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NOTES AND REFERENCES. Initially, the term projection methods was used mainly to describe one-

dimensional techniques such as those presented in Section 5.3. An excellent account of what has been

done in the late 1950s and early 1960s can be found in Householder™s book [122] as well as Gastinel

[101]. For more general, including nonlinear, projection processes, a good reference is Kranoselskii

and co-authors [138].

Projection techniques are present in different forms in many other areas of scienti¬c computing

and can be formulated in abstract Hilbert functional spaces. The terms Galerkin and Petrov-Galerkin

techniques are used commonly in ¬nite element methods to describe projection methods on ¬nite

element spaces. The principles are identical to those seen in this chapter.

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where . When there is no ambiguity, will be denoted by . The

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different versions of Krylov subspace methods arise from different choices of the subspace

and from the ways in which the system is preconditioned, a topic that will be covered

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in detail in later chapters.

Viewed from the angle of approximation theory, it is clear that the approximations

obtained from a Krylov subspace method are of the form T

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in which is a certain polynomial of degree . In the simplest case where ,

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Although all the techniques provide the same type of polynomial approximations, the

choice of , i.e., the constraints used to build these approximations, will have an im-

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portant effect on the iterative technique. Two broad choices for give rise to the best- ¢

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known techniques. The ¬rst is simply and the minimum-residual variation ¢ ¢

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chapter. The second class of methods is based on de¬ning to be a Krylov subspace ¢

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method associated with , namely, . Methods of this class will be ¨(

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covered in the next chapter. There are also block extensions of each of these methods

termed block Krylov subspace methods, which will be discussed only brie¬‚y. Note that

a projection method may have several different implementations, giving rise to different

algorithms which are all mathematically equivalent.