The symmetric,positive semidefinite,and positive definite real solutions of the matrix equation XA=YAD from an inverse problem of vibration theory are considered.When D=T the necessary and sufficient conditions fo...The symmetric,positive semidefinite,and positive definite real solutions of the matrix equation XA=YAD from an inverse problem of vibration theory are considered.When D=T the necessary and sufficient conditions for the existence of such solutions and their general forms are derived.展开更多
QL(QR) method is an efficient method to find eigenvalues of a matrix. Especially we use QL(QR) method to find eigenvalues of a symmetric tridiagonal matrix. In this case it only costs O(n2) flops, to find all eigenval...QL(QR) method is an efficient method to find eigenvalues of a matrix. Especially we use QL(QR) method to find eigenvalues of a symmetric tridiagonal matrix. In this case it only costs O(n2) flops, to find all eigenvalues. So it is one of the most efficient method for symmetric tridiagonal matrices. Many experts have researched it. Even the method is mature, it still has many problems need to be researched. We put forward five problems here. They are: (1) Convergence and convergence rate; (2) The convergence of diagonal elements; (3) Shift designed to produce the eigenvalues in monotone order; (4) QL algorithm with multi-shift; (5) Error bound. We intoduce our works on these problems, some of them were published and some are new.展开更多
Let G =(Y, E) be a primitive digraph. The vertex exponent of G at a vertex v E V, denoted by expG(v), is the least integer p such that there is a v → u walk of length p for each u E V. We choose to order the vert...Let G =(Y, E) be a primitive digraph. The vertex exponent of G at a vertex v E V, denoted by expG(v), is the least integer p such that there is a v → u walk of length p for each u E V. We choose to order the vertices of G in such a way that expG(v1) ≤ expG(v2) ≤... ≤expG(vn). Then expG(vk) is called the k-point exponent of G and is denoted by expG(k), 1 〈 k 〈 n. We define the k-point exponent set E(n, k) := {expG(k)|G= G(A) with A E CSP(n)|, where CSP(n) is the set of all n × n central symmetric primitive matrices and G(A) is the associated graph of the matrix A. In this paper, we describe E(n, k) for all n, k with 1 〈 k 〈 n except n ≡ l(mod 2) and 1 〈 k 〈 n - 4. We also characterize the extremal graphs when k = 1.展开更多
In this paper, a sufficient and necessary condition is presented for existence of a class of exact solutions to N-dimensional incompressible magnetohydrodynamic (MHD) equations. Such solutions can be explicitly expr...In this paper, a sufficient and necessary condition is presented for existence of a class of exact solutions to N-dimensional incompressible magnetohydrodynamic (MHD) equations. Such solutions can be explicitly expressed by appropriate formulae. Once the required matrices are chosen, solutions to the MHD equations axe directly constructed.展开更多
From the formulas of the conjugate gradient, a similarity between a symmetric positive definite (SPD) matrix A and a tridiagonal matrix B is obtained. The elements of the matrix B are determined by the parameters of t...From the formulas of the conjugate gradient, a similarity between a symmetric positive definite (SPD) matrix A and a tridiagonal matrix B is obtained. The elements of the matrix B are determined by the parameters of the conjugate gradient. The computation of eigenvalues of A is then reduced to the case of the tridiagonal matrix B. The approximation of extreme eigenvalues of A can be obtained as a 'by-product' in the computation of the conjugate gradient if a computational cost of O(s) arithmetic operations is added, where s is the number of iterations This computational cost is negligible compared with the conjugate gradient. If the matrix A is not SPD, the approximation of the condition number of A can be obtained from the computation of the conjugate gradient on AT A. Numerical results show that this is a convenient and highly efficient method for computing extreme eigenvalues and the condition number of nonsingular matrices.展开更多
