Iterative Methods for Solving Linear Systems

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Iterative Methods for Solving Linear Systems

• Leo Magallon Morgan Ulloa

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What is an Iterative Method

• - An alternative method for solving the linear
  systems problem Axb where A is an nxn matrix
  with n equations and n unknowns.
• - These methods use a repetitive algorithm to
  generate sequences of vectors that steadily
  approach your solution.
• We will discuss two types of iterative methods
• ? Jacobi
• ? Gauss-Seidel

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Convergence

• Convergence is when the outputs of your functions
  stop changing significantly. In other words they
  converge towards a solution.
• Iterative methods rely on the functions ability
  to converge towards a solution, if they dont
  converge then you cannot find an answer.
• We can say that a system will converge if it is
  diagonally dominant. When the absolute value of
  each diagonal entry is greater than the absolute
  value of the sum of every component in its row.

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The linear systems problemAxb

• Lets look at the following system
• From what weve learned in class we know how to
  solve this system by taking rref(A) in an
  augmented matrix.
• So lets solve our system using elementary row
  operations.

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• As we can see by taking rref(A)
  the solution to our system is
• x1 1 x2 2
• Lets check our answer
• 7(1) 1(2) 5
• 3(1) 5(2) -7

• ?Correct!

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So why use iterative methods?
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• When using elementary row operations by hand it
  is likely that calculating errors will be made
  (especially when the matrix is full of fractions)
• Iterative methods can be quicker if the matrix is
  simple or has many zero components.
• You can stop calculating when your answers start
  to converge, whereas with row operations you must
  work all the way to rref(A)
• Round off errors may actually accelerate
  convergence because you are jumping more quickly
  towards your solution.

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Lets look at Jacobis method

• From our 2x2 matrix we can see that there are 2
  equations with 2 unknowns, and the matrix is
  diagonally dominant therefore it will converge.
• 7x1 1x2 5
• 3x1 5x2 -7
• Step 1 solve the first equation for x1 and solve
  the second equation for x2
• x1 5 x2 x2 7 3x1
• 7 5
• Step 2 choose an initial approximation. We
  choose
• x1 0 x2 0

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• Step 3 plug in your chosen values and solve for
  x1 and x2.
• X1 5 1(0) 5/7 X2 7 3(0) 7/5
• 7 5
• Step 4 use new values for x1 and x2 and input
  them into your original functions. continue this
  process until it converges to your solution.
• Here we can see the solution is converging to
• x11 x2 2 at around 6 iterations.
• the same as our answer from rref(A).

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Gauss-Seidel

• Almost identical to Jacobis method except it
  converges faster because we use our new outputs
  as soon as we can.
• Lets see how it works
• Step1 from our two equations, solve for x1 and
  x2
• 7x1 1x2 5
• 3x1 5x2 -7
• x1 5 x2 x2 7 3x1
• 7 5

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• Step 2 choose an initial approximation. we
  choose
• Step 3 plug in your initial value and solve for
  x1
• x1 5 1(0) 5/7
• 7
• Step 4 now use this new value for x1 to solve
  for x2
• x2 7 3(5/7) 64/35 1.829
• 5
• solve again for x1 x 5
  1(64/35) 0.976
• 
  7

x1 0 x2 0
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• Step 5 continue to input your newest values for
  x1 and x2 into the functions until they converge
  towards a solution.
• Here we can see that the solution is converging
  to x1 1 x2 2 at around
• 4 iterations.

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iteration shown graphically

• David Strong's applet for solving Axb using
  iterative methods

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• Iterative methods rock!