Sparse Matrices (Mathematics in Science and Engineering 99) by Reginald P. Tewarson

By Reginald P. Tewarson

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The elementary matrix at the kth stage of back substitution. 3. 12) tik)= -a$+ '), i

8) * B)ej = 1, then columns i and j of B belong to the same diagonal block of B. 4)we have B’ * F * B - * ( B * B’)]* B = B’ * [ ( B * B’) * ( B * B’) * * * = (B’ * B ) * (B’ * B) * . . * (B’ * B) = (B’*B)b, Q > v. 1, it follows that ei’(B’ * F * B)ej = ei‘(B’* B y e j = 1 implies that columns i and j are connected by a path and therefore belong to the same diagonal block. This completes the proof of the theorem. 7 for permuting the columns into block diagonal form.

This is relatively simple and easy to do, and is therefore recommended in many practical applications (Tinney and Walker, 1967; Spillers and Hickerson, 1968; Churchill, 1971). One of the main reasons for choosing only the diagonal elements as pivots is that if A is symmetric, then quite often only the upper triangular part of A along with the main diagonal is stored and the diagonal pivot choice during the forward course of the Gaussian elimination maintains the symmetry. Furthermore the v(,)s can be easily obtained from the upper triangular matrix obtained at the end of the forward course.

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