Linear SystemsPivoting in Gaussian Elim.

CSE 541

Roger Crawfis

The naïve implementation of Gaussian Elimination is not robust and can suffer from severe round-off errors due to:Dividing by zeroDividing by small numbers and adding.

Both can be solved with pivoting

Limitations of Gaussian Elimation

What if at step i, Aii = 0?

Simple Fix:If Aii = 0

Find Aji 0 j > i

Swap Row j with i

Factored Portion

iiA

jiA

Row i

Row j

Partial Pivoting

75

25.6

5.125.12

25.11025.1

2

14

x

x

Forward Elimination

52

1

5

4

1025.675

25.6

1025.15.120

25.11025.1

x

x

9999.4

0001.1

52

1

digitsx

x

Example – Partial Pivoting

52

1

5

4

1025.6

25.6

1025.10

25.11025.1

x

x

Rounded to 3 digits

Example – Partial Pivoting

75

25.6

5.125.12

25.11025.1

2

14

x

x

Forward Elimination

52

1

5

4

1025.675

25.6

1025.15.120

25.11025.1

x

x

9999.4

0001.1

52

1

digitsx

x

5

0

32

1

digitsx

x

Partial Pivoting to mitigate round-off error

Adds an O(n) search.

If | | < max | |

Swap row with arg (max | |)

ii jij i

ijj i

A A

i A

Avoids Small Multipliers

Better Pivoting

Forward Elimination

5

2

1

5 107525.6

75

105.1225.10

5.125.12

x

x

5

1

32

1

digitsx

x

25.6

75

25.10

5.125.12

2

1

x

x

Rounded to 3 digits

75

25.6

5.125.12

25.11025.1

2

14

x

x

25.6

75

25.11025.1

5.125.12

2

14 x

x

swap

Partial Pivoting

Pivoting strategies

k

k

k

k

Partial Pivoting:Only row interchange

Complete (Full) PivotingRow and Column interchange

Threshold PivotingOnly if prospective pivot is found

to be smaller than a certain threshold

Pivoting With Permutations

Adding permutation matrices in the mix:

However, in Gaussian Elimination we will only swap rows or columns below the current pivot point. This implies a global reordering of the equations will work:

bPMPMPMAxPMPMPM nnnnnnnn 112211112211

bxAM

MPbMPAx

Pivoting

Again, the pivoting is strictly a function of the matrix A, so once we determine P it is trivial to apply it to many problems bk.

For LU factorization we have:LU = PALy = PbUx = y