University of Guilan Journal of Algebra and Related Topics 2345-3931 10 2 2022 12 01 Construction of symmetric pentadiagonal matrix from three interlacing spectrum 89 98 6003 10.22124/jart.2022.19706.1276 EN K. Ghanbari Department of Mathematics, Sahand University of Technology, Tabriz, IRAN M. Rahimnevasi Moghaddam Department of Mathematics, Sahand University of Technology, Tabriz, Iran Journal Article 2021 05 23 ‎In this paper‎, ‎we introduce a new algorithm for constructing a‎ ‎symmetric pentadiagonal matrix by using three interlacing spectrum‎, ‎say $(\lambda_i)_{i=1}^n$‎, ‎$(\mu_i)_{i=1}^n$ and $(\nu_i)_{i=1}^n$‎ ‎such that‎<br />‎\begin{eqnarray*}‎<br />‎0<\lambda_1<\mu_1<\lambda_2<\mu_2<...<\lambda_n<\mu_n,\\‎<br />‎\mu_1<\nu_1<\mu_2<\nu_2<...<\mu_n<\nu_n‎,<br />‎\end{eqnarray*}‎<br />‎where $(\lambda_i)_{i=1}^n$ are the eigenvalues of pentadiagonal‎ ‎matrix $A$‎, ‎$(\mu_i)_{i=1}^n$ are the eigenvalues of $A^*$ (the‎   ‎matrix $A^*$ differs from $A$ only in the $(1,1)$ entry) and‎ ‎$(\nu_i)_{i=1}^n$ are the eigenvalues of $A^{**}$ (the matrix‎ ‎$A^{**}$ differs from $A^*$ only in the $(2,2)$ entry)‎. ‎From the‎<br />‎interlacing 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. https://jart.guilan.ac.ir/article_6003_ba3dd30e6883dbfabe73cd10068b9fe7.pdf