Rank-deficient and discrete ill-posed problems
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.
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