The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the bas...The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the basic function and of the weight function,and is mainly determined by that of the weight function.Therefore,the weight function greatly affects the accuracy of results obtained.Different kinds of weight functions,such as the spline function, the Gauss function and so on,are proposed recently by many researchers.In the present work,the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method.The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed.Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and α in Gauss and exponential weight functions are in the range of reasonable values,respectively,and the higher the smoothness of the weight function,the better the features of the solutions.展开更多
This paper presents a new approach to the structural topology optimization of continuum structures. Material-point independent variables are presented to illustrate the existence condition,or inexistence of the materi...This paper presents a new approach to the structural topology optimization of continuum structures. Material-point independent variables are presented to illustrate the existence condition,or inexistence of the material points and their vicinity instead of elements or nodes in popular topology optimization methods. Topological variables field is constructed by moving least square approximation which is used as a shape function in the meshless method. Combined with finite element analyses,not only checkerboard patterns and mesh-dependence phenomena are overcome by this continuous and smooth topological variables field,but also the locations and numbers of topological variables can be arbitrary. Parameters including the number of quadrature points,scaling parameter,weight function and so on upon optimum topological configurations are discussed. Two classic topology optimization problems are solved successfully by the proposed method. The method is found robust and no numerical instabilities are found with proper parameters.展开更多
In this paper an introduction of the moving least squares approach is presented in the context of data approximation and interpolation problems in Geodesy.An application of this method is presented for geoid height ap...In this paper an introduction of the moving least squares approach is presented in the context of data approximation and interpolation problems in Geodesy.An application of this method is presented for geoid height approximation and interpolation using different polynomial basis functions for the approximant and interpolant,respectively,in a regular grid of geoid height data in the region 16.0417°≤φ≤47.9583°and 36.0417°≤λ≤69.9582°,with increment 0.0833°in both latitudal and longitudal directions.The results of approximation and interpolation are then compared with the geoid height data from GPS-Levelling approach.Using the standard deviation of the difference of the results,it is shown that the planar interpolant,with reciprocal of distance as weight function,is the best choice in this local approximation and interpolation problem.展开更多
Based on the moving least square （MLS） approximations and the boundary integral equations （BIEs）, a meshless algorithm is presented in this paper for elliptic Signorini problems. In the algorithm, a projection ope...Based on the moving least square （MLS） approximations and the boundary integral equations （BIEs）, a meshless algorithm is presented in this paper for elliptic Signorini problems. In the algorithm, a projection operator is used to tackle the nonlinear boundary inequality conditions. The Signorini problem is then reformulated as BIEs and the unknown boundary variables are approximated by the MLS approximations. Accordingly, only a nodal data structure on the boundary of a domain is required. The convergence of the algorithm is proven. Numerical examples are given to show the high convergence rate and high computational efficiency of the presented algorithm.展开更多
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 val...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.展开更多
In order to get an approximation with better effect of pararneterization of Bezier curves, we proposed a method for arc-length parameterization and the corresponding algorithms by square approximation for the discrete...In order to get an approximation with better effect of pararneterization of Bezier curves, we proposed a method for arc-length parameterization and the corresponding algorithms by square approximation for the discrete even de-parameterization of the curves. This method is simple and easy to implement, and the property of the approximation has no change compared with the original curve. A quantitative criterion for estimating the effect of parameterization is also built to quantitatively characterize the parameterization effect of the algorithms. As a result, the nearly arc-length parameterized curve has a smaller relative deviation using either the algorithm with point constraint at endpoints or the algorithm with point constraint plus the first derivative constraint at endpoints. Experiments show that after re-parameterization with our algorithms, the relative deviation will have at least a 20% reduction.展开更多
