In this paper,the generalized inverse eigenvalue problem for the（P,Q）-conjugate matrices and the associated approximation problem are discussed by using generalized singular value decomposition（GSVD）.Moreover,the ...In this paper,the generalized inverse eigenvalue problem for the（P,Q）-conjugate matrices and the associated approximation problem are discussed by using generalized singular value decomposition（GSVD）.Moreover,the least residual problem of the above generalized inverse eigenvalue problem is studied by using the canonical correlation decomposition（CCD）.The solutions to these problems are derived.Some numerical examples are given to illustrate the main results.展开更多
This paper researches the following inverse eigenvalue problem for arrow-like matrices. Give two characteristic pairs, get a generalized arrow-like matrix, let the two characteristic pairs are the characteristic pairs...This paper researches the following inverse eigenvalue problem for arrow-like matrices. Give two characteristic pairs, get a generalized arrow-like matrix, let the two characteristic pairs are the characteristic pairs of this generalized arrow-like matrix. The expression and an algorithm of the solution of the problem is given, and a numerical example is provided.展开更多
In this paper, an inverse problem on Jacobi matrices presented by Shieh in 2004 is studied. Shieh＇s result is improved and a new and stable algorithm to reconstruct its solution is given. The numerical examples is al...In this paper, an inverse problem on Jacobi matrices presented by Shieh in 2004 is studied. Shieh＇s result is improved and a new and stable algorithm to reconstruct its solution is given. The numerical examples is also given.展开更多
Given a list of real numbers ∧={λ1,…, λn}, we determine the conditions under which ∧will form the spectrum of a dense n × n singular symmetric matrix. Based on a solvability lemma, an algorithm to compute th...Given a list of real numbers ∧={λ1,…, λn}, we determine the conditions under which ∧will form the spectrum of a dense n × n singular symmetric matrix. Based on a solvability lemma, an algorithm to compute the elements of the matrix is derived for a given list ∧ and dependency parameters. Explicit computations are performed for n≤5 and r≤4 to illustrate the result.展开更多
The main aim of this paper is to discuss the following two problems: Problem I: Given X ∈ Hn×m (the set of all n×m quaternion matrices), A = diag(λ1,…, λm) EEEEE Hm×m, find A ∈ BSHn×n≥such th...The main aim of this paper is to discuss the following two problems: Problem I: Given X ∈ Hn×m (the set of all n×m quaternion matrices), A = diag(λ1,…, λm) EEEEE Hm×m, find A ∈ BSHn×n≥such that AX = X(?), where BSHn×n≥ denotes the set of all n×n quaternion matrices which are bi-self-conjugate and nonnegative definite. Problem Ⅱ2= Given B ∈ Hn×m, find B ∈ SE such that ||B-B||Q = minAE∈=sE ||B-A||Q, where SE is the solution set of problem I , || ·||Q is the quaternion matrix norm. The necessary and sufficient conditions for SE being nonempty are obtained. The general form of elements in SE and the expression of the unique solution B of problem Ⅱ are given.展开更多
is gained by deleting the k<sup>th</sup> row and the k<sup>th</sup> column （k=1,2,...,n） from T<sub>n</sub>.We put for-ward an inverse eigenvalue problem to be that:If we don’t k...is gained by deleting the k<sup>th</sup> row and the k<sup>th</sup> column （k=1,2,...,n） from T<sub>n</sub>.We put for-ward an inverse eigenvalue problem to be that:If we don’t know the matrix T<sub>1,n</sub>,but weknow all eigenvalues of matrix T<sub>1,k-1</sub>,all eigenvalues of matrix T<sub>k+1,k</sub>,and all eigenvaluesof matrix T<sub>1,n</sub> could we construct the matrix T<sub>1,n</sub>.Let μ<sub>1</sub>,μ<sub>2</sub>,…,μ<sub>k-1</sub>,μ<sub>k</sub>,μ<sub>k+1</sub>,…,μ<sub>n-1</sub>,展开更多
