Algorithms and Parallel Computing

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 B ⊂ . 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)

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 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)

(11.24)

Subtracting the two equations, ...