Lecture 13 LU Decomposition

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Lecture 13 - LU Decomposition

• CVEN 302
• September 24, 2001

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Lectures Goals

• LU Decomposition
• Crouts technique
• Doolittles technique
• Choleskys technique

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LU Decomposition

• A modification of the elimination method, called
  the LU decomposition. The technique will rewrite
  the matrix as the product of two matrices.
• A LU

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LU Decomposition

• The technique breaks the matrix into a product of
  two matrices, L and U, L is a lower triangular
  matrix and U is an upper triangular matrix.

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LU Decomposition

• There are variation of the technique using
  different methods.
• Crouts reduction (U has ones on the diagonal)
• Doolittles method( L has ones on the diagonal)
• Choleskys method ( The diagonal terms are the
  same value for the L and U matrices)

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Decomposition
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LU Decomposition Solving

• Using the LU decomposition
• Ax LUx LUx b
• Solve
• Ly b
• and then solve
• Ux y

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LU Decomposition

• The matrices are represented by

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Equation Solving

• What is the advantage of breaking up one linear
  set into two successive ones?
• The advantage is that the solution of triangular
  set of equations is trivial to solve.

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Equation Solving

• First step - forward substitution

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Equation Solving

• Second step - back substitution

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LU Decomposition (Crouts reduction)

• Matrix decomposition

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LU Decomposition (Doolittles method)

• Matrix decomposition

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Choleskys method

• Matrix is decomposed into
• where, lii uii

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LU Decomposition (Crouts reduction)

• Matrix decomposition

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Crouts Reduction

• The method alternates from solving from the lower
  triangular to the upper triangular

Step 1 sweep down
Step 2 sweep across
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Crouts Reduction

• Second set of steps through the reduction

Step 3 sweep down
Step 4 sweep across
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General formulation of Crouts

• These are the general equations for the
  component of the two matrices

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Example

• The matrix is broken into a lower and upper
  triangular matrices.

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Example

• Solve problem using LU decomposition

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LU Programs

• There are two programs
• LU_crout_factor - the program does a Crout
  decomposition of a matrix and returns the L and U
  matrices
• LU_solve uses an L and U matrix combination to
  solve the system of equations.

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Summary

• Defined a LU decomposition.
• Setup of the LU Crouts decomposition technique.
• Showed an example

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Homework

• Check the Homework webpage