In CAD/CAM, mesh rather than smooth surface is only needed sometimes. A mesh-generating method from permanence principle of Coons patch is developed. A new mesh point is defined through local small subpatch and all me...In CAD/CAM, mesh rather than smooth surface is only needed sometimes. A mesh-generating method from permanence principle of Coons patch is developed. A new mesh point is defined through local small subpatch and all mesh points are computed by a linear system with special symmetric block tridiagonal coefficient matrix. By simplification, the determinant of coefficient matrix is determined by determinants of submatrices. Condition of existence of solution is given. Whether coefficient matrix is singular can be judged by a simple polynomial function with the eigenvalue of submatrix as variable. Numerical examples demonstrate the effects of shape parameters.展开更多
Least squares solution of F=PG with respect to positive semidefinite symmetric P is considered,a new necessary and sufficient condition for solvablity is given,and the expression of solution is derived in the some spe...Least squares solution of F=PG with respect to positive semidefinite symmetric P is considered,a new necessary and sufficient condition for solvablity is given,and the expression of solution is derived in the some special cases. Based on the expression, the least spuares solution of an inverse eigenvalue problem for positive semidefinite symmetric matrices is also given.展开更多
In this paper, the following two problems are considered:Problem Ⅰ. Given S∈E Rn×p,X,B 6 Rn×m, find A ∈ SRs,n such that AX = B, where SR8,n = {A∈ Rn×n|xT(A - AT) = 0, for all x ∈ R(S)}.Problem Ⅱ. ...In this paper, the following two problems are considered:Problem Ⅰ. Given S∈E Rn×p,X,B 6 Rn×m, find A ∈ SRs,n such that AX = B, where SR8,n = {A∈ Rn×n|xT(A - AT) = 0, for all x ∈ R(S)}.Problem Ⅱ. Given A* ∈ Rn×n, find A ∈ SE such that ||A-A*|| = minA∈sE||A-A*||, where SE is the solution set of Problem Ⅰ.The necessary and sufficient conditions for the solvability of and the general form of the solutions of problem Ⅰ are given. For problem Ⅱ, the expression for the solution, a numerical algorithm and a numerical example are provided.展开更多
By applying the generalized singular value decomposition and the canonical correlation decomposition simultaneously, we derive an analytical expression of the optimal approximate solution ^-X, which is both a least-sq...By applying the generalized singular value decomposition and the canonical correlation decomposition simultaneously, we derive an analytical expression of the optimal approximate solution ^-X, which is both a least-squares symmetric orthogonal anti-symmetric solu- tion of the matrix equation A^TXA = B and a best approximation to a given matrix X^*. Moreover, a numerical algorithm for finding this optimal approximate solution is described in detail, and a numerical example is presented to show the validity of our algorithm.展开更多
We study the symmetric positive semidefinite solution of the matrix equation AX_1A^T + BX_2B^T=C. where A is a given real m×n matrix. B is a given real m×p matrix, and C is a given real m×m matrix, with...We study the symmetric positive semidefinite solution of the matrix equation AX_1A^T + BX_2B^T=C. where A is a given real m×n matrix. B is a given real m×p matrix, and C is a given real m×m matrix, with m, n, p positive integers: and the bisymmetric positive semidefinite solution of the matrix equation D^T XD=C, where D is a given real n×m matrix. C is a given real m×m matrix, with m. n positive integers. By making use of the generalized singular value decomposition, we derive general analytic formulae, and present necessary and sufficient conditions for guaranteeing the existence of these solutions.展开更多
A class of regularized conjugate gradient methods is presented for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned symmetric positive definite matrix. The conv...A class of regularized conjugate gradient methods is presented for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned symmetric positive definite matrix. The convergence properties of these methods are discussed in depth, and the best possible choices of the parameters involved in the new methods are investigated in detail. Numerical computations show that the new methods are more efficient and robust than both classical relaxation methods and classical conjugate direction methods.展开更多
