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 1. 对称正交对称矩阵反问题的最小二乘解  被引次数：18 戴华《计算数学》,2003年第25卷第1期 Let P ∈ Rn×n be a symmetric orthogonal matrix. A∈Rn×n is called a symmetric orthogonal symmetric matrix if AT = A and (PA) T = PA. The set of all n × n symmetric orthogonal symmetric matrices is denoted by SRnxnp. This paper discusses the following problems: Problem I. Given X,B∈ Rn×m, find A ∈SRn×np such that||AX - B|| = min Problem II. Given A∈ Rn×n, find A∈SL such thatwhere ||·|| is the Frobenius norm, and SL is the solution set of Problem I.The general form of SL is given. The solvability conditions for the inverseproblem AX = B in SRn×nP are obtained. The expression of the solution toProblem II is presented. 2. LEAST—SQUARES SOLUTIONS OF X^TAX=B OVER POSITIVE SEMIDEFINITE MATRIXES A Dongxiu Xie+ 《计算数学(英文版)》,2003年第21卷第2期 This paper is mainly concerned with solving the following two problems: Problem Ⅰ. Given X ∈ Rn×m, B . Rm×m. Find A ∈ Pn such thatwhereProblem Ⅱ. Given A ∈Rn×n. Find A ∈ SE such thatwhere F is Frobenius norm, and SE denotes the solution set of Problem I.The general solution of Problem I has been given. It is proved that there exists a unique solution for Problem II. The expression of this solution for corresponding Problem II for some special case will be derived. 3. THE INVERSE PROBLEM FOR PART SYMMETRIC MATRICES ON A SUBSPACE  被引次数：1 Zhen-yun Peng《计算数学(英文版)》,2003年第21卷第4期 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. 4. THE SOLVABILITY CONDITIONS FOR INVERSE EIGENVALUE PROBLEM OF ANTI-BISYMMETRIC MATRICES  被引次数：3 Dong-xiu Xie 《计算数学(英文版)》,2002年第3期 AbstractThis paper is mainly concerned with solving the following two problems: Problem I. Given X Cnxm, A = diag( 1, 2, ..... , m) Cmxm . Find A ABSRnxn such thatAX = XAwhere ABSRnxn is the set of all real n x n anti-bisymmetric matrices. Problem II. Given A RnXn. Find A SE such thatwhere || || is Frobenius norm, and SE denotes the solution set of Problem I.The necessary and sufficient conditions for the solvability of Problem I have been studied. The general form of SB has been given. For Problem II the expression of the solution has been provided. 5. A Class of Constrained Inverse Eigenproblem and Associated Approximation Problem for Symmetric Reflexive Matrices Xiaoping Pan  Xiyan Hu and Lei Zhang College of Mathematics and Econometrics  Hunan University  Changsha 410082  China.《高等学校计算数学学报(英文版)》,2006年第15卷第3期 Let S∈Rn×n be a symmetric and nontrival involution matrix. We say that A∈E R n×n is a symmetric reflexive matrix if AT = A and SAS = A. Let S R r n×n(S)={A|A= AT,A = SAS, A∈Rn×n}. This paper discusses the following two problems. The first one is as follows. Given Z∈Rn×m (m < n),∧= diag(λ1,...,λm)∈Rm×m, andα,β∈R withα<β. Find a subset (?)(Z,∧,α,β) of SRrn×n(S) such that AZ = Z∧holds for any A∈(?)(Z,∧,α,β) and the remaining eigenvaluesλm 1 ,...,λn of A are located in the interval [α,β], Moreover, for a given B∈Rn×n, the second problem is to find AB∈(?)(Z,∧,α,β) such that where ||.|| is the Frobenius norm. Using the properties of symmetric reflexive matrices, the two problems are essentially decomposed into the same kind of subproblems for two real symmetric matrices with smaller dimensions, and then the expressions of the general solution for the two problems are derived. 