Variations of Zhang's Lanczos-type product method
Applied Numerical Mathematics - Developments and trends in iterative methods for large systems of equations—in memoriam Rüdiger Weiss
The Chebyshev iteration revisited
Parallel Computing - Parallel matrix algorithms and applications
Accurate conjugate gradient methods for families of shifted systems
Applied Numerical Mathematics - Numerical algorithms, parallelism and applications
Limiting accuracy of segregated solution methods for nonsymmetric saddle point problems
Journal of Computational and Applied Mathematics
VPAStab(J,L): An iterative method with look-ahead for the solution of large sparse linear systems
Journal of Computational and Applied Mathematics
Improved Balanced Incomplete Factorization
SIAM Journal on Matrix Analysis and Applications
On Efficient Numerical Approximation of the Bilinear Form $c^*A^{-1}b$
SIAM Journal on Scientific Computing
A variant of IDRstab with reliable update strategies for solving sparse linear systems
Journal of Computational and Applied Mathematics
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Many conjugate gradient-like methods for solving linear systems $Ax=b$ use recursion formulas for updating residual vectors instead of computing the residuals directly. For such methods it is shown that the difference between the actual residuals and the updated approximate residual vectors generated in finite precision arithmetic depends on the machine precision $\epsilon$ and on the maximum norm of an iterate divided by the norm of the true solution. It is often observed numerically, and can sometimes be proved, that the norms of the updated approximate residual vectors converge to zero or, at least, become orders of magnitude smaller than the machine precision. In such cases, the actual residual norm reaches the level $\epsilon \| A \| \| x \|$ times the maximum ratio of the norm of an iterate to that of the true solution. Using exact arithmetic theory to bound the size of the iterates, we give a priori estimates of the size of the final residual for a number of algorithms.