Fast Algorithms for Integral Equations and Least Squares Identification Problems

Sammanfattning: This work is concerned with fast algorithms for integral equations and least squares identification problems.The presentation is divided into three parts. In the first part a fast algorithm for solving systems oflinear equations with a matrix that is alm ost Toeplitz is derived and applied to Fredholm integral equations with stationary kernels. The algorithm is investigated numerically with simulations. Also some areas, where this kind of integral equations arise ha ve been considered. A summary of applications is given in part two. In particular, image restoration problems and boundary element methods from stress analysis have been treated in this respect.In the third part a fast algorithm for computing the gain vector for recursive least squares estimation schemes is the main issue. A comparison to conventional, square root and lattice algorithms is established. The investigation aims at deciding whether the algorithms are stable by performing a simple step response technique. All computations are assumed to be ideal, i. e. the arithmetic operations are performed exactly. An error is then introduced in important quantities ofthe algorithm and the propagation ofthis error is studied as function of time and forgetting factor. The study is. both theoretical and experimental. I fan algorithm tums out to be exponentially stable, a simple mode! for estimating the discrepancy of the computed quantities from the exact ones is performed.The fast algorithms of parts one and three are apparently different. However, there is an immediate interrelation between them. The introduction includes a discussion around this topic.