A Comparison of Some Iterative
Methods in Scientific Computing
Shawn Sickel
Man-Cheng Yeung
Jon Held
Department of Mathematics
Introduction:
Large sparse linera systems affect our lives from every direction.
In High School, everybody is taught the Gaussian Elimination
method. If a 1000x1000 matriz was solved using Gaussian
Elimination, it would take 334,333,000 steps, which would take
well more than 300 years to complete. In this presentation, I will
compare and contrast the methods used today for solving large sparse
linera equations.
Discussion:
The top graph, which compares the iteration speed between the
Jacobian Method and the Gauss-Seidel Method, clearly shows that
the Gauss-Seidel Method is better. Both of those methods require
the magnitude of the largest number to be <1. Gauss-Seidel
converges faster because the EigenValue is smaller than Jacobi’s.
The lower the eigenvalue is, the iterations will converge faster.
The second graph, which compares the Conjugate Gradient
Method and the BiConjugate Gradient Method in solving a non-SPD
matrix, shows that CG does not converge, but BiCG can. BiCG was
made to be able to converge with any linear system, whereas CG
was designed to solve SPD matrices only.
Purpose:
How is the GE method outdated?
Gaussian Elimination
What is an Iterative Method?
An Iterative Method uses educated guesses to find closer and
more accurate guesses each step.
Which algorithms are compared in this work?
In this work, GS, Jacobi, Conjugate Gradient, BiConjugate
Gradient, and BiConjugate Gradient Stabilized Methods are
evaulated.
Which Iterative Method is the fastest?
Each Krylov Subspace method and each Basic Iterative Method
has unique conditions that specifically limit the kinds of systems
that may be solved.
The third graph compares the BiCgonjugate Gradient Method
and the BiConjugate Gradient Stabilized Method. In this case, both
equations converge. BiCG was made to converge with any linear
system, but if the A-Transpose is not present, it cannot proceed.
The BiCGSTAB was created for situations which lack such
information.
Acknowledgements:
For the success in my paper, I would like to thank my fellow SRAPers
and the SRAP staff for encouragement and support. I would not have been
able to complete this research paper without the help and guidance of ManChung Yeung and Jon Held. They taught me how to use Matlab computer
programming, they helped edit my paper, and they showed me how the
different iterative methods work.
Methods:
My professor and graduate student guided me through learning the different methods, as well as scientific
computing. First, I learned how to solve linear systems by hand using GE. Then I was introduced into using the GS
and Jacobi method. Once I have experienced the time it takes to solve a simple 10x10 matriz with those, I learned that
in real life, people deal with matrices everyday, some having dimensions of over millions by millions. Matlab 7.0 was
a tool I was taught how to use. This programming tool was used to map these itration graphs, and solves for
eigenvalues instantly.