An improved moving least square meshless method is developed for the numerical solution of the nonlinear improved Boussinesq equation. After the approximation of temporal derivatives, nonlinear systems of discrete alg...An improved moving least square meshless method is developed for the numerical solution of the nonlinear improved Boussinesq equation. After the approximation of temporal derivatives, nonlinear systems of discrete algebraic equations are established and are solved by an iterative algorithm. Convergence of the iterative algorithm is discussed. Shifted and scaled basis functions are incorporated into the method to guarantee convergence and stability of numerical results. Numerical examples are presented to demonstrate the high convergence rate and high computational accuracy of the method.展开更多
A nonlinear problem of mean-square approximation of a real nonnegative continuous function with respect to two variables by the modulus of double Fourier integral dependent on two real parameters with use of the smoot...A nonlinear problem of mean-square approximation of a real nonnegative continuous function with respect to two variables by the modulus of double Fourier integral dependent on two real parameters with use of the smoothing functional is studied. Finding the optimal solutions of this problem is reduced to solution of the Hammerstein type two-dimensional nonlinear integral equation. The numerical algorithms to find the branching lines and branching-off solutions of this equation are constructed and justified. Numerical examples are presented.展开更多
In this article, some properties of matrices of moving least-squares approximation have been proven. The used technique is based on known inequalities for singular-values of matrices. Some inequalities for the norm of...In this article, some properties of matrices of moving least-squares approximation have been proven. The used technique is based on known inequalities for singular-values of matrices. Some inequalities for the norm of coefficients-vector of the linear approximation have been proven.展开更多
Curvature estimation is a basic step in many point relative applications such as feature recognition, segmentation,shape analysis and simplification.This paper proposes a moving-least square(MLS) surface based method ...Curvature estimation is a basic step in many point relative applications such as feature recognition, segmentation,shape analysis and simplification.This paper proposes a moving-least square(MLS) surface based method to evaluate curvatures for unorganized point cloud data.First a variation of the projection based MLS surface is adopted as the underlying representation of the input points.A set of equations for geometric analysis are derived from the implicit definition of the MLS surface.These equations are then used to compute curvatures of the surface.Moreover,an empirical formula for determining the appropriate Gaussian factor is presented to improve the accuracy of curvature estimation.The proposed method is tested on several sets of synthetic and real data.The results demonstrate that the MLS surface based method can faithfully and efficiently estimate curvatures and reflect subtle curvature variations.The comparisons with other curvature computation algorithms also show that the presented method performs well when handling noisy data and dense points with complex shapes.展开更多
Extracting approximate symmetry planes is a challenge due to the difficulty of accurately measuring numerical values. Introducing the approximate symmetry planes of a 3D point set, this paper presents a new method by ...Extracting approximate symmetry planes is a challenge due to the difficulty of accurately measuring numerical values. Introducing the approximate symmetry planes of a 3D point set, this paper presents a new method by gathering normal vectors of potential of the planes, clustering the high probability ones, and then testing and verifying the planes. An experiment showed that the method is effective, robust and universal for extracting the complete approximate planes of symmetry of a random 3D point set.展开更多
The neural network partial least square (NNPLS) method was used to establish a robust reaction model for a multi-component catalyst of methane oxidative coupling. The details, including the learning algorithm, the num...The neural network partial least square (NNPLS) method was used to establish a robust reaction model for a multi-component catalyst of methane oxidative coupling. The details, including the learning algorithm, the number of hidden units of the inner network, activation function, initialization of the network weights and the principal components, are discussed. The results show that the structural organizations of inner neural network are 1-10-5-1, 1-8-4-1, 1-8-5-1, 1-7-4-1, 1-8-4-1, 1-8-6-1, respectively. The Levenberg-Marquardt method was used in the learning algorithm, and the central sigmoidal function is the activation function. Calculation results show that four principal components are convenient in the use of the multi-component catalyst modeling of methane oxidative coupling. Therefore a robust reaction model expressed by NNPLS succeeds in correlating the relations between elements in catalyst and catalytic reaction results. Compared with the direct network modeling, NNPLS model can be adjusted by experimental data and the calculation of the model is simpler and faster than that of the direct network model.展开更多
