Fast and Numerically Stable Algorithms for Discrete Hartley Transforms and Applications to Preconditioning

Abstract

The discrete Hartley transforms (DHT) of types I -- IV and the related matrix algebras are discussed. We prove that any of these DHTs of length $N=2^t$ can be factorized by means of a divide--and--conquer strategy into a product of sparse, orthogonal matrices where in this context sparse means at most two nonzero entries per row and column. The sparsity joint with orthogonality of the matrix factors is the key for proving that these new algorithms have low arithmetic costs equal to $\frac52 N\log_2 (N)+O(N)$ arithmetic operations and an excellent normwise numerical stability. Further, we consider the optimal Frobenius approximation of a given symmetric Toeplitz matrix generated by an integrable symbol in a Hartley matrix algebra. We give explicit formulas for computing these optimal approximations and discuss the related preconditioned conjugate gradient (PCG) iterations. By using the matrix approximation theory, we prove the strong clustering at unity of the preconditioned matrix sequences under the sole assumption of continuity and positivity of the generating function. The multilevel case is also briefly treated. Some numerical experiments concerning DHT preconditioning are included.