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CONTENTS: Implementation of a conjugate-gradient type method for solving sparse linear equations: Solve\begin {align*} Ax= b\\text {or}\(A-sI) x= b.\end {align*} The matrix\(A-sI\) must be symmetric but it may be definite or indefinite or singular. The scalar\(s\) is a shifting parameter--it may be any number. The method is based on Lanczos tridiagonalization. You may provide an SPD preconditioner M.(Matlab version minres20. zip allows for an indefinite M to cause graceful exit and possible re-entry with a more positive-definite M.)

MINRES is really solving one of the least-squares problems\begin {align*}\text {minimize}\| Ax-b\|\\text {or}\\|(A-sI) x-b\|.\end {align*} If\(A\) is singular (and\(s= 0\)), MINRES returns a least-squares solution with small\(\| Ar\|\)(where\(r= b-Ax\)), but in general it is not the minimum-length solution. To get the min-length solution, use MINRES-QLP [2, 3].