General neural network inverse adaptive controller has two flaws: the first is the slow convergence speed; the second is the invalidation to the non-minimum phase system. These defects limit the scope in which the neu...General neural network inverse adaptive controller has two flaws: the first is the slow convergence speed; the second is the invalidation to the non-minimum phase system. These defects limit the scope in which the neural network inverse adaptive controller is used. We employ Davidon least squares in training the multi-layer feedforward neural network used in approximating the inverse model of plant to expedite the convergence, and then through constructing the pseudo-plant, a neural network inverse adaptive controller is put forward which is still effective to the nonlinear non-minimum phase system. The simulation results show the validity of this scheme.展开更多
A relaxation least squares-based learning algorithm for neual networks is proposed. Not only does it have a fast convergence rate, but it involves less computation quantity. Therefore, it is suitable to deal with the ...A relaxation least squares-based learning algorithm for neual networks is proposed. Not only does it have a fast convergence rate, but it involves less computation quantity. Therefore, it is suitable to deal with the case when a network has a large scale but the number of training data is very limited. It has been used in converting furnace process modelling, and impressive result has been obtained.展开更多
We study iterative processes of stochastic approximation for finding fixed points of weakly contractive and nonexpansive operators in Hilbert spaces under the condition that operators are given with random errors. We ...We study iterative processes of stochastic approximation for finding fixed points of weakly contractive and nonexpansive operators in Hilbert spaces under the condition that operators are given with random errors. We prove mean square convergence and convergence almost sure (a.s.) of iterative approximations and establish both asymptotic and nonasymptotic estimates of the convergence rate in degenerate and non-degenerate cases. Previously the stochastic approximation algorithms were studied mainly for optimization problems.展开更多
In this paper, least-squaxes mirrorsymmetric solution for matrix equations (AX = B, XC = D) and its optimal approximation is considered. With special expression of mirrorsymmetric matrices, a general representation of...In this paper, least-squaxes mirrorsymmetric solution for matrix equations (AX = B, XC = D) and its optimal approximation is considered. With special expression of mirrorsymmetric matrices, a general representation of solution for the least-squares problem is obtained. In addition, the optimal approximate solution and some algorithms to obtain the optimal approximation are provided.展开更多
Least squares projection twin support vector machine(LSPTSVM)has faster computing speed than classical least squares support vector machine(LSSVM).However,LSPTSVM is sensitive to outliers and its solution lacks sparsi...Least squares projection twin support vector machine(LSPTSVM)has faster computing speed than classical least squares support vector machine(LSSVM).However,LSPTSVM is sensitive to outliers and its solution lacks sparsity.Therefore,it is difficult for LSPTSVM to process large-scale datasets with outliers.In this paper,we propose a robust LSPTSVM model(called R-LSPTSVM)by applying truncated least squares loss function.The robustness of R-LSPTSVM is proved from a weighted perspective.Furthermore,we obtain the sparse solution of R-LSPTSVM by using the pivoting Cholesky factorization method in primal space.Finally,the sparse R-LSPTSVM algorithm(SR-LSPTSVM)is proposed.Experimental results show that SR-LSPTSVM is insensitive to outliers and can deal with large-scale datasets fastly.展开更多
This note explores the relations between two different methods. The first one is the Alternating Least Squares (ALS) method for calculating a rank<em>-k</em> approximation of a real <em>m</em>&...This note explores the relations between two different methods. The first one is the Alternating Least Squares (ALS) method for calculating a rank<em>-k</em> approximation of a real <em>m</em>×<em>n</em> matrix, <em>A</em>. This method has important applications in nonnegative matrix factorizations, in matrix completion problems, and in tensor approximations. The second method is called Orthogonal Iterations. Other names of this method are Subspace Iterations, Simultaneous Iterations, and block-Power method. Given a real symmetric matrix, <em>G</em>, this method computes<em> k</em> dominant eigenvectors of <em>G</em>. To see the relation between these methods we assume that <em>G </em>=<em> A</em><sup>T</sup> <em>A</em>. It is shown that in this case the two methods generate the same sequence of subspaces, and the same sequence of low-rank approximations. This equivalence provides new insight into the convergence properties of both methods.展开更多
In this paper the new notion of multivariate least-squares orthogonal poly-nomials from the rectangular form is introduced. Their existence and uniqueness isstudied and some methods for their recursive computation are...In this paper the new notion of multivariate least-squares orthogonal poly-nomials from the rectangular form is introduced. Their existence and uniqueness isstudied and some methods for their recursive computation are given. As an applica-is constructed.展开更多
