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Module 4 : Solving Linear Algebraic Equations

Section 4 : Direct Methods for Solving Sparse Linear Systems

4.2 Thomas Algorithm for Tridiagonal and Block Tridiagonal Matrices [2]
Consider system of equation given by following equation
	-------(34)
where matrix $\QTR{bf}{A}$ is a tridiagonal matrix. Thomas algorithm is the Gaussian elimination algorithm tailored to solve this type of sparse system. Step 1:Triangularization: Forward sweep with normalization
	-------(35)
	-------(36)
	-------(37)
	-------(38)
This sequence of operations finally results in the following system of equations Step 2: Backward sweep leads to solution vector
	-------(39)
	-------( 40)
$\;$ Total no of multiplications and divisions in Thomas algorithm is which is significantly smaller than the $\;n^{3}/3$ operations (approximately) necessary for carrying out the Gaussian elimination and backward sweep for a dense matrix.

Total no of multiplications and divisions in Thomas algorithm is which is significantly smaller than the operations (approximately) necessary for carrying out the Gaussian elimination and backward sweep for a dense matrix.