Lecture 13 Iterative of nonlinear equations

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Lecture 13Iterative of nonlinear equations LU
technique

• February 27, 2001
• CVEN 302

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Lectures Goals

• Iterative methods using a scalar gradient
  techniques
• LU Methods

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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

• 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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Example 1-D problem

• Minimize the function to find the roots.
• f(x) x2 - 2

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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

• f(x,y) x2 y2 - 1
• g(x,y) x2 - y

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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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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 (Couts 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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Summary

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