| This paper is seperated into three parts:the first is the introduction of the inverse eigenvalue problem,then we give a numerical method to the nonnegative prescribed entries inverse eigenvalue problem,finaly ,numerical examples are showed to illustate the effectiveness of our approach.In mathematical physics,the direct probem is to seek changes in the system according to the system equation and boundary conditions;on the other hand,inverse problem in mathematical physics is to find the border, coefficients or some parameters from the solution and some other conditions. Inverse eigenvalue problem ,IEP for short,refers to reconstruct matrices from eigenvalues or partial eigenvalues and eigenvectors. The source of the IEP is extensive.The reseach of IEP mainly contains the following three aspects :(1)Solvability,including sufficient the necessary conditions of solutions.(2)Posedness,that is the existence, uniqueness and stability of solutions.(3)Numerical methods.Classical additictive inverse eigenvalue problem is much more mature and there are some complete results about it. Here we make a brief presentation of classical additictive inverse eigenvalue problem. It consists of the following major issues :Additictive inverse eigenvalue problem.Let matrix A and A = diag{λ1,…λn} belong to field Pn×n,try to find X = diag{x1,x2,…,xn}∈Pn×n such thatλ(A + X) =λ(∧).Multiplicative inverse eigenvalue problem.Let matrix A and A = diag{λ1,…λn} belong to field Pn×n ,try to find X = diag{x1,x2,…,xn}∈Pn×nsuch that Best approximation with spectra constraints. Let matrix diag{λ1,…λn} is the spectra of A. find A|∈Rn×n such thatwhereandwhere xi is the eigenvector ofλi, such thatThere are some more kinds of classical IEPs,such as Jacobi matrix inverse eigenvalue problem, real symmetric inverse eigenvalue banded matrix,generalized matrix inverse eigenvalue problem, and inverse singular value problem, more details can be found in [20].Next non-negative inverse spectral problem will be talked. Non-negative inverse spectral problem has a wide range of applications in physics,economics, engineering, operating and many other areas , But in theory, it is not solved entirely. Here ,we briefly talk about some of the results those have been done.Theorem(Perfect and Mirsky[14]).If 1≥λ2≥…≥λn≥-1, andthen there exists a symmetric doubly stochastic matrix X, such that X(X) = diag{1,λ2,…,λn}. Later in 1983,Soules[15]generalizad the above thoerem by proving the following theorem. Theorem (Soules) .if , andwhere m=[(n-1)/2], repesents the integer part of (n-1)/2, then there exists a symmetric doubly stochastic matrix X, such thatλ(X) = diag{1,λ2,…,λn}Theorem(Kellogg).Letλ= {λ0,λ1,…,λN}∈Rn be a set of real numbers withand let M be the greatest index j(0≤j < N)for whichλj≥0.Let the set of indicesIfandprovided that N≥2M + 1, then there exists an (N + 1)×(N -1)non-negative matrix A, such thatλ(A) =λ.Borobia[24]obtained a generaliztion of the Kellogg result,as follow: Theorem(Borobia).Letλ={λ0,λ1,…,λN}∈Rn be a set of real numbers withand let M be the greatest index j(0≤j < N) for whichλj≥0. If there exists a partion of , for some 1≤S≤N - M, such thatsatisfies the Kellogg condition , then there exists an (N + 1)×(N + 1) non-negative matrix A, such thatλ(A) =λ.Ricardo L.Soto[11]improved Borobia condition, as follow: Theorem(Ricardo L.Soto).Letσ={λ0,λ1,…,λN}∈Rn, such thatIf there exists a partionσ=σ1∪σ2∪…∪σt Withand for kpk oddandsatisfyingthen there exists an generalized stochastic matrix A, such thatλ(A) =λ.More results can be seen in the raletive papers.Chapter II gives an algorithm to calculate the nonnegative prescribed entries inverse eigenvalue problem.This method derives from transforming the nonnegative PEIEP into an optimization problem which then will be solved by the deepest descent algorithm.Nonnegative PEIEP can be described as follow:Given a certain subset L—of pairs of subscripts, , acertainset of value over a field R such that , and a set of n values {λ1,…,λn} over the algebraically closed extention of C,find a matrix X∈Rn×n such thatGiven the setthat is called isospectral surface,where (?) denote the general group of n×n nonsin-gular matrices in Rn×n ;and the setwhich contains all matrices with the prescribed entries at the desired location,where A (?) B = (aij×bij) is called matrices Hadamard product, if A=(aij), B = (bij). Define project P as followwhere LC denotes the positions which satisfy L∪LC equal to the position of all matrix, RLC denotes that the elements of the LC position remain, and let the elements of the LC position equal to zero.Define AL is constant , where , the elements of the LC are zero.So,nonnegative PEIEP can be transformed as follow:Minimize , The Frechet derivative of (?) at points V∈(?), R∈Rn×n actives on H,K∈Rn×n can be calculated as wheredenotes matrices Frobenius product, through the calculation the characteristics of Frobenius product is used:which is easy to proved. VT denotes the transpose of matrix V.So we haveThus nonnegative PEIEP can be transform into an optimization problem from gradient as follow:the initia value can be chosen arbitrarily,say,V(0) = I, R(0) = R0∈G Rn×n.Along the flow the desired matrix can be abtained.The second part of Chapter II we introduce two ways to let the calculation much more stable. Finaly,with the numerical test results,the validity of our approach is proved.Prom the numerical test results it can be seen that non-negative inverse spectral problem has not uniqueness. It is hoped that this paper will provide some new ideas on the theoretical researches of the non-negative inverse spectral problem.whether we could try to restrict the structure of the matrix in order to get it. Also,our approach could support some effective numerical basis on non-negative inverse spectral problem.
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