P. Hansen. SIAM Monographs on Mathematical Modeling and Computation Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, (1998)Numerical aspects of linear inversion.
This research monograph describes the numerical treatment of certain linear
systems of equations which we characterize as either rank-deficient problems or
discrete ill-posed problems. Both classes of problems are characterized by
having a coefficient matrix that is very ill conditioned; i.e., the condition number
of the matrix is very large, and the problems are effectively under determined.
Given a very ill conditioned problem, the advice usually sounds something
like "do not trust the computed solution, because it is unstable and most likely
dominated by rounding errors." This is good advice for general ill-conditioned
problems, but the situation is different for rank-deficient and discrete ill-posed
problems. These particular ill-conditioned systems can be solved by numerical
regularization methods in which the solution is stabilized by including
appropriate additional information. Since the two classes of problems share many
of the same regularization algorithms, it is natural to discuss the numerical
aspects of both problem classes in the same book.