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m] be a fractional splitting set for some U, V, and W, and let S be defined as in Lemma 4.
An important benefit of using a fractional splitting to define an orthonormal stretching (S, [?
be an orthonormal stretching defined by a fractional splitting as in Lemmas 4.
In most of the applications of the splitting lemma [9, 1, 14, 16], there is also a symmetric-product representation of A and B, a representation using edge and vertex vectors, and an implicit W.
Before we conclude this section, we show that for orthonormal stretchings derived from fractional splitting sets, one of the norm-bounds on [?
This remains valid for any reordering of the spaces in the splitting.
Its condition number equals the condition number [kappa] of the splitting (2.
8), the corresponding chain of subspaces for the splitting (4.
In this splitting, the anisotropy of the problem is "built in".
The coarsening for the induced splitting is given by the arrows between the grids in Figure 5.
nxn], the splitting A = M - N is called P-regular if the matrix [M.
Let A = M - N be a P-regular splitting of a Hermitian matrix A.
Since A = P - Q is a convergent splitting, there exists an induced matrix norm [parallel]*[parallel] such that [rho]([P.
and let the P-regular splitting A = P - Q be given by
j] is a P-regular splitting of the Hermitian positive definite matrix [M.