DEFLATED DECOMPOSITION OF SOLUTIONS OF NEARLY SINGULAR SYSTEMS.

The concept of a deflated solution to the linear system Ax equals b, where A may be nearly singular and b is not consistent with A is generalized to that of a deflated decomposition. Such decompositions are treated in a uniform framework, and some new deflated solutions based on LU-factorization are introduced. In particular, it is proved that the difference between one of the LU-based deflated solutions and the SVD-based deflated solution tends to zero as A tends to exactly singular. In addition, noniterative implicit algorithms are given for computing the LU-based decompositions. Numerical results verifying the accuracy and stability of the algorithms are presented.