On a Matrix Algebra Related to the Discrete Hartley Transform

Fast and numerically stable algorithms for discrete Hartley transforms and applications to preconditioning

“…We will see that for every X ∈ {I, II, III, IV}, the sets Diag(H X N ) consist of special symmetric Toeplitz-plusHankel matrices. More specifically, in this section we improve previous results by Bini and Favati [4] and by Bortoletti and Di Fiore [5]. Bini and Favati [4] have considered the Hartley matrix algebra of type I and, recently, Bortoletti and Di Fiore [5] have analyzed the Hartley matrix algebras of types X ∈ {II, III, IV}.…”

“…By ⊕ we denote the orthogonal sum of linear subspaces of R N ×N . The following theorem improves a corresponding result of Bini and Favati [4], more specifically, the orthogonal sum representation (4.2) and formula (4.3) are new.…”

Matrix structures and applications

“…The discrete Hartley transform is defined as x → y = Gx where G = 1 √ n (cos(ij π n ) + sin(ij π n )). The algebra associated with this transform has been studied in [13] where in particular it is shown that the Hartley algebra contains symmetric circulant matrices.…”

Matrix algebras in Quasi-Newton methods for unconstrained minimization

“…We point out that the latter property was already implemented in preconditioning techniques to reduce the number of steps of Conjugate Gradient (CG) methods for pd linear systems Bx = v [4], [12], [13], [22], [28], [29], [36], [40]. Moreover, the class of algebras of L-type, for which the above property holds, has been also used to define more efficient direct methods solving the system Bx = v. By utilizing simple displacement decompositions of B −1 in terms of matrices of L-type, it is, in fact, possible to compute B −1 v via a finite number of fast unitary transforms [5], [8], [17], [21], [30].…”