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Lecture 12 Nonlinear systems of equations

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The technique is a simple iterative method for nonlinear systems. A (k) x (k 1) = b (k) ... dx = - dh/dx. Minimization algorithm. Multiple loops to convergence ... – PowerPoint PPT presentation

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Title: Lecture 12 Nonlinear systems of equations


1
Lecture 12 - Nonlinear systems of equations
  • CVEN 302
  • September 21, 2001

2
Lectures Goals
  • Nonlinear Systems of equations
  • Successive Substitution
  • Minimization algorithm

3
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)

4
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) )

5
For the example problem using successive
substitution
6
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

7
Successive Substitution
8
Successive Substitution
  • This method may also favor one solution over
    another solution.
  • The method does not work for highly nonlinear
    problems.

9
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

10
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.

11
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

12
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

13
Program FFMIN
  • The program is adapted from the book to do a
    minimization of scalar and uses a gradient
    technique to find the roots.

14
Example of the 2-D Problem
  • f1(x,y) x2 y2 - 1
  • f2(x,y) x2 - y

15
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))

16
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

17
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

18
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.

19
Homework
  • Check the Homework webpage
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