Unformatted text preview: ectors w 2 W . This is illustrated for two- and three-dimensional spaces in Figure 9.4.1.
Two choices of the test space W are important: W = AV and W = V . In the
rst case, the solution of (9.4.2b) (or (9.4.3)) is optimal in the sense of minimizing k~k2,
8w 2 AV . To show this, let x satisfy (9.4.2b), add a perturbation x, and consider the
krk2 = (b ; A(~ + x))T (b ; A(~ + x)):
krk2 = k~k2 ; 2(A x)T (b ; Ax) + xAT A x:
With x 2 V and w = A x 2 W , we have from (9.4.2b)
(A x)T (b ; Ax) = 0:
Thus, (9.4.4) krk2 = k~k2 + xAT A x:
Since xAT A x is non-negative de nite, we establish k~k2 as the minimum. The conr2
verse, i.e., that minimizers of krk satisfy (9.4.2b), is also true and its proof follows similar arguments.
Using (9.4.2) ~ = b ; A(x(0) ; v) = r(0) ; Av:
r 9.4. Krylov Subspace Methods 57 Condition (9.4.4) enforces orthogonality between ~ and all vectors in AV . Using the
above relation, we say that Av is the orthogonal projection of r(0) onto the subspace V
(Figure 9.4.1). Methods of this type are called residual projection methods.
When W = V , the method (9.4.2b) is called Galerkin's method. When A is symmetric
and positive de nite, Galerkin's method is optimal in the sense of minimizing kx ; yk2 = (x ; y)T A(x ; y)
A (9.4.5a) for all y = x(0) + v, v 2 V . The appropriate inner product for use with this strain energy
(v w) := wT Av: (9.4.5b) ~
Thus, if x satis es (9.4.2b), we have
wT AT (b ; Ax) = 0 8w 2 V : (9.4.5c) ~
Proving that x also minimizes (9.4.5a), and conversely, follows the lines of the proof for
residual projection methods and is left as an exercise (cf. Problem 1 at the end of this
If our aim is to address nonsymmetric systems, we will primarily be interested in
residual projection methods however, let us proceed in a more general manner for the
moment and examine solution schemes for (9.4.2b). Let V = v1 v2 : : : vM ] W = w1 w2 : : : wM ] (9.4.6a) be N M matrices whose columns form bases for V and W , respectively....
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