**AAS Meeting #193 - Austin, Texas, January 1999**

*Session 77. Astrophysical Processes and Computational Techniques*

Display, Friday, January 8, 1999, 9:20am-6:30pm, Exhibit Hall 1
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## [77.04] Improved Quasi-discrete Hankel Transform

*Li Yu, Wei Ji (Dept. of Physics, Nat'l Univ. of Singapore)*

Hankel transform (HT) is very useful in geophysics,
astronomy, electromagnetics, optics and other branches of
physics. Previously there were many discussions on
algorithms of HT. In general, these algorithms were not
exact discrete-HT (DHT), for the reason that the sampled
points are not directly decided by Bessel function.
Quasi-discrete HT (QDHT), originally developed by us
[Opt.Lett.23, 409,1998], is not only a very efficient
algorithm, but also a theoretical framework of DHT in which
sample theorem and discrete Parseval's theorem are included.
However, QDHT is not an exact DHT, because the product of
two QDHT matrixes is not an unitary matrix. This leads to
increasing errors in the iterative calculations like phase
retrieval problem. Our purpose is to introduce an operation
of improving the QDHT matrix numerically, whose process is
descibed as below. Let M_{0} and M_{0}^{-1} denote a
QDHT matrix and its inverse matrix respectively. One such
operation includes two steps: first, let a new matrix\
M_{1} equal (M_{0}+M_{0}^{-1})/2; second, get its inverse
matrix M_{1}^{-1}. M_{1} or M_{1}^{-1} is the improved
matrix. This operation can be repeated again and again until
the improved matrix meets the prescribed exactness. In fact,
the product of two improved-QDHT matrixes is much more
closed to unitary matrix than = that of QDHT matrixes. For
example, the error of 4 dimensional QDHT matrix is
10^{-7}. And the error of improved matrix is about of
10^{-15} after 5 times of improving operations. The
inverse matrix is easily got by software like QTR *
mathematica*.

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