Hartley-type algebras in displacement and optimization strategies.
The purpose of this paper is to show that a class of structural algebras, the Hartley-type (Ht) algebras linked in different ways to the Hartley transform, are extremly useful in solving both linear and nonlinear (minimizaton) problems. The matrix representations obtained in this paper are classical displacement formulas and lead to efficient Toeplitz inversion formulas (the inverse is rewritten in terms of a number of simple matrices which can be fastly multiplied by vectors) competitive with the well known Ammar-Gader formula based on circulant algebras. The best least squares approximation is exploited as a preconditioner in quasi-Newton methods (QN) for unconstrained minimization. For novel \(L\)QN minimization methods an algebra \(L\) (\(L=\text{Ht})\) of matrices simultnenously diagonalized by a fast unitary transform is used in the iterative scheme of generalized BFGS-type algorithm. The complexity per step of \(L\)QN is \(O(n\log n)\) and only \(O(n)\) memory allocationsare required. It is shown that \(L\)QN methods are linearly convergent.
scientific article; zbMATH DE number 1915806
scientific article