Lecture 9 Nonlinear systems of equations

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Lecture 9 - Nonlinear systems of equations

• CVEN 302
• June 15, 2001

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Lectures Goals

• Iterative Techniques
• Jacobian method
• Gaus-Siedel method
• Relaxation technique

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Lectures Goals

• Nonlinear Systems of equations
• Newton method
• fixed point iterations

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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 Newton method comes from a Taylor series
  expansion
• The series is truncated and the root is set equal
  to zero.

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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

• The Taylor series expansion for two equations
  will result in

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Newton Method

• The Taylor series is truncated at the first
  derivative and resulting set equations

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Newton Method

• Rearrange the equations in matrix format

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Newton 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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Newton Method

• The Taylor series expansion for two equations
  will result in

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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

• Example of a 3 x 3 matrix (demoNewtonSys4.m)
• 10 X1 - X12 - X2
  X32 4
• - X12 10X2 - X22 - X3
  5
• -X1 - X22 10X3 -
  X32 6

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Newton Example

• The Jacobian matrix of the example is

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Nonlinear Equations

• Newtons method for systems of equations uses the
  Taylor Series expansion to help find the
  intersections of the equations.
• The problem of finding the solution of a set of
  nonlinear equations is much more difficult than
  for linear equations.

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Nonlinear Equations

• In nonlinear systems the expression
• x A-1(x) b(x)
• indicates that the right hand side of the
  equation can
• not be evaluated until x is known. The best one
  can
• do is devise a method that gives a sequence of
• increasingly good guesses at the x that satisfies
  the
• equation.

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Iterative methods for nonlinear equations

• Direct iteration solution of nonlinear system in
  which the equations are selected.
• Successive substitution
• Minimization of a scalars functions.

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Direct iterative process

• Does not require computation of partial
  derivatives of the function.
• The equations are set up to iterate to find the
  solution.

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Iterative Solutions

• General rules of the thumb for convergence are
• xf(x,y,z,) , yg(x,y,z,..), zh(x,y,z,)
• fx fy fz lt 1
• gx gy gz lt 1
• hx hy hz lt 1

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Consider the following example using an
iterative method

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

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Iterative Example

• From the Newton method we could find the results
  with iterative techniques.
• The functions are selected to get the values to
  converge.

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Test Iterations

• Case 1

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Test Iterations

• Case 2

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Test Iterations

• Case 3

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Test Iterations

• Case 4

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Iteration Example Program

• The program demoIteration demonstrates different
  sets of equations to find the roots.
• The problem depend on the partial derivative to
  find out whether or not the program will
  converge.

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Programs for Fixed Point Iteration

• X12 50 X1 X22 X32 - 200 0
• X12 20 X2
  X32 - 50 0
• -X12 - X22 40 X3 75 0
• Which becomes
• X1 -0.02X12 - 0.02X22 - 0.02X32 4
• X2 -0.05X12 - 0.05X32 2.5
• X3 0.025X12 0.025X22 - 1.875

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Iterative methods

• When they do converge they may favor one solution
  over the other.
• The solution to nonlinear equations may converge
  at different rates depending on how you select
  the values.

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Successive Substitution

• The technique is a simple iterative method for
  nonlinear systems.
• A (k) x (k1) b (k)
• x (k1) A (k) -1 b (k)

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For the example problem using successive
substitution

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

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For the example problem using successive
substitution
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Programs for Successive Substitution

• The set of equations are
• X12 50 X1 X22 X32 - 200 0
• X12 20 X2
  X32 - 50 0
• -X12 - X22 40 X3 75 0

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Successive Substitution
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Successive Substitution

• This method may also favor one solution over
  another solution.
• The method does not work for highly nonlinear
  problems.

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Successive Substitution

• The program demoSSub will show how the iterative
  technique works for the mildly nonlinear set of
  equations.
• The book has a set of programs which will do the
  same thing

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Summary

• Iteration on nonlinear equations are limited to
  set of conditions.
• The iterations can converge on a solution but not
  all solutions, it may favor on solution to
  another.
• Newtons method uses PDE to help solve the
  problem and can be used for nonlinear functions.

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Homework

• Check the Homework webpage