Solution of large dense complex matrix equations using a fast Fourier transform (FFT)-based wavelet-like methodology

Wavelet-like transformations have been used in the past to compress dense large matrices into a sparse system. However, they generally are implemented through a finite impulse response filter realized through the formulation of Daubechies. In this paper, a method is proposed to use a very high order filter (namely an ideal one) and use the computationally efficient fast Fourier transform (FFT) to carry out the multiresolution analysis. The goal here is to reduce the redundancy in the system and also guarantee that the wavelet coefficients drop off much faster. Hence, the efficiency of the new procedure becomes clear for very high order filters. The advantage of the FFT-based procedure utilizing ideal filters is that it can be computationally efficient and for very large matrices may yield a sparse matrix. However, this is achieved, as well known in the literature, at the expense of robustness, which may lead to a larger reconstruction error due to the presence of the Gibb's phenomenon. Numerical examples are presented to illustrate the efficiency of this procedure as conjectured in the literature.