Lecture 7 Gaussian Elimination

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Lecture 7 - Gaussian Elimination

• CVEN 302
• September 10, 2001

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Lectures Goals

• Introduction of Solving Equations of 1 Variable
• Fixed Point Iteration
• Gaussian Elimination

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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)

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Chapter 3
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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(n) 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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Summary

• Fixed Point method looking at a point a iterate.
• Gaussian Elimination

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Homework

• Check the Homework webpage