Upon using the denotative theorem of anti-Hermitian generalized Hamiltonian matrices,we solve effectively the least-squares problem min‖AX-B‖over anti-Hermitian generalized Hamiltonian matrices.We derive some necess...Upon using the denotative theorem of anti-Hermitian generalized Hamiltonian matrices,we solve effectively the least-squares problem min‖AX-B‖over anti-Hermitian generalized Hamiltonian matrices.We derive some necessary and sufficient conditions for solvability of the problem and an expression for general solution of the matrix equation AX=B.In addition,we also obtain the expression for the solution of a relevant optimal approximate problem.展开更多
文摘The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the basic function and of the weight function,and is mainly determined by that of the weight function.Therefore,the weight function greatly affects the accuracy of results obtained.Different kinds of weight functions,such as the spline function, the Gauss function and so on,are proposed recently by many researchers.In the present work,the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method.The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed.Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and α in Gauss and exponential weight functions are in the range of reasonable values,respectively,and the higher the smoothness of the weight function,the better the features of the solutions.
文摘This paper presents a new approach to the structural topology optimization of continuum structures. Material-point independent variables are presented to illustrate the existence condition,or inexistence of the material points and their vicinity instead of elements or nodes in popular topology optimization methods. Topological variables field is constructed by moving least square approximation which is used as a shape function in the meshless method. Combined with finite element analyses,not only checkerboard patterns and mesh-dependence phenomena are overcome by this continuous and smooth topological variables field,but also the locations and numbers of topological variables can be arbitrary. Parameters including the number of quadrature points,scaling parameter,weight function and so on upon optimum topological configurations are discussed. Two classic topology optimization problems are solved successfully by the proposed method. The method is found robust and no numerical instabilities are found with proper parameters.
文摘In this paper an introduction of the moving least squares approach is presented in the context of data approximation and interpolation problems in Geodesy.An application of this method is presented for geoid height approximation and interpolation using different polynomial basis functions for the approximant and interpolant,respectively,in a regular grid of geoid height data in the region 16.0417°≤φ≤47.9583°and 36.0417°≤λ≤69.9582°,with increment 0.0833°in both latitudal and longitudal directions.The results of approximation and interpolation are then compared with the geoid height data from GPS-Levelling approach.Using the standard deviation of the difference of the results,it is shown that the planar interpolant,with reciprocal of distance as weight function,is the best choice in this local approximation and interpolation problem.
基金supported by the National Natural Science Foundation of China(Grant No.11101454)the Natural Science Foundation of Chongqing CSTC,China(Grant No.cstc2014jcyjA00005)the Program of Innovation Team Project in University of Chongqing City,China(Grant No.KJTD201308)
文摘Based on the moving least square （MLS） approximations and the boundary integral equations （BIEs）, a meshless algorithm is presented in this paper for elliptic Signorini problems. In the algorithm, a projection operator is used to tackle the nonlinear boundary inequality conditions. The Signorini problem is then reformulated as BIEs and the unknown boundary variables are approximated by the MLS approximations. Accordingly, only a nodal data structure on the boundary of a domain is required. The convergence of the algorithm is proven. Numerical examples are given to show the high convergence rate and high computational efficiency of the presented algorithm.
文摘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.
基金The National Natural Science Foundationof China (No.60672135)the Natural Science Foundation of Department of Education of Shaanxi Province, China(No.09JK809)
文摘In order to get an approximation with better effect of pararneterization of Bezier curves, we proposed a method for arc-length parameterization and the corresponding algorithms by square approximation for the discrete even de-parameterization of the curves. This method is simple and easy to implement, and the property of the approximation has no change compared with the original curve. A quantitative criterion for estimating the effect of parameterization is also built to quantitatively characterize the parameterization effect of the algorithms. As a result, the nearly arc-length parameterized curve has a smaller relative deviation using either the algorithm with point constraint at endpoints or the algorithm with point constraint plus the first derivative constraint at endpoints. Experiments show that after re-parameterization with our algorithms, the relative deviation will have at least a 20% reduction.
