Construction of symmetric pentadiagonal matrix from three interlacing spectrum

Abstract. in this paper, we introduce a new algorithm for constructing a symmetric pentadiagonal matrix by using three interlacing spectrum, say (λi) n i=1, (µi) n i=1 and (νi) n i=1 such that 0 < λ1 < µ1 < λ2 < µ2 < ... < λn < µn, µ1 < ν1 < µ2 < ν2 < ... < µn < νn, where (λi) n i=1 are the eigenvalues of pentadiagonal matrix a, (µi) n i=1 are the eigenvalues of a∗ (the matrix a∗ differs from a only in the (1, 1) entry) and (νi) n i=1 are the eigenvalues of a∗∗ (the matrix a∗∗ differs from a∗ only in the (2, 2) entry). from the 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.