Lecture 10 Nonlinear gradient techniques and LU Decomposition

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Lecture 10 - Nonlinear gradient techniques and LU
Decomposition

• CVEN 302
• June 18, 2001

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Lectures Goals

• Nonlinear Gradient technique
• LU Decomposition
• Crouts technique
• Doolittles technique
• Choleskys technique

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Nonlinear Equations

• The nonlinear equations can be solved using a
  gradient technique.
• The minimization technique calculates a positive
  scalar value and use a gradient to find the zero
  of multiple functions.

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Minimization algorithm

• Calculate the square function.
• h(x) S (f(x))2
• Calculate a scalar value
• z0 h(x)
• Calculate the gradient
• dx - dh/dx

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Minimization algorithm

• Multiple loops to convergence
• xnew xold dx z1 h(xnew ) dif z1 -
  z0
• if dif gt 0
• dx dx/2
• xnew xold dx
• else
• end loop
• endif

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Program FFMIN

• The program is adapted from the book to do a
  minimization of scalar and uses a gradient
  technique to find the roots.

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Example of the 2-D Problem

• f1(x,y) x2 y2 - 1
• f2(x,y) x2 - y

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Example of the 2-D Problem

• The gradient function
• h(x,y) ( x2 y2 - 1)2 ( x2 - y) 2
• The derivative of the function
• dh-(4(x2 y2-1)x 4( x2 - y)x)
• -(4(x2 y2 - 1)y - 2( x2 - y))

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Example of the 3-D Problem

• f1(x,y,z) x2 2y2 4z2 - 7
• f2(x,y,z) 2x2 y3 6z2 - 10
• f3(x,y,z) xyz 1

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Example of the 3-D Problem

• The gradient function
• h(x,y,z) (x2 2y2 4z2 - 7)2
• (2x2 y3 6z2 - 10)2
• (xyz 1) 2

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End of material on Exam 1

• Exam 1
• Chapter 1 through 5
• Monday June 25, 2001
• open book and open notes

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Chapter 6

• LU Decomposition of Matrices

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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

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Crouts Reduction

• Second step through the reduction

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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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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 Crouts reduction,
  starting with the upper triangular matrix and
  going to the lower triangular matrix.

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LU Programs

• There are two programs
• LU_factor - the program does a Doolittle
  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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Example

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

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Summary

• Nonlinear scalar gradient method uses a simple
  step to find the crossing terms.
• Setup of the LU decomposition techniques.

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Homework

• Check the Homework webpage