple eigenvalue are introduced. T...

Updating a finite element model to match measured spectral information has been a important task for engineers. In the process, it is desirable to match only the measured spectral information without tampering with the other unmeasured and unknown eigeninformation in the original model and to maintain positive definiteness (semidefiniteness) in the...

Linear undamped gyroscopic systems are defined by three real matrices, M>0,K>0, and G(G^T=-G); the mass, stiffness, and gyroscopic matrices, respectively. In this paper an inverse problem is considered: given complete information about eigenvalues and eigenvectors, @L=diag{@l"1,@l"2,...,@l"2"n"-"1,@l"2"n}@?C^2^n^x^2^n and X=[x"1,x"2,...,x"2"n"-"1,x...

The global bi-conjugate gradient (Gl-BCG) method is an attractive matrix Krylov subspace method for solving nonsymmetric linear systems with multiple right-hand sides, but it often show irregular convergence behavior in many applications. In this paper, we present a new family of global A-biorthogonal methods by using short two-term recurrences and...

Given n+1 pairs of complex numbers and vectors (closed under complex conjugation), the inverse quadratic eigenvalue problem is to construct real symmetric or anti-symmetric matrix C and real symmetric matrix K of size n×n so that the quadratic pencil Q(λ)=λ2In+λC+K has the given n+1 pairs as eigenpairs. Necessary and sufficient conditions under whi...

We derive new and tight bounds about the eigenvalues and certain sums of the eigenvalues for the unique symmetric positive definite solutions of the discrete algebraic Riccati equations. These bounds considerably improve the existing ones and treat the cases that have not been discussed in literatures. Besides, they also result in completions for t...

In this paper, we first give the representation of the general solution of the following inverse monic quadratic eigenvalue problem (IMQEP): given matrices Λ=diag{λ1,…,λp}∈Cp×p, λi≠λj for i≠j, i,j=1,…,p, X=[x1,…,xp]∈Cn×p, rank(X)=p, and both Λ and X are closed under complex conjugation in the sense that λ2j=λ̄2j−1∈C, x2j=x̄2j−1∈Cn for j=1,…,l, and...

This paper deals with the normwise perturbation theory for linear (Hermitian) matrix equations. The definition of condition number for the linear (Hermitian) matrix equations is presented. The lower and upper bounds for the condition number are derived. The estimation for the optimal backward perturbation bound for the Hermitian matrix equations is...

The block Davidson method is a preconditioned variant of the block Lanczos method for solving large symmetric eigenvalue problems. In order to accelerate the convergence of the block Davidson method, we combine the block Chebyshev iteration with the block Davidson method and present the block Chebyshev-Davidson method with deflation for computing t...

The numerical solutions for eigenvalue problems of gyroscopic systems are considered in this paper. By using Lanczos algorithm for skew-symmetric matrices, the second-order Lanczos method for solving eigenvalue problems of gyroscopic systems is presented. Based on the proposed non-equivalence low-rank deflation technique for eigenvalue problems of...

We study the smooth LU decomposition of a given analytic functional λ-matrix A(λ) and its block-analogue. Sufficient conditions for the existence of such matrix decompositions are given, some differentiability about certain elements arising from them are proved, and several explicit expressions for derivatives of the specified elements are provided...

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 sho...

In this paper we develop an efficient numerical method for the finite element model updating of damped gyroscopic systems. This model updating of damped gyroscopic systems is proposed to incorporate the measured modal data into the finite element model to produce an adjusted finite element model on the damping and gyroscopic matrices that closely m...

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,...

Updating an existing but inaccurate structural dynamics model with measured data can be mathematically reduced to the problem of the best approximation to a given matrix pencil in the Frobenius norm under a given spectral constraint and a submatrix pencil constraint. In this paper, a direct method and the associated mathematical theories for solvin...

New methods are presented for computing the derivatives of multiple eigenvalues and the corresponding eigenvectors of unsymmetrical quadratic eigenvalue problems. The expressions of eigenpair derivatives are derived in terms of the eigenvalues and eigenvectors of quadratic eigenvalue problems, and the use of rather undesirable state-space represent...

Let R∈Cm×m and S∈Cn×n be nontrivial unitary involutions, i.e., RH=R=R−1≠Im and SH=S=S−1≠In. We say that G∈Cm×n is a generalized reflexive matrix if RGS=G. The set of all m×n generalized reflexive matrices is denoted by GRCm×n. In this paper, a sufficient and necessary condition for the matrix equation AXB=D, where A∈Cp×m,B∈Cn×q and D∈Cp×q, to have...

