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Books > Science & Mathematics > Mathematics > Algebra
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Krylov Subspace Methods - Principles and Analysis (Hardcover)
Loot Price: R3,098
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Krylov Subspace Methods - Principles and Analysis (Hardcover)
Series: Numerical Mathematics and Scientific Computation
Expected to ship within 10 - 15 working days
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The mathematical theory of Krylov subspace methods with a focus on
solving systems of linear algebraic equations is given a detailed
treatment in this principles-based book. Starting from the idea of
projections, Krylov subspace methods are characterised by their
orthogonality and minimisation properties. Projections onto highly
nonlinear Krylov subspaces can be linked with the underlying
problem of moments, and therefore Krylov subspace methods can be
viewed as matching moments model reduction. This allows
enlightening reformulations of questions from matrix computations
into the language of orthogonal polynomials, Gauss-Christoffel
quadrature, continued fractions, and, more generally, of Vorobyev's
method of moments. Using the concept of cyclic invariant subspaces,
conditions are studied that allow the generation of orthogonal
Krylov subspace bases via short recurrences. The results motivate
the important practical distinction between Hermitian and
non-Hermitian problems. Finally, the book thoroughly addresses the
computational cost while using Krylov subspace methods. The
investigation includes effects of finite precision arithmetic and
focuses on the method of conjugate gradients (CG) and generalised
minimal residuals (GMRES) as major examples. There is an emphasis
on the way algebraic computations must always be considered in the
context of solving real-world problems, where the mathematical
modelling, discretisation and computation cannot be separated from
each other. The book also underlines the importance of the
historical context and demonstrates that knowledge of early
developments can play an important role in understanding and
resolving very recent computational problems. Many extensive
historical notes are included as an inherent part of the text as
well as the formulation of some omitted issues and challenges which
need to be addressed in future work. This book is applicable to a
wide variety of graduate courses on Krylov subspace methods and
related subjects, as well as benefiting those interested in the
history of mathematics.
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