Thomas algorithm to solve tridiagonal matrices

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Thomas algorithm to solve tridiagonal matrices
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Basically sets up an LU decomposition three
parts 1) decomposition 2) forward substitution 3)
backward substitution
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1) decomposition loop from rows 2 to n ai
ai/bi-1 bi bi-aici-1 end loop 2) forward
substitution loop from 2 to n ri ri -
airi-1 end loop
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3) back substitution xn rn/bn loop from n-1 to
1 xi (ri-cixi1)/bi end loop
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Example
First decompose T
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loop from rows 2 to 5 ai ai/bi-1 bi
bi-aici-1 end loop
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forward substitution loop from 2 to n ri ri -
airi-1 end loop
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back substitution xn rn/bn loop from n-1 to
1 xi (ri-cixi1)/bi end loop
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Crout algorithm - alternate LU decomposition Inste
ad of 1s on diagonal of L, get 1s on diagonal
of U Operate on rows and columns sequentially,
narrowing down to single element
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Algorithm
for j2 to n-1
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Cholesky decomposition A decomposition for
symmetric matrices Symmetric matrix e.g. a
covariance matrix
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For symmetric matrices, can write
Can develop relationships for l
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Example
1st row
Skip this equation
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Row 2
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Row 3
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Row 4
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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