In this paper we first consider the existence and the general form of solution to the following generalized inverse eigenvalue problem(GIEP): given a set of n-dimension complex vectors {x j}m j=1 and a set of co...In this paper we first consider the existence and the general form of solution to the following generalized inverse eigenvalue problem(GIEP): given a set of n-dimension complex vectors {x j}m j=1 and a set of complex numbers {λ j}m j=1, find two n×n centrohermitian matrices A,B such that {x j}m j=1 and {λ j}m j=1 are the generalized eigenvectors and generalized eigenvalues of Ax=λBx, respectively. We then discuss the optimal approximation problem for the GIEP. More concretely, given two arbitrary matrices, , ∈C n×n, we find two matrices A and B such that the matrix (A*,B*) is closest to (,) in the Frobenius norm, where the matrix (A*,B*) is the solution to the GIEP. We show that the expression of the solution of the optimal approximation is unique and derive the expression for it.展开更多
This paper focuses on the direct and inverse problems for a third-order self-adjoint differential operator with non-local potential and anti-periodic boundary conditions.Firstly,we obtain the expressions for the chara...This paper focuses on the direct and inverse problems for a third-order self-adjoint differential operator with non-local potential and anti-periodic boundary conditions.Firstly,we obtain the expressions for the characteristic function and resolvent of this third-order differential operator.Secondly,by using the expression for the resolvent of the operator,we prove that the spectrum for this operator consists of simple eigenvalues and a finite number of eigenvalues with multiplicity 2.Finally,we solve the inverse problem for this operator,which states that the non-local potential function can be reconstructed from four spectra.Specially,we prove the Ambarzumyan theorem and indicate that odd or even potential functions can be reconstructed by three spectra.展开更多
Applying constructed homotopy and its properties,we gel some sufficient conditions for the solvability of algebraic inverse eigenvalue problems,which are better than that of the paper [4] in some cases. Inverse eigenv...Applying constructed homotopy and its properties,we gel some sufficient conditions for the solvability of algebraic inverse eigenvalue problems,which are better than that of the paper [4] in some cases. Inverse eigenvalue problems,solvability,sufficient conditions.展开更多
In this paper the unsolvability of generalized inverse eigenvalue problems almost everywhere is discussed.We first give the definitions for the unsolvability of generalized inverse eigenvalue problems almost everywher...In this paper the unsolvability of generalized inverse eigenvalue problems almost everywhere is discussed.We first give the definitions for the unsolvability of generalized inverse eigenvalue problems almost everywhere.Then adopting the method used in [14],we present some sufficient conditions such that the generalized inverse eigenvalue problems are unsohable almost everywhere.展开更多
The inverse design method of a dynamic system with linear parameters has been studied. For some specified eigenvalues and eigenvectors, the design parameter vector which is often composed of whole or part of coefficie...The inverse design method of a dynamic system with linear parameters has been studied. For some specified eigenvalues and eigenvectors, the design parameter vector which is often composed of whole or part of coefficients of spring and mass of the system can be obtained and the rigidity and mass matrices of an initially designed structure can be reconstructed through solving linear algebra equations. By using implicit function theorem, the conditions of existence and uniqueness of the solution are also deduced. The theory and method can be used for inverse vibration design of complex structure system.展开更多
An Inverse perturbation method is described for solving the general inverse eigenvalue problem. By taking the analysis of the rotor system as example based upon FEM, the new inverse perturbation method for structural ...An Inverse perturbation method is described for solving the general inverse eigenvalue problem. By taking the analysis of the rotor system as example based upon FEM, the new inverse perturbation method for structural design with specified low-order natural frequencies or frequency constraint bands is detailed as well as its complete theoretical basis. Moreover, formulations to calculate the inverse perturbation parameter ε and method to select the corresponding ε's value properly are also proposed. The proposed method is characterized in reducing frequency analysis and suitable for large and small structrual changes alike. Finally, several different numerical examples for inverse cigenvalue problem are discussed to illustrate the method, which show that this inverse perturbation method Is general and can be applied to other type of structure or dement.展开更多
