Lecture 14 LU Decomposition

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Lecture 14LU Decomposition

• March 1, 2001
• CVEN 302

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Lectures Goals

• Iterative methods using a scalar gradient
  techniques
• LU Methods

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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.
• Couts 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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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 (Couts reduction)

• Matrix decomposition

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Couts Reduction

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

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Couts Reduction

• Second step through the reduction

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General formulation of Couts

• 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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LU Decomposition (Doolittles method)

• Matrix decomposition

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Doolittes method

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

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General formulation of Doolittles

• The problem is reverse of the Couts reduction,
  starting with the upper triangular matrix and
  going to the lower triangular matrix.

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Example

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

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Choleskys method

• Matrix is decomposed into
• where, lii uii

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Cholesky Method

• The method does not alternate but does it from
  outside rows in.

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Choleskys Method

• The second row in.

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Choleskys Method

• General method

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Example

• The matrices can contain imaginary values.

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Tridiagonal Matrix

• For a banded matrix using Dolittles method, i.e.
  a tridiagonal matrix.

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Tridiagonal LU Decomposition

• The tridiagonal solver first step
• The second step

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Tridiagonal LU Decomposition

• The tridiagonal solver for LU decomposition
  breaks down into form

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Tridiagonal LU Decomposition

• The tridiagonal solver for LU decomposition
  breaks down into form

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Summary

• LU Decomposition.