Programs in Physics & Physical Chemistry
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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.|
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.
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.
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