A Conjugate-Gradients Based Method 
                for Harmonic Retrieval Problems that
         Does Not Use Explicit Signal Subspace Computation

          F. S. V. Bazan, Ph. L. Toint and M. C. Zambaldi

                           Report 97-16

The harmonic retrieval problem consists of estimating frequencies and decay
factors of multiple exponential signals from experimental measurements.  At
variance with  most available algorithms,  which are  based on the singular
value decomposition (SVD) and  the explicit identification of the so-called
``signal subspace'',  we present    an  alternative method in    which this
explicit identification is not needed.  We propose to replace it by an {\em
implicit} estimation  using      the concept of  predictor     matrices and
algorithmically realized using conjugate gradients. We finally discuss some
numerical examples, both synthetic  and  from a real application,   showing
that the new method produces results  of an accuracy  comparable to that of
SVD-based subspace techniques, but  without requiring a priori knowledge of
the dimension of the signal subspace and at a lower computational cost.