1 IntroductionLet R<sup>n×n</sup> be the set of all n×n real matrices.R<sup>n</sup>=R<sup>n×1</sup>.C<sup>n×n</sup>denotes the set of all n×n co...1 IntroductionLet R<sup>n×n</sup> be the set of all n×n real matrices.R<sup>n</sup>=R<sup>n×1</sup>.C<sup>n×n</sup>denotes the set of all n×n complex matrices.We are interested in solving the following inverse eigenvalue prob-lems:Problem A （Additive inverse eigenvalue problem） Given an n×n real matrix A=(a<sub>ij</sub>),and n distinct real numbers λ<sub>1</sub>,λ<sub>2</sub>,…,λ<sub>n</sub>,find a real n×n diagonal matrix展开更多
In this paper,the physics informed neural network(PINN)deep learning method is applied to solve two-dimensional nonlocal equations,including the partial reverse space y-nonlocal Mel'nikov equation,the partial reve...In this paper,the physics informed neural network(PINN)deep learning method is applied to solve two-dimensional nonlocal equations,including the partial reverse space y-nonlocal Mel'nikov equation,the partial reverse space-time nonlocal Mel'nikov equation and the nonlocal twodimensional nonlinear Schr?dinger(NLS)equation.By the PINN method,we successfully derive a data-driven two soliton solution,lump solution and rogue wave solution.Numerical simulation results indicate that the error range between the data-driven solution and the exact solution is relatively small,which verifies the effectiveness of the PINN deep learning method for solving high dimensional nonlocal equations.Moreover,the parameter discovery of the partial reverse space-time nonlocal Mel'nikov equation is analysed in terms of its soliton solution for the first time.展开更多
This paper is devoted to the Polynomial Preserving Recovery (PPR) based a posteriori error analysis for the second-order elliptic non-symmetric eigenvalue problem. An asymptotically exact a posteriori error estimator ...This paper is devoted to the Polynomial Preserving Recovery (PPR) based a posteriori error analysis for the second-order elliptic non-symmetric eigenvalue problem. An asymptotically exact a posteriori error estimator is proposed for solving the convection-dominated non-symmetric eigenvalue problem with non-smooth eigenfunctions or multiple eigenvalues. Numerical examples confirm our theoretical analysis.展开更多
In this paper, we give solvability conditions for three open problems of nonnegative inverse eigenvalues problem (NIEP) which were left hanging in the air up to seventy years. It will offer effective ways to judge an ...In this paper, we give solvability conditions for three open problems of nonnegative inverse eigenvalues problem (NIEP) which were left hanging in the air up to seventy years. It will offer effective ways to judge an NIEP whether is solvable.展开更多
The structural dynamics problems,such as structural design,parameter identification and model correction,are considered as a kind of the inverse generalized eigenvalue problems mathematically.The inverse eigenvalue pr...The structural dynamics problems,such as structural design,parameter identification and model correction,are considered as a kind of the inverse generalized eigenvalue problems mathematically.The inverse eigenvalue problems are nonlinear.In general,they could be transformed into nonlinear equations to solve.The structural dynamics inverse problems were treated as quasi multiplicative inverse eigenalue problems which were solved by homotopy method for nonlinear equations.This method had no requirements for initial value essentially because of the homotopy path to solution.Numerical examples were presented to illustrate the homotopy method.展开更多
