INVERSE UNITARY EIGENPROBLEMS AND RELATED ORTHOGONAL FUNCTIONS

This paper explores the relationship between certain inverse unitary e igenvalue problems and orthogonal functions, In particular, the invers e eigenvalue problems for unitary Hessenberg matrices and for Schur pa rameter pencils are considered. The Szego recursion is known to be ide ntical to the Arnoldi process and can be seen as an algorithm for solv ing an inverse unitary Hessenberg eigenvalue problem. Reformulation of this inverse unitary Hessenberg eigenvalue problem yields an inverse eigenvalue problem for Schur parameter pencils, It is shown that solvi ng this inverse eigenvalue problem is equivalent to computing Laurent polynomials orthogonal on the unit circle. Efficient and reliable algo rithms for solving the inverse unitary eigenvalue problems are given w hich require only O(mn) arithmetic operations as compared with O(mn(2) ) operations needed for algorithms that ignore the structure of the pr oblem.