With the rapid development of the shipping industry,the safety and comfort of ship transportation have been paid more and more attention,and the pitch and heave motion of ships are the most serious factors.In this pap...With the rapid development of the shipping industry,the safety and comfort of ship transportation have been paid more and more attention,and the pitch and heave motion of ships are the most serious factors.In this paper,the longitudinal motion mathematical model of YuKun is esta b lished.By assigning the zero-pole to the left half-plane and using the properties of the symmetric matrix,the shaping weighting functions matrix is designed to stabilize the multi-input multi-output(MIMO)system of YuKun.Finally,a new concise robust controller is designed using the steady output of the shaped system.The simulation results show that under the control of the concise robust controller,the pitch angle and heave of YuKun decrease by 79.9% and 86.2%,respectively.Theoretical analysis and simulation results show that the concise robust controller has a good control effect on the longitudinal motion of YuKun,and is simple and easy to use,with clear engineering significance.展开更多
In this paper,we prove the existence of general Cartesian vector solutions u=b(t)+A(t)x for the Ndimensional compressible Navier–Stokes equations with density-dependent viscosity,based on the matrix and curve integra...In this paper,we prove the existence of general Cartesian vector solutions u=b(t)+A(t)x for the Ndimensional compressible Navier–Stokes equations with density-dependent viscosity,based on the matrix and curve integration theory.Two exact solutions are obtained by solving the reduced systems.展开更多
Let T1,n be an n x n unreduced symmetric tridiagonal matrix with eigenvaluesand is an (n - 1) x (n - 1) submatrix by deleting the kth row and kth column, k = 1, 2,be the eigenvalues of T1,k andbe the eigenvalues of Tk...Let T1,n be an n x n unreduced symmetric tridiagonal matrix with eigenvaluesand is an (n - 1) x (n - 1) submatrix by deleting the kth row and kth column, k = 1, 2,be the eigenvalues of T1,k andbe the eigenvalues of Tk+1,nA new inverse eigenvalues problem has put forward as follows: How do we construct anunreduced symmetric tridiagonal matrix T1,n, if we only know the spectral data: theeigenvalues of T1,n, the eigenvalues of Ti,k-1 and the eigenvalues of Tk+1,n?Namely if we only know the data: A1, A2, An,how do we find the matrix T1,n? A necessary and sufficient condition and an algorithm ofsolving such problem, are given in this paper.展开更多
In this paper, we discuss an inverse eigenvalue problem for constructing a 2n × 2n Jacobi matrix T such that its 2n eigenvalues are given distinct real values and its leading principal submatrix of order n is a g...In this paper, we discuss an inverse eigenvalue problem for constructing a 2n × 2n Jacobi matrix T such that its 2n eigenvalues are given distinct real values and its leading principal submatrix of order n is a given Jacobi matrix. A new sufficient and necessary condition for the solvability of the above problem is given in this paper. Furthermore, we present a new algorithm and give some numerical results.展开更多
Presents preconditioning matrices having parallel computing function for the coefficient matrix and a class of parallel hybrid algebraic multilevel iteration methods for solving linear equations. Solution to elliptic ...Presents preconditioning matrices having parallel computing function for the coefficient matrix and a class of parallel hybrid algebraic multilevel iteration methods for solving linear equations. Solution to elliptic boundary value problem; Discussion on symmetric positive definite matrix; Computational complexities.展开更多
文摘The symmetric,positive semidefinite,and positive definite real solutions of the matrix equation XA=YAD from an inverse problem of vibration theory are considered.When D=T the necessary and sufficient conditions for the existence of such solutions and their general forms are derived.
文摘QL(QR) method is an efficient method to find eigenvalues of a matrix. Especially we use QL(QR) method to find eigenvalues of a symmetric tridiagonal matrix. In this case it only costs O(n2) flops, to find all eigenvalues. So it is one of the most efficient method for symmetric tridiagonal matrices. Many experts have researched it. Even the method is mature, it still has many problems need to be researched. We put forward five problems here. They are: (1) Convergence and convergence rate; (2) The convergence of diagonal elements; (3) Shift designed to produce the eigenvalues in monotone order; (4) QL algorithm with multi-shift; (5) Error bound. We intoduce our works on these problems, some of them were published and some are new.
基金Foundation item:The NSF(04JJ40002)of Hunan and the SRF of Hunan Provincial Education Department.