We present a differential geometric perspective of the IEP for symmetric matrices in the framework of a fibre bundle with structure group SO（n）. In particular, a Newton type algorithm is developed to construct a non...We present a differential geometric perspective of the IEP for symmetric matrices in the framework of a fibre bundle with structure group SO（n）. In particular, a Newton type algorithm is developed to construct a non singular symmetric matrix for given target eigenvalues using a singular symmetric matrix as the initial matrix for the iteration. Explicit computations are performed for 2 x 2 non singular symmetric matrix to illustrate the result.展开更多
By using the characteristic properties of the anti-Hermitian generalized anti-Hamiltonian matrices, we prove some necessary and sufficient conditions of the solvability for algebra inverse eigenvalue problem of anti-H...By using the characteristic properties of the anti-Hermitian generalized anti-Hamiltonian matrices, we prove some necessary and sufficient conditions of the solvability for algebra inverse eigenvalue problem of anti-Hermitian generalized anti-Hamiltonian matrices, and obtain a general expression of the solution to this problem. By using the properties of the orthogonal projection matrix, we also obtain the expression of the solution to optimal approximate problem of an n× n complex matrix under spectral restriction.展开更多
We present new sufficient conditions on the solvability and numericalmethods for the following multiplicative inverse eigenvalue problem: Given an n × nreal matrix A and n real numbers λ1 , λ2 , . . . ,λn, fin...We present new sufficient conditions on the solvability and numericalmethods for the following multiplicative inverse eigenvalue problem: Given an n × nreal matrix A and n real numbers λ1 , λ2 , . . . ,λn, find n real numbers c1 , c2 , . . . , cn suchthat the matrix diag(c1, c2,..., cn)A has eigenvalues λ1, λ2,..., λn.展开更多
基金Supported by the Key Discipline Construction Project of Tianshui Normal University
文摘In this paper,the generalized inverse eigenvalue problem for the（P,Q）-conjugate matrices and the associated approximation problem are discussed by using generalized singular value decomposition（GSVD）.Moreover,the least residual problem of the above generalized inverse eigenvalue problem is studied by using the canonical correlation decomposition（CCD）.The solutions to these problems are derived.Some numerical examples are given to illustrate the main results.
文摘This paper researches the following inverse eigenvalue problem for arrow-like matrices. Give two characteristic pairs, get a generalized arrow-like matrix, let the two characteristic pairs are the characteristic pairs of this generalized arrow-like matrix. The expression and an algorithm of the solution of the problem is given, and a numerical example is provided.
基金Project supported by the National Natural Science Foundation of China （Grant No.10271074）
文摘In this paper, an inverse problem on Jacobi matrices presented by Shieh in 2004 is studied. Shieh＇s result is improved and a new and stable algorithm to reconstruct its solution is given. The numerical examples is also given.
文摘Given a list of real numbers ∧={λ1,…, λn}, we determine the conditions under which ∧will form the spectrum of a dense n × n singular symmetric matrix. Based on a solvability lemma, an algorithm to compute the elements of the matrix is derived for a given list ∧ and dependency parameters. Explicit computations are performed for n≤5 and r≤4 to illustrate the result.
基金This work is supported by the NSF of China (10471039, 10271043) and NSF of Zhejiang Province (M103087).
文摘The main aim of this paper is to discuss the following two problems: Problem I: Given X ∈ Hn×m (the set of all n×m quaternion matrices), A = diag(λ1,…, λm) EEEEE Hm×m, find A ∈ BSHn×n≥such that AX = X(?), where BSHn×n≥ denotes the set of all n×n quaternion matrices which are bi-self-conjugate and nonnegative definite. Problem Ⅱ2= Given B ∈ Hn×m, find B ∈ SE such that ||B-B||Q = minAE∈=sE ||B-A||Q, where SE is the solution set of problem I , || ·||Q is the quaternion matrix norm. The necessary and sufficient conditions for SE being nonempty are obtained. The general form of elements in SE and the expression of the unique solution B of problem Ⅱ are given.