This paper gives a method of parallel block Jacobi-Davidson for computing large generalized eigenvalue problem AX=λBX, in which matrix A and B is symmetric. The large eigenvalue problem is transformed into an eigenvalue problem in a low dimension subspace by using orthogonal projection technique, and the Neumann series is used in the correction equ...

In this paper, we first give the representation of the general solution of the following least-squares problem (LSP): given a matrix X∈Rn×p and symmetric matrices B∈Rp×p, A0∈Rr×r, find an n×n symmetric matrix A such that ∥XTAX-B∥=min,s.t.A([1,r])=A0, where A([1,r]) is the r×r leading principal submatrix of the matrix A. We then consider a best appr...

A parallel block Davidson method is presented for solving extreme eigenpairs of large sparse symmetric matrix based on PC network parallel environment and shared memory parallel environment. The row blocks of matrix A are distributed on each processor. The individual processors run under the control of the program based on the orthogonal basis of p...

In this paper the following problems are considered:Problem I(a):Given matrices X∈Rn×p with full column rank, B∈Rp×p and A0∈Rr×r, find a matrix A∈Rn×n such thatXTAX=B,A([1,r])=A0, where A([1,r]) is the r×r leading principal submatrix of the matrix A.Problem I(b):Given matrices X∈Rn×p, B∈Rp×p and A0∈Rr×r, find a matrix A∈Rn×n such that‖XTAX-B‖=mins....

A procedure is presented for computing the derivatives of repeated eigenvalues and the corresponding eigenvectors of damped systems. The derivatives are calculated in terms of the eigenvalues and eigenvectors of the second-order system, and the use of rather undesirable state space representation is avoided. Hence the cost of computation is greatly...

This paper is concerned with the following problems:Problem I(a). Given a full column rank matrix X∈Rn×p and symmetric matrices B∈Rp×p and A0∈Rr×r, find an n×n symmetric matrix A such thatXTAX=B,A([1,r])=A0, where A([1,r]) is the r×r leading principal submatrix of the matrix A.Problem I(b). Given a matrix X∈Rn×p and symmetric matrices B∈Rp×p, A0∈Rr...

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.

In this paper an interesting property of the restarted FOM algorithm for solving large nonsymmetric linear systems is presented and studied. By establishing a relationship between the convergence of its residual vectors and the convergence of Ritz values in the Arnoldi procedure, it is shown that some important information of previous FOM(m) cycles...

A kind of inverse eigenvalue problem in structural dynamics design is considered. The problem is formulated as an optimization problem. The properties of this problem are analyzed, and the existence of the optimum solution is proved. The directional derivative of the objective function is obtained and a necessary condition for a point to be a local...

Least-squares solution for the inverse problem of real matrices with a submatrix constraint is proposed. The expression of general solution is given. The best approximation to a given matrix is considered. The existence and uniqueness of the optimal approximation are proved. A numerical method for finding the optimal approximation is included. Thes...

A method based on the Davidson method for solving eigenvalue problems is presented to compute eigenpair derivatives. This algorithms calculates eigenpairs and their derivatives simultaneously. The systems of equations for solving eigenvector derivatives can be greatly reduced from the original matrix sizes. This algorithm can be very useful for the...

Let (λ, x) be an eigenpair of a simply connected spring-mass system. Suppose that a simple oscillator of mass m and/or spring k is attached to one end of the system and let (μ, y) be an eigenpair of the modified system. Three classes of the inverse mode problems for constructing the physical elements of the system from (λ, x), (μ, y), m and/or k ar...

Based on Davidson method for solving generalized eigenvalue problems, a new method for synchro calculation of eigenpairs and their partial derivatives of generalized eigenvalue problems is presented. Eigenpairs and their partial derivatives are computed simultaneously. The equation systems that are solved for eigenvector partial derivatives can be...

Derivatives of eigenvalues and eigenvectors with respect to parameters in symmetric quadratic eigenvalue problem are studied. The first and second order deriva- tives of eigenpairs are given. The derivatives are calculated in terms of the eigenvalues and eigenvectors of the quadratic eigenvalue problem, and the use of state space repre- sentation i...

Let P be an n x n symmetric orthogonal matrix. A real n x n matrix A is called P-symmetric nonnegative definite if A is symmetric nonnegative definite and (PA)(T) = PA. This paper is concerned with a kind of inverse problem for P-symmetric nonnegative definite matrices: Given a real n x n matrix, real n x m matrices X and B, find an n x n P-symmetr...

This paper considers the sensitivity of semisimple multiple eigenvalues and corresponding generalized eigenvector matrices of a nonsymmetric matrix pencil analytically dependent on several parameters. The directional derivatives of the multiple eigenvalues are obtained, and the average of eigenvalues and corresponding generalized eigenvector matric...