Construction of symmetric pentadiagonal matrix from three interlacing spectrum

‎In this paper‎, ‎we introduce a new algorithm for constructing a‎ ‎symmetric pentadiagonal matrix by using three interlacing spectrum‎, ‎say (\lambda_i)_{i=۱}^n‎, ‎(\mu_i)_{i=۱}^n and (\nu_i)_{i=۱}^n‎ ‎such that‎‎\begin{eqnarray*}‎‎۰<\lambda_۱<\mu_۱<\lambda_۲<\mu_۲<...<\lambda_n<\mu_n,\\‎‎\mu_۱<\nu_۱<\mu_۲<\nu_۲<...<\mu_n<\nu_n‎,‎\end{eqnarray*}‎‎where (\lambda_i)_{i=۱}^n are the eigenvalues of pentadiagonal‎ ‎matrix A‎, ‎(\mu_i)_{i=۱}^n are the eigenvalues of A^* (the‎   ‎matrix A^* differs from A only in the (۱,۱) entry) and‎ ‎(\nu_i)_{i=۱}^n are the eigenvalues of A^{**} (the matrix‎ ‎A^{**} differs from A^* only in the (۲,۲) entry)‎. ‎From the‎‎interlacing spectrum‎, ‎we find the first and second columns of‎ ‎eigenvectors‎. ‎Sufficient conditions for the solvability of the problem‎ ‎are given‎. ‎Then we construct the pentadiagonal matrix A from these‎ ‎eigenvectors and given eigenvalues by using the block Lanczos algorithm‎. ‎We‎ ‎also give an example to demonstrate the efficiency of the algorithm‎.

Keywords:

Inverse eigenvalue problem , Pentadiagonal matrix , Interlacing‎ ‎property , Lanczos algorithm