基金Project supported by the National Natural Science Foundation of China(Grant No.11971085)the Fund from the Chongqing Municipal Education Commission,China(Grant Nos.KJZD-M201800501 and CXQT19018)the Chongqing Research Program of Basic Research and Frontier Technology,China(Grant No.cstc2018jcyjAX0266)。
文摘An improved moving least square meshless method is developed for the numerical solution of the nonlinear improved Boussinesq equation. After the approximation of temporal derivatives, nonlinear systems of discrete algebraic equations are established and are solved by an iterative algorithm. Convergence of the iterative algorithm is discussed. Shifted and scaled basis functions are incorporated into the method to guarantee convergence and stability of numerical results. Numerical examples are presented to demonstrate the high convergence rate and high computational accuracy of the method.
文摘A nonlinear problem of mean-square approximation of a real nonnegative continuous function with respect to two variables by the modulus of double Fourier integral dependent on two real parameters with use of the smoothing functional is studied. Finding the optimal solutions of this problem is reduced to solution of the Hammerstein type two-dimensional nonlinear integral equation. The numerical algorithms to find the branching lines and branching-off solutions of this equation are constructed and justified. Numerical examples are presented.
文摘In this article, some properties of matrices of moving least-squares approximation have been proven. The used technique is based on known inequalities for singular-values of matrices. Some inequalities for the norm of coefficients-vector of the linear approximation have been proven.
基金the National Natural Science Foundation of China(No.60903111)
文摘Curvature estimation is a basic step in many point relative applications such as feature recognition, segmentation,shape analysis and simplification.This paper proposes a moving-least square(MLS) surface based method to evaluate curvatures for unorganized point cloud data.First a variation of the projection based MLS surface is adopted as the underlying representation of the input points.A set of equations for geometric analysis are derived from the implicit definition of the MLS surface.These equations are then used to compute curvatures of the surface.Moreover,an empirical formula for determining the appropriate Gaussian factor is presented to improve the accuracy of curvature estimation.The proposed method is tested on several sets of synthetic and real data.The results demonstrate that the MLS surface based method can faithfully and efficiently estimate curvatures and reflect subtle curvature variations.The comparisons with other curvature computation algorithms also show that the presented method performs well when handling noisy data and dense points with complex shapes.
文摘Extracting approximate symmetry planes is a challenge due to the difficulty of accurately measuring numerical values. Introducing the approximate symmetry planes of a 3D point set, this paper presents a new method by gathering normal vectors of potential of the planes, clustering the high probability ones, and then testing and verifying the planes. An experiment showed that the method is effective, robust and universal for extracting the complete approximate planes of symmetry of a random 3D point set.
文摘The neural network partial least square (NNPLS) method was used to establish a robust reaction model for a multi-component catalyst of methane oxidative coupling. The details, including the learning algorithm, the number of hidden units of the inner network, activation function, initialization of the network weights and the principal components, are discussed. The results show that the structural organizations of inner neural network are 1-10-5-1, 1-8-4-1, 1-8-5-1, 1-7-4-1, 1-8-4-1, 1-8-6-1, respectively. The Levenberg-Marquardt method was used in the learning algorithm, and the central sigmoidal function is the activation function. Calculation results show that four principal components are convenient in the use of the multi-component catalyst modeling of methane oxidative coupling. Therefore a robust reaction model expressed by NNPLS succeeds in correlating the relations between elements in catalyst and catalytic reaction results. Compared with the direct network modeling, NNPLS model can be adjusted by experimental data and the calculation of the model is simpler and faster than that of the direct network model.
基金Tianjin Natural Science Foundation !983602011National 863/CIMS Research Foundation !863-511-945-010
文摘General neural network inverse adaptive controller has two flaws: the first is the slow convergence speed; the second is the invalidation to the non-minimum phase system. These defects limit the scope in which the neural network inverse adaptive controller is used. We employ Davidon least squares in training the multi-layer feedforward neural network used in approximating the inverse model of plant to expedite the convergence, and then through constructing the pseudo-plant, a neural network inverse adaptive controller is put forward which is still effective to the nonlinear non-minimum phase system. The simulation results show the validity of this scheme.
