The Gauss Jordan method

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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
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Example
First pivot
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Normalize pivot row
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Multiply 1st row by 3 and subtract from 2nd row
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Do the other two rows
Now pivot again
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Normalize
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Multiply 2nd row by 0.75 and subtract from first
row
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For first row
and after all eliminated
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No need to pivot, so normalize
Work on rows 1,2 and 4 with row 3
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No rows below row 4 to pivot with, so normalize
and eliminate column 4
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We now have our answer, since backsubstitution is
trivial
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LU decomposition - another method for solving
matrix equations Idea behind LU decomposition -
start with
or
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We know (because we did it in G.E.) we can write
i.e
or
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Assume there exists L
such that
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means that
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LU method 1) factor (decompose) A into L and U 2)
given b, determine d from Ldb 3) using Uxd and
backsubstitution, solve for x Advantage Once you
have L and U, can use many different bs
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How do you get L and U?
Gauss elimination gives you U. It also gives you
L.
The factors are the entries in L
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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 end loop A is now upper triangular
U make all Lkk1
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A fancier way of storing L and U
Good if n is large More overhead to sort out
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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
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Initialize P as identity matrix and pivot when A
is pivoted. Also pivot L
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Example
Starting out
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No pivot
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Now exchange rows 2 and 4
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The pivot factors are
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No pivot again, factor
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Now make the diagonal elements of L1
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Recall
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LU method 1) factor (decompose) A into L and U 2)
given b, determine d 3) using Uxd and
backsubstitution, solve for x Advantage Once you
have L and U, can use many different bs
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Example (no pivoting)
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Get d
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Use Uxd and backsubstitute
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Now change b
We dont have to do elimination again Use the
same L and U
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Get d
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Use Uxd and backsubstitute