Re: pivot = leading entry



On Tue, 7 Feb 2006, G Patel wrote:

My Lin Alg prof uses "pivot column" and "leading entry column"
interchangeably all the time. She even says "pivot, or leading entry
if you like" vise versa a lot.

The text we use doesn't really explain pivots, and when it did lightly
touch on it, it was explained as a number used to zap zeros in all the
entries below it in the column. If that is the case, then the pivot
entries should be a subset (sometimes proper subset) of the set of
leading entries.

Am I right? Because sometimes when doing the pivotting, other entries
in other columns are zapped to zero, creating a leading entry
automatically with zeros below it (thus not needing to use this leading
entry at any time for pivotting)

Is my reasoning correct? Thanks for the help.

I detect loose use of terminology (or conventions).

"Leading" often applies to the first non-zero entry in a list (the entry 2
in [0, 0, 2, 77, 338, 5]), and on other occasions it applies to the zeros
at the beginning of the list (the expansion .00006303 has 4 leading
zeros). Personally, I saw it best applied to "leading ones" in the echelon
form of a matrix.

"Pivot" is, in computational linear algebra, a chosen entry below which
the matrix entries are to be "zapped" to zero, after a possible
interchange of rows, either actually performed or symbolically stored.

In some orthogonal transformation methods, one can talk about a "pivot
column", the one with largest magnitude within a submatrix to be
processed.

The distinction: A frequently recommended pivot is the largest entry in
the column that is being processed (it may reduce the risk of large
errors). But there can be other criteria (in linear programming, it is the
result of other magnitude decisions).

The distinction is obvious: if rounding errors are not an issue, a pivot
may (or may not) be the leading non-zero entry. (And in introductory
linear algebra, they are indeed not an issue.)

As a fresh graduate, I listened to a lecture by a German professor about
the pivoting strategies; he pronounced "pivot" with a silent "t". In
Slavic languages, "pivo" means beer, so we (Czechs and Slovaks in the
audience) were delighted that he emphasized the importance of a proper
choice of a "pivo element".

Cheers, ZVK(Slavek).
.

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