文摘Let G =(Y, E) be a primitive digraph. The vertex exponent of G at a vertex v E V, denoted by expG(v), is the least integer p such that there is a v → u walk of length p for each u E V. We choose to order the vertices of G in such a way that expG(v1) ≤ expG(v2) ≤... ≤expG(vn). Then expG(vk) is called the k-point exponent of G and is denoted by expG(k), 1 〈 k 〈 n. We define the k-point exponent set E(n, k) := {expG(k)|G= G(A) with A E CSP(n)|, where CSP(n) is the set of all n × n central symmetric primitive matrices and G(A) is the associated graph of the matrix A. In this paper, we describe E(n, k) for all n, k with 1 〈 k 〈 n except n ≡ l(mod 2) and 1 〈 k 〈 n - 4. We also characterize the extremal graphs when k = 1.
文摘In this paper, a sufficient and necessary condition is presented for existence of a class of exact solutions to N-dimensional incompressible magnetohydrodynamic (MHD) equations. Such solutions can be explicitly expressed by appropriate formulae. Once the required matrices are chosen, solutions to the MHD equations axe directly constructed.
文摘From the formulas of the conjugate gradient, a similarity between a symmetric positive definite (SPD) matrix A and a tridiagonal matrix B is obtained. The elements of the matrix B are determined by the parameters of the conjugate gradient. The computation of eigenvalues of A is then reduced to the case of the tridiagonal matrix B. The approximation of extreme eigenvalues of A can be obtained as a 'by-product' in the computation of the conjugate gradient if a computational cost of O(s) arithmetic operations is added, where s is the number of iterations This computational cost is negligible compared with the conjugate gradient. If the matrix A is not SPD, the approximation of the condition number of A can be obtained from the computation of the conjugate gradient on AT A. Numerical results show that this is a convenient and highly efficient method for computing extreme eigenvalues and the condition number of nonsingular matrices.
基金Supported by National Natural Science Foundation of China(No.60970097,No.11271376)
文摘In CAD/CAM, mesh rather than smooth surface is only needed sometimes. A mesh-generating method from permanence principle of Coons patch is developed. A new mesh point is defined through local small subpatch and all mesh points are computed by a linear system with special symmetric block tridiagonal coefficient matrix. By simplification, the determinant of coefficient matrix is determined by determinants of submatrices. Condition of existence of solution is given. Whether coefficient matrix is singular can be judged by a simple polynomial function with the eigenvalue of submatrix as variable. Numerical examples demonstrate the effects of shape parameters.
文摘Least squares solution of F=PG with respect to positive semidefinite symmetric P is considered,a new necessary and sufficient condition for solvablity is given,and the expression of solution is derived in the some special cases. Based on the expression, the least spuares solution of an inverse eigenvalue problem for positive semidefinite symmetric matrices is also given.
基金Research supported by National Natural Science Foundation of China(10171031)by Hunan Province Education Foundation(02C025)
文摘In this paper, the following two problems are considered:Problem Ⅰ. Given S∈E Rn×p,X,B 6 Rn×m, find A ∈ SRs,n such that AX = B, where SR8,n = {A∈ Rn×n|xT(A - AT) = 0, for all x ∈ R(S)}.Problem Ⅱ. Given A* ∈ Rn×n, find A ∈ SE such that ||A-A*|| = minA∈sE||A-A*||, where SE is the solution set of Problem Ⅰ.The necessary and sufficient conditions for the solvability of and the general form of the solutions of problem Ⅰ are given. For problem Ⅱ, the expression for the solution, a numerical algorithm and a numerical example are provided.
基金The Project supported by Scientific Research Fund of Hunan Provincial Education Department,by National Natural Science Foundation of China (10171031)by Natural Science Fundation of Hunan Province (03JJY6028).
文摘By applying the generalized singular value decomposition and the canonical correlation decomposition simultaneously, we derive an analytical expression of the optimal approximate solution ^-X, which is both a least-squares symmetric orthogonal anti-symmetric solu- tion of the matrix equation A^TXA = B and a best approximation to a given matrix X^*. Moreover, a numerical algorithm for finding this optimal approximate solution is described in detail, and a numerical example is presented to show the validity of our algorithm.
