GEPivot - Gaussian Elimination with Partial Pivoting While...

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Gaussian Elimination with Partial Pivoting While it is true that almost all nonsingular matrices can be triangularized using only Gauss Transforms (add multiple of one row to another), it does not make a good general purpose numerical method. The problem is caused, as you might suspect (?), by small pivot elements. Consider the k th step, zeroing the ( i, k ) entries with multipliers m ik = - a ( k - 1) ik /a ( k - 1) kk , k +1 , k +2 , . . . , n : a ( k ) i = a ( k - 1) i + m ik a ( k - 1) k , giving A ( k ) = ˆ U ( k ) X 0 ˆ A ( k ) . If a ( k - 1) kk is small, then | m ik | will be large, and two bad things will happen: (i) information in the entries of A ( k - 1) gets swamped when the large vector m ik a ( k - 1) k gets added to a ( k - 1) i , and (ii) that information is replaced by basically the same value for each row: a ( k ) i will be mostly in the direction a ( k - 1) k for all of the rows k +1 , k +2 , . . . , n of A ( k ) , moving ˆ A ( k ) relatively closer to the set of singular matrices. So it’s time to bring back row operation R2: Before zeroing the elements in column k , we find max k j n | a ( k - 1) jk | (the | biggest |
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