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The book offers a mathematically rigorous introduction to fundamental iterative methods for systems of linear algebraic equations. The book distinguishes itself from other texts on the topic by

Providing a straightforward yet comprehensive analysis of the Krylov subspace methods,

Approaching the development and analysis of algorithms from various algorithmic and mathematical perspectives, and

Going beyond the standard description of iterative methods by connecting them in a natural way to the idea of preconditioning.

The book supplements standard texts on numerical mathematics for first-year graduate and advanced undergraduate courses and is suitable for advanced graduate classes covering numerical linear algebra and Krylov subspace and multigrid iterative methods. It will be useful to researchers interested in numerical linear algebra and engineers who use iterative methods for solving large algebraic systems.