Let ∑, Г be two n-by-n diagonal matrices with σi,γi as their diagonals. For the inverse eigenvalue problem: look for y∈Rn such that Г + yyT is similar to ∑, we prove thatu also the sufficient condition for the ...Let ∑, Г be two n-by-n diagonal matrices with σi,γi as their diagonals. For the inverse eigenvalue problem: look for y∈Rn such that Г + yyT is similar to ∑, we prove thatu also the sufficient condition for the solvability of this inverse problem. Its solution (set) is given explicitly. In some case, the problem is unstable. But we prove that the sums of the square of some contigious components keep stable, i.e., small sum keeps small, large sum has a small relative perturbation, see Theorem 3.展开更多
The parallel multisection method for solving algebraic eigenproblem has been presented in recent years with the developing of the parallel computers, but all the research work is limited in standard eigenproblem of sy...The parallel multisection method for solving algebraic eigenproblem has been presented in recent years with the developing of the parallel computers, but all the research work is limited in standard eigenproblem of symmetric tridiagonal matrix. The multisection method for solving generalized eigenproblem applied significantly in many secience and engineering domains has not been studied. The parallel region--preserving multisection method (PRM for shotr) for solving generalized eigenproblem of large sparse real symmetric matrix is presented in this paper. This method not only retains the advantages of the conventional determinant search method (DS for short), but also overcomes its disadvantages such as leaking roots and disconvergence. We tested the method on the YH--1 vector computer,and compared with the parallel region-preserving determinant search method (parallel region--preserving bisection method)(PRB for short). The numerical results show that PRM has a higher speed-up, for instance it attains the speed-up of 7.7 when the scale of the problem is 2114 and the eigenpair found is 3; and PRM is superior to PRB when scale of the problem is large.展开更多
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 dis...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.展开更多
In this article, we study the solvability of nonlinear problem for p-Laplacian with nonlinear boundary conditions. We give some characterization of the first eigenvalue of an intermediary eigenvalne problem as simplic...In this article, we study the solvability of nonlinear problem for p-Laplacian with nonlinear boundary conditions. We give some characterization of the first eigenvalue of an intermediary eigenvalne problem as simplicity, isolation and its strict monotonicity. Afterward, we character also the second eigenvalue and its strictly partial monotony. On the other hand, in some sense, we establish the non-resonance below the first and furthermore between the first and second eigenvalues of nonlinear Steklov-Robin.展开更多
文摘Let ∑, Г be two n-by-n diagonal matrices with σi,γi as their diagonals. For the inverse eigenvalue problem: look for y∈Rn such that Г + yyT is similar to ∑, we prove thatu also the sufficient condition for the solvability of this inverse problem. Its solution (set) is given explicitly. In some case, the problem is unstable. But we prove that the sums of the square of some contigious components keep stable, i.e., small sum keeps small, large sum has a small relative perturbation, see Theorem 3.
文摘The parallel multisection method for solving algebraic eigenproblem has been presented in recent years with the developing of the parallel computers, but all the research work is limited in standard eigenproblem of symmetric tridiagonal matrix. The multisection method for solving generalized eigenproblem applied significantly in many secience and engineering domains has not been studied. The parallel region--preserving multisection method (PRM for shotr) for solving generalized eigenproblem of large sparse real symmetric matrix is presented in this paper. This method not only retains the advantages of the conventional determinant search method (DS for short), but also overcomes its disadvantages such as leaking roots and disconvergence. We tested the method on the YH--1 vector computer,and compared with the parallel region-preserving determinant search method (parallel region--preserving bisection method)(PRB for short). The numerical results show that PRM has a higher speed-up, for instance it attains the speed-up of 7.7 when the scale of the problem is 2114 and the eigenpair found is 3; and PRM is superior to PRB when scale of the problem is large.
基金Research supported by the National Natural Science Foundation of China. (10571047)
文摘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.
文摘In this article, we study the solvability of nonlinear problem for p-Laplacian with nonlinear boundary conditions. We give some characterization of the first eigenvalue of an intermediary eigenvalne problem as simplicity, isolation and its strict monotonicity. Afterward, we character also the second eigenvalue and its strictly partial monotony. On the other hand, in some sense, we establish the non-resonance below the first and furthermore between the first and second eigenvalues of nonlinear Steklov-Robin.
