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Manuscript Title: A program for the solution of ill-posed linear systems arising from the discretization of Fredholm integral equation of the first kind.
Authors: N.N. Abdelmalek
Program title: MLS
Catalogue identifier: ABLZ_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 58(1990)285
Programming language: Fortran.
Computer: IBM 370.
Word size: 64
Keywords: General purpose, Matrix, Fredholm integral Equation of the First kind, Least squares solution, Minimum length Least squares solution, Ill-posed problems, Ill-conditioned problems.
Classification: 4.8.

Nature of problem:
Fredholm integral equation of the first kind arises in the mathematical analysis of many physical problems. Among these are in optics, the restoration of blurred images.

Solution method:
The discretization of Fredholm integral equation of the first kind results in an ill-posed system of linear equations of the form Ax=b. A smooth solution of this system is obtained by a novel algorithm, the details of which are given in a paper by the author. Briefly, the system Ax=b is premultiplied by 'A**T', where the super- script T refers to the transpose. One gets the consistent system of say M linear equations A**TAx=A**Tb, in M unknowns. The algorithm works in an iterative manner. In iteration k, k=1, 2, ..., it properly permutes the last (M-k+1) equations and their updates, of the system A**TAx= A**Tb. Then it calculates the minimum length least squares solution of the first k <= M of the permuted equations. The program stops after k <= M iterations when a certain simple criterion is satisfied and k will be the estimated rank of matrix A. Linear programming techniques are used in which the basic solution in the final simplex tableau is smooth solution of the system Ax=b. Numerical results show that the present program gives comparable accuracy to the truncated singular value decomposition method. Yet it is 2 to 5 times faster.