The Research On Inverse Eigenvalue Problems For Special Matrices

Inverse eigenvalue problem is a hot research orientation.It plays an important role in vibration structure design7 physical characters detection,inversion,singal re-construction and so on.The trend of research on this problem is concerned much by industry and national defence.It has high value in application.This paper mainly researches two inverse eigenvalue problems of Jacobi matrix.Problem Ⅰ.Given a n order Jacobi matrix Jn and a set of distinct eigenvaluesλ1,λ2,…,λ2n,try to construct a 2n order Jacobi matrix J2n.Its eigenvalues are the given values {λ}i=1 2n and the leading n order principal submatrix is exactly Jn.Problem Ⅱ.Given a n order Jacobi matrix Jn and a set of distinct eigenvalues{λi}i=1 2N-2n,try to construct a N order Jacobi matrix JN.Its eigenvalues are the givenvalues {λi}i=1 2N-2n and the leading n order principal submatrix is exactly Jn with N/2<n<N.For the problem Ⅰ,the chapter 2,based on current theory,gives an advanced algorithm by using properties of Jacobi matrix.The algorithm can neither reconstruct leading principal submatrix Jn,nor compute eigenvalues of tail principal submatrix Jn+1,N.The stability,efficiency and precision are improved.For the problem Ⅱ,the chapter 37 based on the algorithm in chapter 27 propos-es two new algorithms.These algorithms should not to compute eigenvalues of tail principal submatrix Jn+1,N and the precision is improved.Chapter 4 induces problem Ⅰ and Ⅱ by practical background and proposes a ca-pacity increasement problem of vibration system and a numerical method for solving this problem by combining with the new algorithms.This paper creates some new numerical methods about two inverse eigenvalue problems of Jacobi matrix,which aims to further reduce error and brings new challenges and achievements for the research of inverse problem theory.