基金Project 19771020 supported by National Science Foundation of China
文摘is gained by deleting the k<sup>th</sup> row and the k<sup>th</sup> column （k=1,2,...,n） from T<sub>n</sub>.We put for-ward an inverse eigenvalue problem to be that:If we don’t know the matrix T<sub>1,n</sub>,but weknow all eigenvalues of matrix T<sub>1,k-1</sub>,all eigenvalues of matrix T<sub>k+1,k</sub>,and all eigenvaluesof matrix T<sub>1,n</sub> could we construct the matrix T<sub>1,n</sub>.Let μ<sub>1</sub>,μ<sub>2</sub>,…,μ<sub>k-1</sub>,μ<sub>k</sub>,μ<sub>k+1</sub>,…,μ<sub>n-1</sub>,
文摘In this paper we first consider the existence and the general form of solution to the following generalized inverse eigenvalue problem(GIEP): given a set of n-dimension complex vectors {x j}m j=1 and a set of complex numbers {λ j}m j=1, find two n×n centrohermitian matrices A,B such that {x j}m j=1 and {λ j}m j=1 are the generalized eigenvectors and generalized eigenvalues of Ax=λBx, respectively. We then discuss the optimal approximation problem for the GIEP. More concretely, given two arbitrary matrices, , ∈C n×n, we find two matrices A and B such that the matrix (A*,B*) is closest to (,) in the Frobenius norm, where the matrix (A*,B*) is the solution to the GIEP. We show that the expression of the solution of the optimal approximation is unique and derive the expression for it.
基金supported by the Tianjin Municipal Science and Technology Program of China(No.23JCZDJC00070)。
文摘This paper focuses on the direct and inverse problems for a third-order self-adjoint differential operator with non-local potential and anti-periodic boundary conditions.Firstly,we obtain the expressions for the characteristic function and resolvent of this third-order differential operator.Secondly,by using the expression for the resolvent of the operator,we prove that the spectrum for this operator consists of simple eigenvalues and a finite number of eigenvalues with multiplicity 2.Finally,we solve the inverse problem for this operator,which states that the non-local potential function can be reconstructed from four spectra.Specially,we prove the Ambarzumyan theorem and indicate that odd or even potential functions can be reconstructed by three spectra.
文摘Applying constructed homotopy and its properties,we gel some sufficient conditions for the solvability of algebraic inverse eigenvalue problems,which are better than that of the paper [4] in some cases. Inverse eigenvalue problems,solvability,sufficient conditions.
文摘In this paper the unsolvability of generalized inverse eigenvalue problems almost everywhere is discussed.We first give the definitions for the unsolvability of generalized inverse eigenvalue problems almost everywhere.Then adopting the method used in [14],we present some sufficient conditions such that the generalized inverse eigenvalue problems are unsohable almost everywhere.
基金Science Developing Plan of Beijing Educational Committee, Beijing Natural Science Fund (No. 3022003), and NationalNatural Science Fund of China(No.50375002)
文摘The inverse design method of a dynamic system with linear parameters has been studied. For some specified eigenvalues and eigenvectors, the design parameter vector which is often composed of whole or part of coefficients of spring and mass of the system can be obtained and the rigidity and mass matrices of an initially designed structure can be reconstructed through solving linear algebra equations. By using implicit function theorem, the conditions of existence and uniqueness of the solution are also deduced. The theory and method can be used for inverse vibration design of complex structure system.
基金This research is supported by China National Natural Science Foundation (CNNSF), Research Grant No. 50128504
文摘An Inverse perturbation method is described for solving the general inverse eigenvalue problem. By taking the analysis of the rotor system as example based upon FEM, the new inverse perturbation method for structural design with specified low-order natural frequencies or frequency constraint bands is detailed as well as its complete theoretical basis. Moreover, formulations to calculate the inverse perturbation parameter ε and method to select the corresponding ε's value properly are also proposed. The proposed method is characterized in reducing frequency analysis and suitable for large and small structrual changes alike. Finally, several different numerical examples for inverse cigenvalue problem are discussed to illustrate the method, which show that this inverse perturbation method Is general and can be applied to other type of structure or dement.
