Iterative methods for solving Ax = b
GMRES/FOM versus QMR/BiCG
Iterative methods for solving Ax = b
GMRES/FOM versus QMR/BiCG
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Date
1998-10-15
Authors
Cullum, Jane K.
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Abstract
We study the convergence of GMRES/FOM and QMR/BiCG methods for solving
nonsymmetric Az = b. We prove that given the results of a BiCG
computation on Az = b, we can obtain a matrix B with the same eigenvalues
as A and a vector c such that the residual norms generated by a FOM
computation on Bz = c are identical to those generated by the BiCG
computations. Using a unitary equivalence for each of these methods, we
obtain test problems where we can easily vary certain spectral properties
of the matrices. We use these test problems to study the effects of
nonnormality on the convergence of GMRES and QMR, to study the effects of
eigenvalue outliers on the convergence of QMR, and to compare the
convergence of restarted GMRES and QMR across a family of normal and
nonnormal problems. Our GMRES tests on nonnormal test matrices indicate
that nonnormality can have unexpected effects upon the residual norm
convergence, giving misleading indications of superior convergence when
the error norms for GMRES are not significantly different from those for
QMR. Our QMR tests indicate that the convergence of the QMR residual and
error norms is infLuenced predominantly by small and large eigenvalue
outliers and by the character, real, complex, or nearly real, of the
outliers and the other eigenvalues. In our comparison tests QMR
outperformed GMRES(10) and GMRES(20) on both the normal and nonnormal
test matrices.
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http://www.cs.umd.edu/fs/ftp/pub/papers/papers/3587.figures.ps.
This is the second part of the paper in a separate file.
(Also cross-referenced as UMIACS-TR-96-2)