A tridiagonal matrix construction by the quotient difference recursion formula in the case of multiple eigenvalues
The authors consider the inverse eigenvalue problem to find, from (multiple) eigenvalues, a tridiagonal matrix with these eigenvalues, including their multiplicities. For this, they use the classical quotient difference (qd) algorithm, see, e.g. [\textit{M. H. Gutknecht} and \textit{B. N. Parlett}, IMA J. Numer. Anal. 31, No. 3, 741--754 (2011; Zbl 1222.65032)]. This possibility seems to be have known to \textit{H. Rutishauser} [Lectures on numerical mathematics. Ed. by Martin Gutknecht with the assistance of Peter Henrici, Peter Läuchli and Hans-Rudolf Schwarz. Transl. by Walter Gautschi. Boston etc.: Birkhäuser (1990; Zbl 0699.65002)], but the present authors prove that it works. They need, along with the eigenvalues, their multiplicities and also the degrees of the characteristic polynomial and of the minimal polynomial as obtained if the eigenvalues are written in a Jordan form matrix. To start the qd algorithm, they also need two vectors the choice of which is said to be not problematic. The authors prove that the minimal polynomial of the tridiagonal matrix obtained is equal to its characteristic polynomial, containing the prescribed eigenvalues, and that the tridiagonal matrix is diagonalizable if and only if there are no multiple eigenvalues. The algorithm is thought to be used for symbolic computation because of rounding errors and their consequences under floating point computations. Several examples are provided for the algorithm.
scientific article; zbMATH DE number 6506869
scientific article