文摘1 IntroductionLet R<sup>n×n</sup> be the set of all n×n real matrices.R<sup>n</sup>=R<sup>n×1</sup>.C<sup>n×n</sup>denotes the set of all n×n complex matrices.We are interested in solving the following inverse eigenvalue prob-lems:Problem A （Additive inverse eigenvalue problem） Given an n×n real matrix A=(a<sub>ij</sub>),and n distinct real numbers λ<sub>1</sub>,λ<sub>2</sub>,…,λ<sub>n</sub>,find a real n×n diagonal matrix
文摘In this paper,the physics informed neural network(PINN)deep learning method is applied to solve two-dimensional nonlocal equations,including the partial reverse space y-nonlocal Mel'nikov equation,the partial reverse space-time nonlocal Mel'nikov equation and the nonlocal twodimensional nonlinear Schr?dinger(NLS)equation.By the PINN method,we successfully derive a data-driven two soliton solution,lump solution and rogue wave solution.Numerical simulation results indicate that the error range between the data-driven solution and the exact solution is relatively small,which verifies the effectiveness of the PINN deep learning method for solving high dimensional nonlocal equations.Moreover,the parameter discovery of the partial reverse space-time nonlocal Mel'nikov equation is analysed in terms of its soliton solution for the first time.
基金Supported by the National Natural Science Foundation of China (Grant Nos.1236108412001130)。
文摘This paper is devoted to the Polynomial Preserving Recovery (PPR) based a posteriori error analysis for the second-order elliptic non-symmetric eigenvalue problem. An asymptotically exact a posteriori error estimator is proposed for solving the convection-dominated non-symmetric eigenvalue problem with non-smooth eigenfunctions or multiple eigenvalues. Numerical examples confirm our theoretical analysis.
文摘In this paper, we give solvability conditions for three open problems of nonnegative inverse eigenvalues problem (NIEP) which were left hanging in the air up to seventy years. It will offer effective ways to judge an NIEP whether is solvable.
文摘The structural dynamics problems,such as structural design,parameter identification and model correction,are considered as a kind of the inverse generalized eigenvalue problems mathematically.The inverse eigenvalue problems are nonlinear.In general,they could be transformed into nonlinear equations to solve.The structural dynamics inverse problems were treated as quasi multiplicative inverse eigenalue problems which were solved by homotopy method for nonlinear equations.This method had no requirements for initial value essentially because of the homotopy path to solution.Numerical examples were presented to illustrate the homotopy method.
文摘We present a differential geometric perspective of the IEP for symmetric matrices in the framework of a fibre bundle with structure group SO（n）. In particular, a Newton type algorithm is developed to construct a non singular symmetric matrix for given target eigenvalues using a singular symmetric matrix as the initial matrix for the iteration. Explicit computations are performed for 2 x 2 non singular symmetric matrix to illustrate the result.
基金Project(10171031) supported by the National Natural Science Foundation of China
文摘By using the characteristic properties of the anti-Hermitian generalized anti-Hamiltonian matrices, we prove some necessary and sufficient conditions of the solvability for algebra inverse eigenvalue problem of anti-Hermitian generalized anti-Hamiltonian matrices, and obtain a general expression of the solution to this problem. By using the properties of the orthogonal projection matrix, we also obtain the expression of the solution to optimal approximate problem of an n× n complex matrix under spectral restriction.
文摘We present new sufficient conditions on the solvability and numericalmethods for the following multiplicative inverse eigenvalue problem: Given an n × nreal matrix A and n real numbers λ1 , λ2 , . . . ,λn, find n real numbers c1 , c2 , . . . , cn suchthat the matrix diag(c1, c2,..., cn)A has eigenvalues λ1, λ2,..., λn.