6. COMPUTING THE NEAREST BISYMMETRIC POSITIVE SEMIDEFINITE MATRIX UNDER THE SPECTRAL RESTRICTION 谢冬秀  盛炎平  张忠志《高等学校计算数学学报(英文版)》,2003年第12卷第1期 Let A and C denote real n × n matrices. Given real n-vectors x1, ... ,xm, m ≤ n, and a set of numbers L = {λ1,λ2,... ,λm}. We describe (I) the set (?) of all real n × n bisymmetric positive seidefinite matrices A such that Axi is the "best" approximate to λixi, i = 1,2,...,m in Frobenius norm and (II) the Y in set (?) which minimize Frobenius norm of ||C - Y||.An existence theorem of the solutions for Problem I and Problem II is given and the general expression of solutions for Problem I is derived. Some sufficient conditions under which Problem I and Problem II have an explicit solution is provided. A numerical algorithm of the solution for Problem II has been presented. 7. INVERSE EIGENVALUE PROBLEM OF HERMITIAN GENERALIZED ANTI-HAMILTONIAN MATRICES Zhang Zhongzhi Liu ChangrongSchool of Math. Science  Central South Univ.  Changsha 410083  China   Dept. of Math.  Hunan City Univ.  Yiyang 413000  China. Faculty of Mathematics and Econometrics  Hunan Univ.  Changsha 410082  China.《高校应用数学学报(英文版)》,2004年第3期 §1　IntroductionWe considerthe following inverse eigenvalue problem offinding an n-by-n matrix A∈S such thatAxi =λixi,i =1,2 ,...,m,where S is a given set of n-by-n matrices,x1 ,...,xm(m≤n) are given n-vectors andλ1 ,...,λmare given constants.Let X=(x1 ,...,xm) ,Λ=(λ1 ,λ2 ,...,λm) ,then the above inverse eigenvalue problemcan be written as followsProblem Given X∈Cn×m,Λ=(λ1 ,...,λm) ,find A∈S such thatAX =XΛ,where S is a given matrix set.We also discuss the so-called opti… 8. THE BI-SELF-CONJUGATE AND NONNEGATIVE DEFINITE SOLUTIONS TO THE INVERSE EIGENVALUE PROBLEM OF QUATERNION MATRICES 褚玉明《数学物理学报(B辑英文版)》,2005年第25卷第3期 The main aim of this paper is to discuss the following two problems:λm)∈Hm×m, find A ∈ BSH≥n×n such that AX= X∧, where BSH≥n×n denotes the set of all n × n quaternion matrices which are bi-self-conjugate and nonnegative definite.Problem Ⅱ:Given B ∈ Hn×m, find -B∈SE such that ||B- B||Q = minA∈sE ||B - A||Q,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. 9. SUFFICIENT CONDITIONS FOR THE SOLUBILITY OF ADDITIVE INVERSE EIGENVALUE PROBLEMS 《高等学校计算数学学报(英文版)》,2000年第Z1期 1 IntroductionLet R~(n×n) be the set of all n×n real matrices.R~n=R~(n×1).C~(n×n)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_(ij)),and n distinct real numbers λ_1,λ_2,…,λ_n,find a real n×n diagonal matrix D=diag 10. 一类反对称正交反对称矩阵逆特征值问题 周富照  胡锡炎  张磊《数学的实践与认识》,2007年第37卷第4期 设P为一给定的对称正交矩阵,记AARnP={A∈Rn×n‖AT=-A,(PA)T=-PA}.讨论了下列问题:问题给定X∈Cn×m,Λ=diag(λ1,λ2,…,λm).求A∈AARPn使AX=XΛ.问题设A~∈Rn×n,求A*∈SE使‖A~-A*‖=infA∈SE‖A~-A‖,其中SE为问题的解集合,‖.‖表示Frobenius范数.研究了AARPn中元素的通式,给出了问题解的一般表达式,证明了问题存在唯一逼近解A*,且得到了此解的具体表达式. 11. THE EIGENVALUE PROBLEM OF QUASILINEAR ELLIPTIC EQUATION 张正杰《数学物理学报(B辑英文版)》,1991年第1期 In this paper, we are concerned with the following eigenvalue problems Find a function u and a real number λ such that When p=2, above problems are of semilinear eigenvalue problems on R~N 12. THE SOLVABILITY CONDITIONS FOR AN INVERSE EIGEN-PAIR PROBLEM 张磊  胡锡炎《高等学校计算数学学报(英文版)》,1995年第1期 This paper discusses problem IEP:Given n×m matrix X and m×m diagonal matrix A, find an n×n matrix A such that AX=XA.The new solvablily conditions for the problem IEP are obtained. The eigenvalue dislribulaion of the solutions for the problem IEP are described in detail. 