基金This project was supported by the National Natural Science Foundation of China (No. 60174021)the Key Project of Tianjin Natural Science Foundation (No.010115).
文摘A relaxation least squares-based learning algorithm for neual networks is proposed. Not only does it have a fast convergence rate, but it involves less computation quantity. Therefore, it is suitable to deal with the case when a network has a large scale but the number of training data is very limited. It has been used in converting furnace process modelling, and impressive result has been obtained.
文摘We study iterative processes of stochastic approximation for finding fixed points of weakly contractive and nonexpansive operators in Hilbert spaces under the condition that operators are given with random errors. We prove mean square convergence and convergence almost sure (a.s.) of iterative approximations and establish both asymptotic and nonasymptotic estimates of the convergence rate in degenerate and non-degenerate cases. Previously the stochastic approximation algorithms were studied mainly for optimization problems.
基金supported by National Natural Science Foundation of China(1057,1047).
文摘In this paper, least-squaxes mirrorsymmetric solution for matrix equations (AX = B, XC = D) and its optimal approximation is considered. With special expression of mirrorsymmetric matrices, a general representation of solution for the least-squares problem is obtained. In addition, the optimal approximate solution and some algorithms to obtain the optimal approximation are provided.
基金supported by the National Natural Science Foundation of China(6177202062202433+4 种基金621723716227242262036010)the Natural Science Foundation of Henan Province(22100002)the Postdoctoral Research Grant in Henan Province(202103111)。
文摘Least squares projection twin support vector machine(LSPTSVM)has faster computing speed than classical least squares support vector machine(LSSVM).However,LSPTSVM is sensitive to outliers and its solution lacks sparsity.Therefore,it is difficult for LSPTSVM to process large-scale datasets with outliers.In this paper,we propose a robust LSPTSVM model(called R-LSPTSVM)by applying truncated least squares loss function.The robustness of R-LSPTSVM is proved from a weighted perspective.Furthermore,we obtain the sparse solution of R-LSPTSVM by using the pivoting Cholesky factorization method in primal space.Finally,the sparse R-LSPTSVM algorithm(SR-LSPTSVM)is proposed.Experimental results show that SR-LSPTSVM is insensitive to outliers and can deal with large-scale datasets fastly.
文摘This note explores the relations between two different methods. The first one is the Alternating Least Squares (ALS) method for calculating a rank<em>-k</em> approximation of a real <em>m</em>×<em>n</em> matrix, <em>A</em>. This method has important applications in nonnegative matrix factorizations, in matrix completion problems, and in tensor approximations. The second method is called Orthogonal Iterations. Other names of this method are Subspace Iterations, Simultaneous Iterations, and block-Power method. Given a real symmetric matrix, <em>G</em>, this method computes<em> k</em> dominant eigenvectors of <em>G</em>. To see the relation between these methods we assume that <em>G </em>=<em> A</em><sup>T</sup> <em>A</em>. It is shown that in this case the two methods generate the same sequence of subspaces, and the same sequence of low-rank approximations. This equivalence provides new insight into the convergence properties of both methods.
基金This work is supported by NNSF(10271022)of China.
文摘In this paper the new notion of multivariate least-squares orthogonal poly-nomials from the rectangular form is introduced. Their existence and uniqueness isstudied and some methods for their recursive computation are given. As an applica-is constructed.
基金This research was supported by the NSF of China under grant number 10571047.
文摘Upon using the denotative theorem of anti-Hermitian generalized Hamiltonian matrices,we solve effectively the least-squares problem min‖AX-B‖over anti-Hermitian generalized Hamiltonian matrices.We derive some necessary and sufficient conditions for solvability of the problem and an expression for general solution of the matrix equation AX=B.In addition,we also obtain the expression for the solution of a relevant optimal approximate problem.
