Lecture 11 Iterative Techniques and Nonlinear Equations

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Lecture 11Iterative Techniques and Nonlinear
Equations

• February 20, 2001
• CVEN 302

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Lectures Goals

• Correction to the Quintdiagonal terms
• Iterative Techniques - SOR method
• Nonlinear Systems of equations
• Newton method
• fixed point iterations

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Band Solver

• The technique can be applied to higher order
  banded matrices.
• The quintdiagonal matrix has a similar algorithm.

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Quintdiagonal Method

• The first phase starts with the first row of
  coefficients scales the a and r coefficients.
• The second phase solves for x values using the a
  and r coefficients.

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Correction Quintdiagonal Method

• There is a correction for the program, the bi
  term will change and cause problems.
• The b term must be modified before being used in
  the updates of a, c and r terms.

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Convergence Restrictions

• There are two conditions for the iterative method
  to converge.
• Necessary that 1 coefficient in each equation is
  dominate.
• The sufficient condition is that the diagonal is
  dominate.

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Successive over Relaxation

• The technique is a modification on the
  Gauss-Seidel method with an additional parameter,
  w, that may accelerate the convergence of the
  iterations.
• The weighting parameter, w, has two ranges 0 lt w
  lt1, and 1lt w lt2. If w 1, then the problem is
  the Gauss-Seidel technique.

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SOR Method

• The SOR algorithm is defined as
• The difference is the weighting parameter, w.

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The Weighting Parameter

• If the parameter, w is under 1, the residuals
  will be under-relaxed.
• If the parameter, w 1, the residuals are equal
  to a Gauss-Seidel model.
• If 1lt w lt 2 the residuals will be over-relaxed
  and will general help accelerate the convergence
  of the solution.

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Example of SOR

• 4X1 2X2 2
• 2X1 10X2 4X3 6
• 4X2 5X3 5
• Solution (X1 , X2 , X3 ) (0.41379, 0.17241,
  0.86206)

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SOR Example

• Formulation of the SOR Algorithm

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Effects of w Parameter

• Using the SOR program SOR(A,b,w,nmax,tol) with
  nmax50 and tol 0.000001

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Nonlinear Equations

• What is a nonlinear equation?
• Nonlinear equations are assumed to be those which
  contain powers or products of the variables
  and/or transcendental functions.

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Nonlinear Equations

• What methods will we be using to handle nonlinear
  equations?
• Newtons method
• Direct iteration
• Minimum of a nonlinear function of several
  variables.

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Newtons Method

• By assuming that a sufficiently close approximate
  solution x a1 and y b1 is known, the
  equations can be expanded into Taylor series
  about these points. By using Dx and Dy to
  represent the changes in x and y, which need to
  be made in successive cycles of approximation.

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1-D Newtons Method

• The idea for the method comes from a Taylor
  series expansion, where you know the function and
  its first derivative.
• f(xk1) f(xk) (xk1 - xk)fx (xk) ...

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Newtons Method

• The goal of the calculations is to find a f(x)0,
  so set f(xk1) 0 and rearrange the equation.
  f x (xk) is the first derivative of f(x).
• 0 f(xk) (xk1 - xk)f x(xk)
• xk1 _at_ xk - f(xk) / fx (xk)

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Newton Method for 2-D Problem

• Using a two nonlinear functions, we would like to
  find the intersection of the two.
• The problem uses a Taylor series expansion to
  calculate the next value if we have an initial
  estimation.

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Newton Method
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Newtons Method

• The step size will vary with the function of the
  derivatives.
• The equation can be written in-terms of the
  functions and their partial derivatives.

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Newtons Method

• The Jacobian matrix is defined as the matrix of
  partial derivatives of the functions.

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Newtons Method

• The method is exactly the same as the 1-D method
  except that the Jacobian matrix has more partial
  derivatives to use to find the solution.

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Example of Newtons Method

• 1.4X1 - X2 0.6
• X12 - 1.6X1 - X2 4.6
• Solution (X1 , X2 ) ( (-1,-2), (4,5) )

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Newtons Example

• Jacobian matrix
• The functions are given.

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Newton Example Program

• The program demoNewtonSys is a demonstration of
  the Newton method for a two dimensional example.

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Example of Newton

• 3X12 1.7X2 X32
  1.7
• - 4X1 - X22 1.2X2 5X3
  3.3
• - X12 2X1 2.7X2
  -3X3 2.1

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Summary

• SOR (Successive over Relaxation) uses the
  residuals to accelerate the convergence.

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Newton Method
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Newton