A tridiagonal matrix has zero entries outside its main diagonal and the two adjacent diagonals. With nonzero elimination pivots, the Thomas algorithm solves its linear system in linear time by specialized Gaussian elimination and back substitution.
Nonsingularity alone does not ensure that the unpivoted Thomas algorithm is defined: the invertible matrix has a zero first pivot. Conditions such as strict diagonal dominance guarantee suitable nonzero pivots; otherwise row interchanges or another factorization are needed.
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