Tridiagonal matrix algorithm  TDMA (Thomas algorithm)
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#{{referencebookauthor=Conte, S.D., and deBoor, C.year=1972title=Elementary Numerical Analysisrest= McGrawHill, New York.}}  #{{referencebookauthor=Conte, S.D., and deBoor, C.year=1972title=Elementary Numerical Analysisrest= McGrawHill, New York.}}  
+  ==External link==  
+  *[http://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm Wikipedia article]  
''TODO: Add more references, more on the variants, and maybe more performance type info''  ''TODO: Add more references, more on the variants, and maybe more performance type info''  
[[User:JasondJasond]] 16:50, 18 December 2005 (MST)  [[User:JasondJasond]] 16:50, 18 December 2005 (MST) 
Revision as of 12:32, 14 February 2006
Contents 
Introduction
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 of equations. A tridiagonal system may be written as
where and . In matrix form, this system is written as
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 substitution produces the solution. Example of such matrices commonly arise from the discretization of 1D problems (e.g. the 1D Possion problem).
Algorithm
Forward elimination phase
 for k = 2 step until n do

 end loop (k)
Backward substitution phase

 for k = n1 stepdown until 1 do

 end loop (k)
Variants
In some situations, particularly those involving periodic boundary conditions, a slightly perturbed form of the tridiagonal system may need to be solved:
In this case, we can make use of the ShermanMorrison formula to avoid the additional operations of Gaussian elimination and still use the Thomas algorithm.
In other situation, the system of equation 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
 Conte, S.D., and deBoor, C. (1972), Elementary Numerical Analysis, McGrawHill, New York..
External link
TODO: Add more references, more on the variants, and maybe more performance type info Jasond 16:50, 18 December 2005 (MST)