Reduction Algorithm

The following algorithm corrects the one in section 1.6 in the book:

1.

Start assuming that the pivot will be the first unknown, i.e. unknown number u=1, and the first equation, e=1.

2.

If a_eu is nonzero, select it as pivot. Otherwise, scan through the equations below it for an equation e' for which a_e'u is nonzero and interchange it with the e-th equation. If that is not possible since all coefficients below a_eu are also zero, try the next unknown, i.e., increment u by one and repeat. If that is not possible since there are no unknowns left, done.

3.

Use the pivot a_eu to eliminate unknown u from all equations below the pivot. In other words, for all e'>e, add -a_e'u/a_eu times the e-th equation to the e'-th equation.

4.

If there are no equations and/or unknowns left, done. Otherwise, go to the next unknown and equation, (increment u and e by 1), and repeat from step 2.

This will create an echelon matrix. To obtain a canocical matrix from the echelon form, eliminate all nonzero elements above the pivots, starting from the last one, and then divide each equation by its pivot.

What is wrong with the book: Step 1 in the book procedure is impossible if the first row is zero. Deleting equations or unknowns is a no-no with your instructor. And step 4 may not be possible since there may not be any equations left after step 3 following the book.

09/21/01 0:19:43 09/24/01 0:03:14