DIRECT PERTURBATION METHOD FOR REANALYSIS OF MATRIX SINGULAR VALUE DECOMPOSITION

The perturbational reanalysis technique of matrix singular value decomposition is applicable to many theoretical and practical problems in mathematics, mechanics, control theory, engineering, etc.. An indirect perturbation method has previously been proposed by the author in this journal, and now the direct perturbation method has also been presented in this paper. The second-order perturbation results of non-repeated singular values and the corresponding left and right singular vectors are obtained. The results can meet the general needs of most problems of various practical applications. A numerical example is presented to demonstrate the effectiveness of the direct perturbation method.

PERTURBATION METHOD FOR REANALYSIS OF THE MATRIX SINGULAR VALUE DECOMPOSITION

The perturbation method for the reanalysis of the singular value decomposition (SVD) of general real matrices is presented in this paper. This is a simple but efficient reanalysis technique for the SVD, which is of great worth to enhance computational efficiency of the iterative analysis problems that require matrix singular value decomposition repeatedly. The asymptotic estimate formulas for the singular values and the corresponding left and right singular vectors up to second-order perturbation components are derived. At the end of the paper the way to extend the perturbation method to the case of general complex matrices is advanced.

A Singular Values Based Newton Method for Linear Complementarity Problems

The existence condition of the solution of special nonlinear penalized equation of the linear complementarity problems is obtained by the relationship between penalized equations and an absolute value equation. Newton method is used to solve penalized equation, and then the solution of the linear complementarity problems is obtained. We show that the proposed method is globally and superlinearly convergent when the matrix of complementarity problems of its singular values exceeds 0;numerical results show that our proposed method is very effective and efficient.

Wavelet Multi-Resolution Interpolation Galerkin Method for Linear Singularly Perturbed Boundary Value Problems

In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be readily extended to special node generation techniques,such as the Shishkin node.Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients.All the shape functions possess the Kronecker delta property,making the imposition of boundary conditions as easy as that in the finite element method.Four numerical examples are studied to demonstrate the validity and accuracy of the proposedwavelet method.The results showthat the use ofmodified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer.Compared with many other methods,the proposed method possesses satisfactory accuracy and efficiency.The theoretical and numerical results demonstrate that the order of theε-uniform convergence of this wavelet method can reach 5.

A SINGULAR VALUE DECOMPOSITION BASED TRUNCATION ALGORITHM IN SOLVING THE STRUCTURAL DAMAGE EQUATIONS

The structural damage identification through modal data often leads to solving a set of linear equations. Special numerical treatment is sometimes required for an accurate and stable solution owing to the ill conditioning of the equations. Based on the singular value decomposition (SVD) of the coefficient matrix, an error based truncation algorithm is proposed in this paper. By rejection of selected small singular values, the influence of noise can be reduced. A simply-supported beam is used as a simulation example to compare the results to other methods. Illustrative numerical examples demonstrate the good efficiency and stability of the algorithm in the nondestructive identification of structural damage through modal data.

An Approximate Linear Solver in Least Square Support Vector Machine Using Randomized Singular Value Decomposition

In this paper, we investigate the linear solver in least square support vector machine（LSSVM） for large-scale data regression. The traditional methods using the direct solvers are costly. We know that the linear equations should be solved repeatedly for choosing appropriate parameters in LSSVM, so the key for speeding up LSSVM is to improve the method of solving the linear equations. We approximate large-scale kernel matrices and get the approximate solution of linear equations by using randomized singular value decomposition（randomized SVD）. Some data sets coming from University of California Irvine machine learning repository are used to perform the experiments. We find LSSVM based on randomized SVD is more accurate and less time-consuming in the case of large number of variables than the method based on Nystrom method or Lanczos process.

A Radial Basis Function Method with Improved Accuracy for Fourth Order Boundary Value Problems

Accurately approximating higher order derivatives is an inherently difficult problem. It is shown that a random variable shape parameter strategy can improve the accuracy of approximating higher order derivatives with Radial Basis Function methods. The method is used to solve fourth order boundary value problems. The use and location of ghost points are examined in order to enforce the extra boundary conditions that are necessary to make a fourth-order problem well posed. The use of ghost points versus solving an overdetermined linear system via least squares is studied. For a general fourth-order boundary value problem, the recommended approach is to either use one of two novel sets of ghost centers introduced here or else to use a least squares approach. When using either ghost centers or least squares, the random variable shape parameter strategy results in significantly better accuracy than when a constant shape parameter is used.

A SMALLEST SINGULAR VALUE METHOD FOR SOLVING INVERSE EIGENVALUE PROBLEMS

Utilizing the properties of the smallest singular value of a matrix, we propose a new, efficient and reliable algorithm for solving nonsymmetric matrix inverse eigenvalue problems, and compare it with a known method. We also present numerical experiments which illustrate our results.

MODIFIED LEAST SQUARE METHOD ON COMPUTING DIRICHLET PROBLEMS

The singularity theory of dynamical systems is linked to the numerical computation of boundary value problems of differential equations. It turns out to be a modified least square method for a calculation of variational problem defined on Ck(Ω), in which the base functions are polynomials and the computation of problems is transferred to compute the coefficients of the base functions. The theoretical treatment and some simple examples are provided for understanding the modification procedure of the methods. A modified least square method on difference scheme is introduced with a general matrix form of dynamical systems. We emphasize the simplicity of the algorithm and only use Euler algorithm to compute initial value problems of ODEs. A better algorithm is needed to reduce the stiffness of ODEs.

Extracting the interference components of normal modes in shallow water waveguide using singular value decomposition method

The normal mode interference characteristic in shallow water waveguide is a valu- able topic in the fields of underwater acoustic. A method for extracting the interference components of normal modes from broadband acoustic propagation data recorded by a single hy- drophone without any prior information is present in this paper. First, a Hermitian matrix is formed by the power spectral density. Second, a singular value decomposition （SVD） is performed on the Hermitian matrix to obtain the orthonormal eigenvectors, which are proportional to the interference components of normal modes. The fundamental equations of the new extracting method are derived based on normal mode and waveguide invariant theory. And the validity of the present method is verified by the numerical simulation and experimental results. In addition, the extracted results of normal-mode interference components are intended to be used for passive ranging of broadband sources.

Precise integration method for a class of singular two-point boundary value problems

In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the method of variable coefficient dimensional expanding,the non-homogeneous ordinary differential equations（ODEs） are transformed into homogeneous ODEs.Then the interval is divided evenly,and the transfer matrix in every subinterval is worked out using the high order multiple perturbation method,and a set of algebraic equations is given in the form of matrix by the precise integration relation for each segment,which is worked out by the reduction method.Finally numerical examples are elaboratedd to validate the present method.

ON THE SINGULARITY OF LEAST SQUARES ESTIMATOR FOR MEAN-REVERTING α-STABLE MOTIONS

We study the problem of parameter estimation for mean-reverting α-stable motion, dXt = （a0 - θ0Xt）dt ＋ dZt, observed at discrete time instants. A least squares estimator is obtained and its asymptotics is discussed in the singular case （a0, θ0） = （0, 0）. If a0 = 0, then the mean-reverting α-stable motion becomes Ornstein-Uhlenbeck process and is studied in [7] in the ergodic case θ0 〉 0. For the Ornstein-Uhlenbeck process, asymptotics of the least squares estimators for the singular case （θ0 = 0） and for ergodic case （θ0 〉 0） are completely different.

Mean Square Numerical Methods for Initial Value Random Differential Equations

In this paper, the random Euler and random Runge-Kutta of the second order methods are used in solving random differential initial value problems of first order. The conditions of the mean square convergence of the numerical solutions are studied. The statistical properties of the numerical solutions are computed through numerical case studies.

B-Spline Collocation Method for Solving Singularly Perturbed Boundary Value Problems

We use fifth order B-spline functions to construct the numerical method for solving singularly perturbed boundary value problems. We use B-spline collocation method, which leads to a tri-diagonal linear system. The accuracy of the proposed method is demonstrated by test problems. The numerical results are found in good agreement with exact solutions.

