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In many applications, for many reasons,
we have a starting guess of the solution.
It is usually a wise idea to use the starting guess
rather than begin anew from .This presents a problem.
To do preconditioning,
we begin by defining (implicitly)
the new underlying model variables by the substitution .When we are given an initial ,then the corresponding should be
but we might not know .So a noninvertible preconditioner
is in conflict with a nonzero starting model.
You cannot have both.
(If you want to explore a null space by trying various starting
solutions, it might suffice to try various starting ;you need not try various starting .)
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Stanford Exploration Project
2/27/1998