13. OPTIMAL APPROXIMATE SOLUTION OF THE MATRIX EQUATION AXB = C OVER SYMMETRIC MATRICES Anping Liao Yuan Lei《计算数学(英文版)》,2007年第25卷第5期 Let SE denote the least-squares symmetric solution set of the matrix equation A×B = C, where A, B and C are given matrices of suitable size. To find the optimal approximate solution in the set SE to a given matrix, we give a new feasible method based on the projection theorem, the generalized SVD and the canonical correction decomposition. 14. Least-Squares Solution of Inverse Problem for Hermitian Anti-reflexive Matrices and Its Appoximation Zhen Yun PENG Yuan Bei DENG Jin Wang LIU《数学学报(英文版)》,2006年第22卷第2期 In this paper, we first consider the least-squares solution of the matrix inverse problem as follows： Find a hermitian anti-reflexive matrix corresponding to a given generalized reflection matrix J such that for given matrices X, B we have minA ｜｜AX - B｜｜. The existence theorems are obtained, and a general representation of such a matrix is presented. We denote the set of such matrices by SE. Then the matrix nearness problem for the matrix inverse problem is discussed. That is： Given an arbitrary A^＊, find a matrix A E SE which is nearest to A^＊ in Frobenius norm. We show that the nearest matrix is unique and provide an expression for this nearest matrix. 15. Spectral Asymptotics for Laplacian with Mixed Boundary-value Condition 毛仕宽  陈化《东北数学》,2004年第2期 Let Ω be a non-empty bounded open set in Rn(n ≥1) with boundary (?)Ω=Γ1∪Γ2. WedefineIn this paper, we consider the following variational eigenvalue problem:where △ denotes the Laplacian in Ω. We say that the scalar λ is an eigenvalue of (P) if 16. 实对称矩阵广义特征值反问题  被引次数：10 戴华《高校应用数学学报(A辑)》,1992年第7卷第2期 本文研究如下实对称矩阵广义特征值反问题: 问题IGEP,给定X∈R~(n×m),1=diag(λ_II_k_I,…,λ_pI_k_p)∈R~(n×m),并且λ_I,…,λ_p互异,sum from i=1 to p(k_i=m,求K,M∈SR~(n×n),或K∈SR~(n×n),M∈SR_0~(n×m),或K,M∈SR_0~(n×n),或K∈SR~(n×n),M∈SR_+~(n×n),或K∈SR_0~(n×n),M∈SR_+~(n×n),或K,M∈SR_+~(n×m), (Ⅰ)使得 KX=MXA, (Ⅱ)使得 X~TMX=I_m,KX=MXA,其中SR~(n×n)={A∈R~(n×n)|A~T=A},SR_0~(n×n)={A∈SR~(n×n)|X~TAX≥0,X∈R~n},SR_+~(n×n)={A∈SR~(n×n)|X~TAX>0,X∈R~n,X≠0}. 利用矩阵X的奇异值分解和正交三角分解,我们给出了上述问题的解的表达式. 17. MULTIPLE POSITIVE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEMS INVOLVING CONCAVE-CONVEX NONLINEARITIES AND MULTIPLE HARDY-TYPE TERMS Tsing-San HSU《数学物理学报(B辑英文版)》,2013年第5期 In this paper, we deal with the existence and multiplicity of positive solutions for the quasilinear elliptic problem??pu?kX i=1 μi |u|p?2|x?ai|p u=|u|p*?2 u+λ|u|q?2 u, x∈?, where??RN (N ≥3) is a smoot... 18. A New Interior Path Following Method for Nonconvex Nonlinear Programming 黄纯一  于波  王宇  冯果忱《东北数学》,1997年第3期 Consider the following nonlinear programming problem:where f, gi's are sufficiently smooth functions in R~n. LetIt is well known that if x~*∈Ω is a solution of (1) then there exists λ~*=(λ_I~*,…,λ_m~*)∈R~m such that(x~*,λ~*) is a solution of the K-K-T system 19. Spinless Bosons in a 1D Harmonic Trap with Repulsive Delta Function Interparticle Interaction I： General Theory 马中骐  杨振宁《中国物理快报》,2009年第12期 In these two papers, we solve the N body 1D harmonically trapped spinless Boson problem with repulsive δ function interaction in the limit N→∞. The general theory is given in paper I and the numerical solutions will be given in paper II. 20. 实对称矩阵的两类逆特征值问题  被引次数：95 孙继广《计算数学》,1988年第10卷第3期 §gi.两类逆特征值问题先说明一些记号.R~(m×n)是所有m×n实矩阵的全体,R~n=R~(n×1),R=R~1;SR~(n×n)是 所有n×n实对称矩阵的全体;OR~(n×n)是所有n×n实正交矩阵的全体;I~((n))是n阶单位矩阵;A~T是矩阵A的转置;A>0表示A是正定的实对称矩阵.?(A)是矩阵A的列空间;A~+是矩阵A的Moore-Penrose广义逆;P_A=AA~+表示到?(A)的正交投影.λ(A)是A的特征值的全体;λ(K,M)是广义特征值问题K_x=λM_x的特征值的
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