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Algorithms and Parallel Computing by Fayez Gebali

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11.6 NULLSPACE OF DEPENDENCE MATRIX: THE BROADCAST SUBDOMAIN B

We will see in this section that the nullvector of the dependence matrix A of some variable v describes a subdomain Bx1D49F_EuclidMathOne_10n_000100. We will prove that all points in B contain the same instance of v.

11.6.1 The Nullspace of A

If the dependence matrix A is rank deficient, then the number of independent nullvectors associated with A is given by

(11.22) c11e022

where n is the number of indices of the algorithm. These nullvectors define the nullspace of matrix A.

Now assume a specific instance for the variable v(c). The following theorem identifies the points in x1D49F_EuclidMathOne_10n_000100 that use that variable.

Theorem 11.1

Consider a particular nullvector e associated with a variable v. If two distinct points p1 and p2 use the same instance v(c), then the vector connecting the two points is a nullvector of A.

Proof:

Assume the two points use the same instance v(c). Substitute the two points into Eq. 11.21 to get

(11.23) c11e023

(11.24) c11e024

Subtracting the two equations, ...

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