Inverse eigenvalue problems linked to rational Arnoldi, and rational, (non)symmetric Lanczos
Thomas Mach, Raf Vandebril, Marc Van Barel · Lirias · 2013
Two inverse eigenvalue problems are discussed. First, given the eigenvalues and a weight vector an extended Hessenberg matrix is computed. This matrix represents the recurrences linked to a (rational) Arnoldi inverse problem. It is well-known that the matrix capturing the recurrence coefficients is of Hessenberg form in the standard Arnoldi case. Considering, however, rational functions and admitting finite as well as infinite poles we will prove that the recurrence matrix is still of a highly structured form — the extended Hessenberg form. An efficient memory-economical representation is proposed and used to reconstruct the matrix capturing the recurrence coefficients. In the second inverse eigenvalue problem, the above setting is generalized to the biorthogonal case: unitarity is dropped in the similarity transformations. Given the eigenvalues and the first component of each left eigenvector and each right eigenvector we want to retrieve the matrix of recurrences as well as the matrices governing the transformation.