Lecture 6 Single Variable Problems

1
Lecture 6 - Single Variable Problems Systems
of Equations

• CVEN 302
• June 14, 2002

2
Lectures Goals

• Introduction of Solving Equations of 1 Variable
• Newtons Method
• Mullers Method
• Fixed Point Iteration

3
Lectures Goals

• Introduction to Systems of Equations
• Discuss how to solve systems
• Gaussian Elimination
• Gaussian Elimination with Pivoting
• Tridiagonal Solver
• Problems with the technique
• Examples

4
Newtons Method

• Do while x2 - x1 gt tolerance value 1
• or f(x2)gt tolerance
  value 2
• or
  f(x1) 0
• Set x2 x1 - f(x1)/f(x1)
• Set x1 x2
• END loop

5
Newtons Method

• Same example problem
• f(x) x5 x3 4x2 - 3x - 2
• and
• f(x) 5x4 3x2 8x - 3
• roots are between (-1.7,-1.3), (-1,0), (0.5,1.5)

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Newtons Method

• x1 f(x1) f(x1) f(x1)/f(x1)
  x2
• -2.0000 -20.0000 73.0000 -0.27397
  -1.7260
• -1.7260 -5.3656 36.5037 -0.14699
  -1.5790
• -1.5790 -1.0423 22.9290 -0.04546
  -1.5335
• -1.5335 -0.0797 19.4275 -0.00410
  -1.5294
• -1.5294 -0.0005 19.1381 -0.00003
  -1.5294
• etc. etc. etc.
  etc. etc.

7
Mullers Method

• Mullers method is an interpolation method that
  uses quadratic interpolation rather than linear.
  
• A second degree polynomial is used to fit three
  points in the vicinity of the root.

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Mullers Method

• Do while x2 - x1 gt tolerance value 1
• or f(x3) gt
  tolerance value 2
• c1 (f(x2)-f(x1))/(x2 - x1)
• c2 (f(x3)-f(x2))/(x3 - x2)
• d1 (c2-c1)/(x3 - x1)
• s c2 d1 (x3 - x2)
• x4 x3 - 2f(x3)/
• s sign(s)sqrt( s2
  -4f(x3)d1
• Set x1 x2
• Set x2 x3
• Set x3 x4
• END loop

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Mullers Method
This method uses a Newton form of an
interpolating polynomials.
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Example Problem

• f(x) x5 x3 4x2 - 3x - 2
• The roots are around (-2,-1),(-1,0) and (0,1)
• Look at (1,0) choice three points, 1.5, 0.5, 0.25

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Mullers Method

• x1 f(x1) x2 f(x2) x3
  f(x3) c1 c2 d1
  s1 x4
• 1.5000 13.4688 0.2500 -2.4834 0.5000 -2.3438
  12.7617 0.5586 12.2031 3.6094 0.8145
• 0.2500 -2.4834 0.5000 -2.3438 0.8145 -0.8900
  0.5586 4.6205 7.1938 6.8839 0.9300
• 0.5000 -2.3438 0.8145 -0.8900 0.9300 0.1698
  4.6205 9.1854 10.6157 10.4101 0.9134
• 0.8145 -0.8900 0.9300 0.1698 0.9134 -0.0049
  9.1854 10.5319 13.6309 10.3057 0.9139
• etc. etc. etc. etc. etc. etc.
  etc. etc. etc. etc. etc.

12
Fixed-Point Iteration Method

• The method uses an iterative scheme to find the
  root.
• The equation is rewritten to obtain new equation
  in terms of x.
• f(x) a3 x3 a2 x2 a1 x a0 0
• g(x) - ( a3/a1 ) x3 (a2 /a1) x2 (a0 /a1)
  

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Fixed-Point Iteration Method

• Problem is that the method only converge for a
  small range.
• g(x) lt 0.5

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Fixed-Point Iteration Method

• Rewrite f(x) -gt g(x)
• IF g(x) lt 0.5
• Do while xk1 - xk gt tolerance
• value
• xk1 g(xk)
• k k1
• END loop
• ENDIF

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Fixed Point Iteration

• From the book
• f(x) 5x3 -10x 3
• g(x) 0.5x3 0.3
• g(x) 1.5x2
• for 0.1ltxlt0.5
• g(x) lt 0.5

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Fixed Point Iteration Method

• The program demoFixedPoint test the convergence
  of the point.
• The program is limited to small range of values
• demoFixed(0.1,0.0,0.5)

17
Systems of Equations
18
Simple Linear Oscillator

• The spring-mass system can
• be described with a series of
• equations to model the
• physical behavior. The
• displacement of the masses
• are given as u, k represents
• the stiffness of the springs
• and M represents the mass of
• each member.

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Simple Linear Oscillator

• The free body diagrams of
• components of the spring
• mass system can be
• represented. Using the
• equilibrium equations, the
• static behavior of the model
• can be determined.

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Simple Oscillator

• The equations can be written from the free body
  diagrams.
• The matrix and vectors can be obtained from the
  equations.

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Simple 3-D frame

• A force is applied to the apex
• of a simple 3-D frame. We
• would like to determine the
• forces in each of the
• members of the frame. With
• a FBD, the static equilibrium
• equations can be derived.

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Simple 3-D Frame

• The set of 3 equilibrium equations and 3 unknowns
  can be
• obtained from the FBD of the frame. Place the
  equations in a
• matrix format.

