Block LU Decomposition: explained

1
Block LU Decomposition explained

• Keiran OHaire

2
Introduction

• Method for performing LU decomposition on large
  matrices
• Requires more calculations
• Faster due to memory locality

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The Matrix
A11 A12 A13
A21 A22 A23
A31 A32 A33
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Notes

• Each of the boxes represent sub matrices of the
  original matrix
• The sub matrices size are determined by the
  block size
• Example sub matrix with block size 4

4 2
1 8
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Matrix Dimensions
b
n-b
A11 A12 A13
A21 A22 A23
A31 A32 A33
b
n-b
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Step 1

• LU Decomposition of left-most column

L11 A12 A13
L21 A22 A23
L31 A32 A33
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Step 2

• Solve the rest of the top row using the topmost L

L11 U12 U13
L21 A22 A23
L31 A32 A33
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Step 2 Breakdown
Example L
Example U
1 0 3 1
3 1 2 5
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Column Version

• Use one column at a time from the matrix you wish
  to solve

1 0 3
3 1 2
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Column Version Contd
The result!
3
-7
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Column Version Contd

• Now insert the next column
• Repeat until all columns are solved

1 0 1
3 1 5
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Row Version

• Same matrices as before, but this time were doing
  rows
• The version is dependent on the language you are
  using, such as C or FORTRAN due to array
  orientation

1 0 3 1
3 1 2 5
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Row Version Contd

• This time, solve each of these (one cell at a
  time) in a row.
• i.e. solve 1 0 3
• Clearly, for the first, x13, and for the second,
  x11

1 0 3 1
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Row Version Contd

• These calculated xs are now used in the next
  row.
• i.e., 3(x13) x2 2, x2-7
• Then, 3(x11) x2 5, x22

1 0 3 1
3 1 2 5
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Continued

• So now we have calculated this portion. Now..

L11 U12 U13
L21 A22 A23
L31 A32 A33
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The Next Step

• Continue recursively

L11 U12 U13
L21 L22 U23
L31 L32 A33
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The Completed Decomposition

• Eventually you end up with the solution

L11 U12 U13
L21 L22 U23
L31 L32 L33
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Questions?
Questions?