基金Subsidized by the Special Funds for Major State Basic Research Projects G1999032803
文摘We study the symmetric positive semidefinite solution of the matrix equation AX_1A^T + BX_2B^T=C. where A is a given real m×n matrix. B is a given real m×p matrix, and C is a given real m×m matrix, with m, n, p positive integers: and the bisymmetric positive semidefinite solution of the matrix equation D^T XD=C, where D is a given real n×m matrix. C is a given real m×m matrix, with m. n positive integers. By making use of the generalized singular value decomposition, we derive general analytic formulae, and present necessary and sufficient conditions for guaranteeing the existence of these solutions.
基金Subsidized by The Special Funds For Major State Basic Research Projects G1999032803.
文摘A class of regularized conjugate gradient methods is presented for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned symmetric positive definite matrix. The convergence properties of these methods are discussed in depth, and the best possible choices of the parameters involved in the new methods are investigated in detail. Numerical computations show that the new methods are more efficient and robust than both classical relaxation methods and classical conjugate direction methods.
基金This work is partially supported by the National Science Foundation of China(Grant No.51679024)Dalian Innovation TeamSupport Plan in the Key Research Field(Grant No.2020RT08)The Funda mental Research Funds for the Central Universities(Grant No.3132021139)。
文摘With the rapid development of the shipping industry,the safety and comfort of ship transportation have been paid more and more attention,and the pitch and heave motion of ships are the most serious factors.In this paper,the longitudinal motion mathematical model of YuKun is esta b lished.By assigning the zero-pole to the left half-plane and using the properties of the symmetric matrix,the shaping weighting functions matrix is designed to stabilize the multi-input multi-output(MIMO)system of YuKun.Finally,a new concise robust controller is designed using the steady output of the shaped system.The simulation results show that under the control of the concise robust controller,the pitch angle and heave of YuKun decrease by 79.9% and 86.2%,respectively.Theoretical analysis and simulation results show that the concise robust controller has a good control effect on the longitudinal motion of YuKun,and is simple and easy to use,with clear engineering significance.
基金This research is partially supported by the National Science Foundation of China(Grant No.11271079,10671095)RG 11/2015-2016R from the Education University of Hong Kong。
文摘In this paper,we prove the existence of general Cartesian vector solutions u=b(t)+A(t)x for the Ndimensional compressible Navier–Stokes equations with density-dependent viscosity,based on the matrix and curve integration theory.Two exact solutions are obtained by solving the reduced systems.
基金Project 19771020 supported by National Science Foundation of China.
文摘Let T1,n be an n x n unreduced symmetric tridiagonal matrix with eigenvaluesand is an (n - 1) x (n - 1) submatrix by deleting the kth row and kth column, k = 1, 2,be the eigenvalues of T1,k andbe the eigenvalues of Tk+1,nA new inverse eigenvalues problem has put forward as follows: How do we construct anunreduced symmetric tridiagonal matrix T1,n, if we only know the spectral data: theeigenvalues of T1,n, the eigenvalues of Ti,k-1 and the eigenvalues of Tk+1,n?Namely if we only know the data: A1, A2, An,how do we find the matrix T1,n? A necessary and sufficient condition and an algorithm ofsolving such problem, are given in this paper.
基金This work was supported by The National Natural Science Foundation of China, under grant 10271074.
文摘In this paper, we discuss an inverse eigenvalue problem for constructing a 2n × 2n Jacobi matrix T such that its 2n eigenvalues are given distinct real values and its leading principal submatrix of order n is a given Jacobi matrix. A new sufficient and necessary condition for the solvability of the above problem is given in this paper. Furthermore, we present a new algorithm and give some numerical results.
基金Subsidized by the Special Funds for Major State Basic Research Projects G1999032803 and Suported bythe National Natural Scienc
文摘Presents preconditioning matrices having parallel computing function for the coefficient matrix and a class of parallel hybrid algebraic multilevel iteration methods for solving linear equations. Solution to elliptic boundary value problem; Discussion on symmetric positive definite matrix; Computational complexities.