The Idea Behind Krylov Methods
- 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 right-hand 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 CCR-9102853 and CCR-9400921,
and Prof. Meyer's work was supported in
part by the National Science Foundation under grants
DMS-9020915 and DMS-9403224.
- American Mathematical Monthly
- Vol. 105, No. 10, December, 1998, pp. 889-899.
Ilse C. F. Ipsen
Carl D. Meyer
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