Iterative methods for solving matrix equations

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Iterative methods for solving matrix equations 1.
Jacobi 2. Gauss-Seidel 3. Successive
overrelaxation (SOR)
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What are C and d?
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Rewrite matrix equation in same way
becomes
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Then
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Jacobi method is like fixed point
iteration Example Shape of a stretched membrane
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Shape can be described by potential function
Let us give some boundary conditions
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Problem look likes this
Solve for us
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Leads to this system of equations
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Choose an initial u1 1 1 1 1
Iterate using xCxd
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Matlab solution, 49 iterations
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Gauss-Seidel method differs from Jacobi by
sequential updating - use new xis as they become
available
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Example
Jacobi
Gauss-Seidel
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Matlab example jacobidemo02 seideldemo02
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SOR - successive overrelaxation after xnew is
calculated by Gauss-Seidel, get another xnew,2 by
overrelaxation
If
underrelaxation
Problem specific
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Curve fitting - preliminaries Some statistics we
will need
Mean or average standard deviation variance coeffi
cient of variation
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Normal distribution we make the assumption that
errors are distributed normally
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Confidence intervals
Standard normal estimate
If we set z to some confidence level
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Confidence interval for mean is then
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For small n, use Student-t distribution