Thomas algorithm

Variants

In this case the coefficients ${\displaystyle a_{1}}$ and ${\displaystyle c_{n}}$ are, generally speaking, non-zero, so their presence does not allow to apply the Thomas algorithm directly. We can therefore consider ${\displaystyle B\in \mathbb {R} ^{n\times n}}$ and ${\displaystyle u,v\in \mathbb {R} ^{n}}$ as following: $${\displaystyle B={\begin{bmatrix}b_{1}-\gamma &c_{1}&&&0\\a_{2}&b_{2}&c_{2}&&\\&a_{3}&b_{3}&\ddots &\\&&\ddots &\ddots &c_{n-1}\\0&&&a_{n}&b_{n}-{\frac {c_{n}a_{1}}{\gamma }}\end{bmatrix}},u={\begin{bmatrix}\gamma \\0\\0\\\vdots \\c_{n}\end{bmatrix}},v={\begin{bmatrix}1\\0\\0\\\vdots \\a_{1}/\gamma \end{bmatrix}}.}$$ Where ${\displaystyle \gamma \in \mathbb {R} }$ is a parameter to be chosen. The matrix A can be reconstructed as ${\displaystyle A=B+uv^{\mathsf {T}}}$. The solution is then obtained in the following way: first we solve two tridiagonal systems of equations applying the Thomas algorithm: $${\displaystyle By=d\qquad \qquad Bq=u}$$