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Manuscript Title: A spline-based method for experimental data deconvolution.
Authors: I. Beniaminy, M. Deutsch
Program title: DECONV
Catalogue identifier: ABVR_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 21(1980)271
Programming language: Fortran.
Computer: IBM 370/168.
Operating system: OS/MVS.
RAM: 52K words
Word size: 32
Keywords: General purpose, Fit, Deconvolution, Unfolding, Desmearing, Resolution enhancement.
Classification: 4.9.

Nature of problem:
In many types of experiments, notably in scattering and spectroscopic measurements of all kinds, the quantity or distribution J(t) measured in the laboratory can be expressed mathematically as a convolution of two functions: J(t) = integral(-inf,+inf)(K(s)Jo(t-s)ds) where K(t) is the resolution (or response) function of the measuring instrument, and Jo(t) is the same quantity as J(t) but measured using an ideal instrument having infinite resolution. The function Jo(t) is, of course, the real information of interest. K(t) is known either from measurements or through theoretical calculations. Since in practice it is impossible to increase the resolution of an instrument past a given value, one must in general employ a deconvolution method to obtain Jo(t) from the J(t) and K(t) data. Although the equation can be formally solved by Fourier transform methods, a straightforward application of such methods without taking care of the random errors inevitably present in all experimental data, is doomed to failure. Iterative methods developed for solving the equation are also very sensitive to the noise present in the J(t) data, and are in general quite time consuming. In reaction to this situation, we have developed a one-step deconvolution method, which does not suffer from the disadvantages mentioned above. It is simple to use, employs neither Fourier transforms nor iterations and yields high accuracy results even in the presence of a high level of random experimental errors in the J(t) data. The present program is an implementation of our deconvolution method.

Solution method:
Let Jo(t) be represented by a piece-wise cubic spline function. Then, upon substitution of this function in, we obtain for J(t) another piece- wise cubic spline, having the same knots, but different coefficients. These coefficients are related to those of the Jo(t) spline via a set of equations involving the moments of the resolution function. These equations can be inverted to obtain the Jo(t) spline coefficients in terms of the J(t)-spline coefficients. Thus, from fitting a piece-wise cubic spline to the J(t) data, we directly obtain the coefficients of the spline function representing Jo(t). This method, being a one-step procedure, is fast and, due to the excellent smoothing properties of the spline function, it is insensitive to even large amplitude random errors in the J(t) data.

Only 250 J(t) values can be treated by the present version in a single run.