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Simple 3-D Frame

• The frame is represented as a matrix with 3
  unknowns. The
• forces are normalized with respect to the applied
  force, F.

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Basic Principles

• The general description of a set of linear
  equations in the matrix form
• Ax b
• A ( m x n ) matrix
• x ( n x 1 ) vector
• b (m x 1 ) vector

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Description of the linear set of equations

• Write the equations in natural form
• Identify unknowns and order them
• Isolate unknowns
• Write equations in matrix form

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Types of Matrix Formulation

• ( m x n ) Array
• If m n The solution of Ax b with
• n unknowns and m equations
• If m gt n The system is overdetermined system
• (Least Square Problems)
• If m lt n The system is underdetermined system
• (Optimization Problems)

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Matrix Representation
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Consistency

• Ax b
• The problem is consistent, if a solution exists
  for the problem.
• The problem is inconsistent, if there is no
  solution for the problem.

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Rank of Matrix
Rank of a matrix is the number of linearly
independent column vectors in the matrix. For n
x n matrix, A

• If rank(A) n and is consistent, A has an unique
  solution exists
• If rank(A) n and is inconsistent, A has no
  solution exists
• If rank(A) lt n and system is consistent, A has an
  infinite number of solutions

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Matrix

• For an n x n system, rank(A) n automatically
  guarantees
• that the system is consistent.
• The columns of A are linearly independent
• The rows of A are linearly independent
• rank(A) n
• det(A) 0
• A-1 exists
• The solution to Ax b exist and is unique.

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Matrix Definition

• Consider
• y -2.0 x 6
• y 0.5 x 1
• 2 unknowns x , y and
• rank is 2

• Consider
• y -2 x 6
• y -2 x 5
• 2 unknowns x , y and rank is 1 and is
  inconsistent

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Gaussian Elimination

• Gaussian elimination is a fundamental
• procedure for solving linear sets of
• equation general it is applied to a square
• matrix.

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Gaussian Elimination

• There are two phases to the solving technique
• Elimination --- use row operations to convert the
  matrix into an upper diagonal matrix. (The
  elimination phase, which takes the the most
  effort and most susceptible to corruption by
  round off)
• Back substitution -- Solve x using a back
  substitution.

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Gaussian Elimination Algorithm

• Ax b
• Augment the n x n coefficient matrix with the
  vector of right hand sides to form a n x (n1)
• Interchange rows if necessary to make the value
  a11 with the largest magnitude of any coefficient
  in the first row
• Create zero in 2nd through nth row in first row
  by subtracting ai1 / a11 times first row from ith
  row

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Gaussian Elimination Algorithm

• Repeat (2) (3) for second through the (n-1)th
  rows, putting the largest magnitude coefficient
  in the diagonal by interchanging rows (consider
  only row j to n ) and then subtract times the
  jth row from the ith row so as to create zeros in
  all positions of jth column below the diagonal at
  conclusion of this step the system is upper
  triangular
• Solve for n from nth equation xn an,n1 / ann
• Solve for xn-1 , xn-2 , ...x1 from the (n-1)th
  through the first xi (ai,n1 - Sji1,n aj1
  xj ) / aii

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Example 1

• X1 3X2 5
• 2X1 4X2 6

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Example 2

• -3X1 2X2 - X3 -1
• 6X1 - 6X2 7X3 -7
• 3X1 - 4X2 4X3 -6

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Computer Program

• The program GEdemo(A,b) does the Gaussian
• elimination for a square matrix (nxn). It does
• not do any pivoting and works for only one
• b vector.

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Test the Program

• Example 1
• Example 2
• New Matrix
• 2X1 4X2 - 2 X3 - 2 X4 - 4
• 1X1 2X2 4X3 - 3 X4 5
• - 3X1 - 3X2 8X3 - 2X4 7
• - X1 X2 6X3 - 3X4
  7

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Problem with Gaussian Elimination

• The problem can occur when a zero appears in the
  diagonal and makes a simple Gaussian elimination
  impossible.
• Pivoting changes the matrix so that it will
  become diagonally dominate and reduce the
  round-off and truncation errors in the solving
  the matrix.

41
Computer Program

• GEPivotdemo(A,b) is a program, which will do a
  Gaussian elimination on matrix A with pivoting
  technique to make matrix diagonally dominate.
• The program is modification to handle a single
  value of b

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Question?

• How would you modify the programs to handle
  multiple inputs?
• What is diagonal matrix, upper triangular matrix,
  and lower triangular matrix?
• Can you do a column exchange and how would you
  handle the problem if it works?

43
Question?

• What happens with the following example?
• 0.0001X1 0.5 X2 0.5
• 0.4000X1 - 0.3 X2 0.1
• What happens is the second equation becomes
  0.4000X1 - 2000 X2 -2000

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Gaussian Elimination

• If the diagonal is not dominate the problem can
  have round off error and truncation errors.
• The scaling will result in problems

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Summary

• Newtons - need to know f(x) and f(x) and an
  initial guess.
• Mullers method -need to know the f(x) and 3
  values.
• Fixed Point method looking at a point a iterate.

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Summary

• Matrix properties consistency, rank, diagonally
  dominate, upper triangular, lower triangular
• Gaussian elimination is a method to solve for a
  set of linear equations
• Non-pivoting problems
• Setting up the system of linear equations to
  avoid problems

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Homework

• Check the Homework webpage