A TRUST-REGION METHOD FOR SOLVING TRUNCATED COMPLEX SINGULAR VALUE DECOMPOSITION

The truncated singular value decomposition has been widely used in many areas of science including engineering,and statistics,etc.In this paper,the original truncated complex singular value decomposition problem is formulated as a Riemannian optimiza-tion problem on a product of two complex Stiefel manifolds,a practical algorithm based on the generic Riemannian trust-region method of Absil et al.is presented to solve the underlying problem,which enjoys the global convergence and local superlinear conver-gence rate.Numerical experiments are provided to illustrate the efficiency of the proposed method.Comparisons with some classical Riemannian gradient-type methods,the existing Riemannian version of limited-memory BFGS algorithms in the MATLAB toolbox Manopt and the Riemannian manifold optimization library ROPTLIB,and some latest infeasible methods for solving manifold optimization problems,are also provided to show the merits of the proposed approach.

ON THE METHOD OF SOLUTION FOR A KIND OFNONLINEAR SINGULAR INTEGRAL EQUATION

The solutions of the nonlinear singular integral equation , t 6 L, are considered, where L is a closed contour in the complex plane, b ≠ 0 is a constant and f(t) is a polynomial. It is an extension of the results obtained in [1] when f(t) is a constant. Certain special cases are illustrated.

Properties of the total least squares estimation

Through theoretical derivation, some properties of the total least squares estimation are found. The total least squares estimation is the linear transformation of the least squares estimation, and the total least squares estimation is unbiased. The condition number of the total least squares estimation is greater than the least squares estimation, so the total least squares estimation is easier to be affected by the data error than the least squares estimation. Then through the further derivation, the relationships of solutions, residuals and unit weight variance estimations between the total least squares and the least squares are given.

MESHLESS ANALYSIS FOR THREE-DIMENSIONAL ELASTICITY WITH SINGULAR HYBRID BOUNDARY NODE METHOD

The singular hybrid boundary node method （SHBNM） is proposed for solving three-dimensional problems in linear elasticity. The SHBNM represents a coupling between the hybrid displacement variational formulations and moving least squares （MLS） approximation. The main idea is to reduce the dimensionality of the former and keep the meshless advantage of the later. The rigid movement method was employed to solve the hyper-singular integrations. The ＇boundary layer effect＇, which is the main drawback of the original Hybrid BNM, was overcome by an adaptive integration scheme. The source points of the fundamental solution were arranged directly on the boundary. Thus the uncertain scale factor taken in the regular hybrid boundary node method （RHBNM） can be avoided. Numerical examples for some 3D elastic problems were given to show the characteristics. The computation results obtained by the present method are in excellent agreement with the analytical solution. The parameters that influence the performance of this method were studied through the numerical examples.

Least-Squares Solutions of the Matrix Equation A^TXA=B Over Bisymmetric Matrices and its Optimal Approximation

A real n×n symmetric matrix X=(x_(ij))_(n×n)is called a bisymmetric matrix if x_(ij)=x_(n+1-j,n+1-i).Based on the projection theorem,the canonical correlation de- composition and the generalized singular value decomposition,a method useful for finding the least-squares solutions of the matrix equation A^TXA=B over bisymmetric matrices is proposed.The expression of the least-squares solutions is given.Moreover, in the corresponding solution set,the optimal approximate solution to a given matrix is also derived.A numerical algorithm for finding the optimal approximate solution is also described.

THE NESTED TOTAL LEAST SQUARES PROBLEM:FORMULATION,SOLUTIONS AND PERTURBATION ANALYSIS

In this paper, we propose the nested totoal least squatres problem (NTLS), which is an extension of the equality constrained least squares problem (LSE). The formulation of the NTLS problem is given and the solution set of the NTLS problem is obtained. The least squares residuals and the minimal norm correction matrices of the NTLS solution are provided and a perturbation analysis of the NTLS solutions is given.