# Tridiagonal matrix algorithm - TDMA (Thomas algorithm)

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 Revision as of 20:39, 14 December 2005 (view source)Tsaad (Talk | contribs)m (Thomas algorithm moved to Tridiagonal matrix algorithm - TDMA (Thomas algorithm))← Older edit Revision as of 20:43, 15 December 2005 (view source)Tsaad (Talk | contribs) (towards a uniform notation for linear systems : A*Phi = B)Newer edit → Line 2: Line 2: The Thomas Algorithm is a special form of Gauss elimination that can be used to solve tridiagonal The Thomas Algorithm is a special form of Gauss elimination that can be used to solve tridiagonal systems of equations. When the matrix is tridiagonal, the solution can be obtained in O(n) operations, systems of equations. When the matrix is tridiagonal, the solution can be obtained in O(n) operations, - instead of O(n3/3). Example of such matrices are matrices arising from descretisation of 1D problems. + instead of O(n3/3). Example of such matrices are matrices arising from discretization of 1D problems. We can write the tri-diagonal system in the form:
We can write the tri-diagonal system in the form:
:$:[itex] - a_i x_{i - 1} + b_i x_i + c_i x_{i + 1} = d_i + a_i \phi_{i - 1} + b_i \phi_i + c_i \phi_{i + 1} = d_i$
[/itex]
Where $a_1 = 0$ and $c_n = 0$
Where $a_1 = 0$ and $c_n = 0$
Line 17: Line 17: :    end loop (k) :    end loop (k) :    then
:    then
- ::    $x_n = {{d_n^' } \over {b_n }}$
+ ::    $\phi_n = {{d_n^' } \over {b_n }}$
:    for k := n-1 stepdown until 1 do
:    for k := n-1 stepdown until 1 do
- ::    $x_k = {{d_k^' - c_k x_{k + 1} } \over {b_k }}$
+ ::    $\phi_k = {{d_k^' - c_k \phi_{k + 1} } \over {b_k }}$
:    end loop (k) :    end loop (k) ---- ----

## Introduction

The Thomas Algorithm is a special form of Gauss elimination that can be used to solve tridiagonal systems of equations. When the matrix is tridiagonal, the solution can be obtained in O(n) operations, instead of O(n3/3). Example of such matrices are matrices arising from discretization of 1D problems.

We can write the tri-diagonal system in the form:

$a_i \phi_{i - 1} + b_i \phi_i + c_i \phi_{i + 1} = d_i$

Where $a_1 = 0$ and $c_n = 0$

## Algorithm

for k:= 2 step until n do
$m = {{a_k } \over {b_{k - 1} }}$
$b_k^' = b_k - mc_{k - 1}$
$d_k^' = d_k - md_{k - 1}$
end loop (k)
then
$\phi_n = {{d_n^' } \over {b_n }}$
for k := n-1 stepdown until 1 do
$\phi_k = {{d_k^' - c_k \phi_{k + 1} } \over {b_k }}$
end loop (k)

## References

1. Conte, S.D., and deBoor, C. (1972), Elementary Numerical Analysis, McGraw-Hill, New York..

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