Tridiagonal Matrix Algorithm - TDMA (Thomas Algorithm) - CFD-Wiki, The Free CFD Reference

 

5/13/2018

Tridiagonal matrix algorithm - TDMA (Thomas algorithm) -- CFD-Wiki, the free CFD reference

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Tridiagonal matrix algorithm - TDMA (Thomas algorithm) From CFD-Wiki

Contents 1 Intr oduction oduction 2 Algorithm  Algorithm 3 Assessments  Assessments 4 V  Var  ar iants iants 5 Ref erences erences

Introduction Introdu ction The tridiagonal matrix algorithm (TDMA), also known as the Thomas algorithm, is a simplified form of  Gaussian elimination that can be used to solve tridiagonal  systems  systems of equations. equations. A tridiagonal tridiagonal system may be written as   where

and

. In matrix form, this system is written as

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5/13/2018

Tridiagonal matrix algorithm - TDMA (Thomas algorithm) -- CFD-Wiki, the free CFD reference

For such systems, the solution can be obtained in

operations instead of

required by Gaussian

Elimination. A first sweep eliminates the 's, and then an (abbreviated) backward backward substitution produces the solution. Example of such matrices commonly arise from the discretization of 1D problems (e.g. the 1D Poisson problem).

Algorithm The following algorithm performs the TDMA, overwriting the original arrays. In some situations this is not desirable, so some prefer to copy the original arrays beforehand. Forward elimination phase for k = 2 step until n do

end loop (k) Backward substitution phase

  for k = n-1 stepdown until 1 do

end loop (k)

Assessments This algorithm is only applicable to matrices that are diagonally dominant, which is to say

Variants In some situations, particularly those involving periodic boundary conditions, a slightly perturbed form of the tridiagonal system may need to be solved:  

In matrix form, this is

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5/13/2018

Tridiagonal matrix algorithm - TDMA (Thomas algorithm) -- CFD-Wiki, the free CFD reference

In this case, we can make use of the Sherman-Morrison formula to avoid the additional operations of Gaussian elimination and still use the Thomas algorithm. We are now solving a problem of the form

where

 is a slightly different tridiagonal system than above, and the solution to the perturbed system is obtained by solving

and compute

as

In other situations, the system of equations may be block tridiagonal , with smaller submatrices arranged as the individual elements in the above matrix system. Simplified forms of Gaussian elimination have been developed for these situations.

References 1. Thomas, L.H. (1949), Elliptic Problems in Linear Differential Differential Equations over a Network , Watson Sci. Comput. Lab Report, Columbia University, New Yo York.. rk.. 2. Conte, S.D., and deBoor, C. (1972), Elementary Numerical Analysis, Analysis, McGraw-Hill, New York.. Still TODO: Add more references, more on the variants, make things nicer looking, and maybe more erformance type info --Jasond info --Jasond 22:59, 7 April 2006 (MDT) Retrieved from "https://www.cfd-online.com/Wiki/T "https://www.cfd-online.com/Wiki/Tridiagonal_matrix_algorithm_ridiagonal_matrix_algorithm_ _TDMA_(Thomas_algorithm)" This page was last modified on 14 September 2016, at 22:58. Content is available under GNU Free Documentation License 1.2.

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