The Idea Behind Krylov Methods
 ABSTRACT
 We explain why Krylov methods make sense, and why it is natural to
represent a solution to a linear system as a member of a Krylov space.
In particular we show that the solution to a nonsingular linear
system Ax = b lies in a Krylov space whose dimension is the degree of the
minimal polynomial of A. Therefore, if the minimal polynomial of
A has low degree then the space in which a Krylov method searches
for the solution is small. In this case a Krylov method has the
opportunity to converge fast. When the matrix is singular, however,
Krylov methods can fail. Even
if the linear system does have a solution, it may not lie in a
Krylov space. In this case we describe the class of righthand sides
for which a solution lies in a Krylov space. As it happens,
there is only a single solution that lies in a Krylov space, and it
can be obtained from the Drazin inverse.
Prof. Ipsen's work was supported in part by the
National Science Foundation under grants CCR9102853 and CCR9400921,
and Prof. Meyer's work was supported in
part by the National Science Foundation under grants
DMS9020915 and DMS9403224.
 JOURNAL
 American Mathematical Monthly
 Vol. 105, No. 10, December, 1998, pp. 889899.
 COAUTHORS

Ilse C. F. Ipsen

Carl D. Meyer
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