GEPivot

# GEPivot - Gaussian Elimination with Partial Pivoting While...

This preview shows page 1. Sign up to view the full content.

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 |
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern