An Inverse Eigenvalue Problem For Periodic Jacobi Matrix

In this paper, we mainly discuss how to solve a new type of inverse eigenvalue problem of periodic Jacobi Matrix. If we write the periodic Jacobi matrix (?)_n as:A new inverse eigenvalue problem has put forward as follows: Given the data:λ= {λ₁≤λ₂≤...≤λ_n} , v= {v₁ < v₂ < ... < v_n} andβ> 0, how do we construct a periodic Jacobi matrix (?)_n, which must satisfies the following conditions:By permuting the rows and columns of J_n, we may got a new matrix J_n^*Calculating its eigenpolynomial, we get: We substitute v_k forλin this equation; however, due to the overlap betweenλand v, there are two kinds of situations:1. if v_k andγ_i, do not have double root, we substitute v_k forλdirectly.2. if v_k andγ_i, do have double root, we calculate the limit of both sides asλapproaches tov_k.Both of these two situations will derive to a non-linear equation set, which may be solved by Newton iterative method. After getting the numerical solutions, we may calculate the values ofα₁,α_n,β₁,β_n-1. What's more, by applying Lanczos method, we may get the values of elements in J_2,(n-1) Thus, we construct the periodic Jacobi matrix (?)_n .