The algebraic eigenvalue problem
The algebraic eigenvalue problem
Enclosing the solutions of systems of linear equations by interval iterative processes
on Scientific Computational with automatic result verification
Matrix computations (3rd ed.)
Iterative methods for solving linear systems
Iterative methods for solving linear systems
LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares
ACM Transactions on Mathematical Software (TOMS)
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
The lanczos algorithm for solving symmetric linear systems
The lanczos algorithm for solving symmetric linear systems
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Preconditioned Krylov subspace solvers are an important and frequently used technique for solving large sparse linear systems. There are many advantageous properties concerning convergence rates and error estimates. However, implementing such a solver on a computer, we often observe an unexpected and even contrary behavior.The purpose of this paper is to show that this gap between the theoretical and practical behavior can be narrowed by using a problem-oriented arithmetic. In addition we give rigorous error bounds to our computed results.