The Gauss Jordan method Major difference - eliminate unknowns from all rows, not just subsequent ones Normalize matrix so all entries are 1 Leads to identity matrix instead of upper triangular Backsubstitution is easy

Example First pivot

Normalize pivot row

Multiply 1st row by 3 and subtract from 2nd row

Do the other two rows Now pivot again

Normalize

Multiply 2nd row by 0.75 and subtract from first row

For first row and after all eliminated

No need to pivot, so normalize Work on rows 1,2 and 4 with row 3

No rows below row 4 to pivot with, so normalize and eliminate column 4

We now have our answer, since backsubstitution is trivial

LU decomposition - another method for solving matrix equations Idea behind LU decomposition - start with or

We know (because we did it in G.E.) we can write i.e or

Assume there exists [L] such that

means that

LU method 1) factor (decompose) A into L and U 2) given b, determine d from Ld=b 3) using Ux=d and backsubstitution, solve for x Advantage: Once you have L and U, can use many different b’s

How do you get L and U? Gauss elimination gives you U. It also gives you L. The factors are the entries in L

Changes in algorithm for Gauss elimination for LU decomposition loop over all the rows except the last one loop over all the rows below the current one get fik = aik/akk multiply row k by f and subtract from row i put fik in L at row i, column k end loop A is now upper triangular U make all Lkk=1

A fancier way of storing L and U Good if n is large More overhead to sort out

Pivoting in LU decomposition Still need it Messes up order of L What to do? Need to pivot also both L and a permutation matrix P

Initialize P as identity matrix and pivot when A is pivoted Initialize P as identity matrix and pivot when A is pivoted. Also pivot L

Example Starting out

No pivot

Now exchange rows 2 and 4

The pivot factors are

No pivot again, factor

Now make the diagonal elements of L=1

Recall

LU method 1) factor (decompose) A into L and U 2) given b, determine d 3) using Ux=d and backsubstitution, solve for x Advantage: Once you have L and U, can use many different b’s

Example (no pivoting):

Get d

Use Ux=d and backsubstitute

Now change b We don’t have to do elimination again Use the same L and U

Get d

Use Ux=d and backsubstitute