Shawn Sickel

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A Comparison of some Iterative Methods in
Scientific Computing

• Shawn Sickel

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What is Gaussian Elimination?

• Gaussian Elimination, GE, is a method for solving
  linear systems, taught in High School.

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Original Linear System
After adding the equations 1 and 2, the next step
is to eliminate this variable y.
To remove z, the opposite of line 3 will be
added to line 4. This leaves 2w -2.
To eliminate the previous y, the line 3 was
subtracted from line 2. Now z must be removed.
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Backwards Substitution
Once a variable is pinned down, the process of
backwards substitution begins. This process
includes plugging in variables to find the
solution, backwards.
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Why is the Gaussian Elimination method considered
outdated and Inefficient?

• This equation tells us how many steps are
  required in solving a linear system with the
  Gaussian Elimination method.
• Each step creates a new matrix, so a whole new
  set of numbers is formed each time.
• If this method is used on a 10x10 matrix, 430
  steps will be required in the process.

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Nobody uses Gaussian Elimination in real
applications.
Why?
Gaussian Elimination Is too slow,
-And- Requires too much memory.
Real life linear systems can have dimensions that
range from millions to billions. This amount of
numerical figures is overwhelming even to the
most powerful computer made today.
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The goal of this research is to study some
iterative methods, then to compare them.

• They are
• Jacobian Method
• Gauss-Seidel Method (GS)
• Conjugate Gradient Method (CG)
• BiConjugate Gradient Method (BiCG)
• BiConjugate Gradient Stabilized Method (BiCGSTAB)

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Basic Iterative Methods
The Jacobian Method and the GS Method are
considered Basic Iterative methods. For these
Methods, the p(G) Must be lt1. This means, that
the EigenValue with the largest magnitude must be
less than 1 or the iterations will go in the
completely opposite direction, and will never
converge to find the solution.
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Alpha 1 GS Vs. Jacobi
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Here, Alpha is set to 2. The GS method converges
faster than the Jacobi Method because it uses
more recent numbers to make its guess. Due to
this, the EigenValues of the GS method will
always be lower than the Jacobi method. They
both require the Eigenvalues to be lt1, but the
further it is from 1, the faster the iterations
will converge.
Basic Iterative Methods have some Drawbacks
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Krylov Subspace Methods These methods use the
span of the matrix, which is called the Krylov
Subspace.
Conjugate Gradient Method (CG) BiConjugate
Gradient Method (BiCG) BiConjugate Gradient
Stabilized Method (BiCGSTAB)
Basic Iterative Methods are extremely slow at
converging when dealing with large matrices.
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Conditions and Properties of the Krylov Subspace
Methods
CG BiCG BiCGSTAB
This method was designed to solve SPD matrices.
This method is similar to the Cg method, but can
also solve non-SPD matrices.
Not all real life situations give perfect sets of
data. BiCG requires the A-Transpose, which
is ? ? BiCG converges faster, but cannot
converge at all without the
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CG Vs. GS
CG Vs. BiCG
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BiCGSTAB Vs. BiCG
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Conclusion

• Basic Iterative Methods
• Krylov Subspace Methods
• Advantages / Disadvantages of the separate
  algorithms

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