Thomas algorithm

In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations. A tridiagonal system for n unknowns may be written as

where ${\displaystyle a_{1}=0\,}$ and ${\displaystyle c_{n}=0\,}$.

For such systems, the solution can be obtained in ${\displaystyle O(n)}$ operations instead of ${\displaystyle O(n^{3})}$ required by Gaussian elimination. A first sweep eliminates the ${\displaystyle a_{i}}$'s, and then an (abbreviated) backward substitution produces the solution. Examples of such matrices commonly arise from the discretization of 1D Poisson equation and natural cubic spline interpolation; similar systems of matrices arise in tight binding physics or nearest neighbor effects models.

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