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tamaskis authored Mar 27, 2021
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Expand Up @@ -15,9 +15,9 @@ Solves the tridiagonal linear system <img src="https://latex.codecogs.com/svg.la

## Tridiagonal Matrix Convention

For this implementation, I use the following convention for denoting the elements of the tridiagonal matrix <img src="https://latex.codecogs.com/svg.latex?A" title="A" />:
For this implementation, I use the following convention for denoting the elements of the tridiagonal matrix <img src="https://latex.codecogs.com/svg.latex?\mathbf{A}" title="\mathbf{A}" />:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img src="https://latex.codecogs.com/svg.latex?A=\begin{bmatrix}b_{1}&c_{1}\\a_{1}&b_{2}&c_{2}\\&a_{2}&\ddots&\ddots\\&&\ddots&\ddots&c_{n-2}\\&&&&space;a_{n-2}&b_{n-1}&c_{n-1}\\&&&&a_{n-1}&&space;b_{n}\\\end{bmatrix}" title="A=\begin{bmatrix}b_{1}&c_{1}\\a_{1}&b_{2}&c_{2}\\&a_{2}&\ddots&\ddots\\&&\ddots&\ddots&c_{n-2}\\&&& a_{n-2}&b_{n-1}&c_{n-1}\\&&&&a_{n-1}& b_{n}\\\end{bmatrix}" />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img src="https://latex.codecogs.com/svg.latex?\mathbf{A}=\begin{bmatrix}b_{1}&c_{1}\\a_{1}&b_{2}&c_{2}\\&a_{2}&\ddots&\ddots\\&&\ddots&\ddots&c_{n-2}\\&&&&space;a_{n-2}&b_{n-1}&c_{n-1}\\&&&&a_{n-1}&&space;b_{n}\\\end{bmatrix}" title="\mathbf{A}=\begin{bmatrix}b_{1}&c_{1}\\a_{1}&b_{2}&c_{2}\\&a_{2}&\ddots&\ddots\\&&\ddots&\ddots&c_{n-2}\\&&& a_{n-2}&b_{n-1}&c_{n-1}\\&&&&a_{n-1}& b_{n}\\\end{bmatrix}" />

Most references, including the two listed below, have <img src="https://latex.codecogs.com/svg.latex?a_{i}" title="a_{i}" />'s ranging from <img src="https://latex.codecogs.com/svg.latex?a_{2}" title="a_{2}" /> to <img src="https://latex.codecogs.com/svg.latex?a_{n}" title="a_{n}" /> both in the definition of the tridiagonal matrix and in the algorithm used to solve the corresponding linear system. In this implementation, I have the <img src="https://latex.codecogs.com/svg.latex?a_{i}" title="a_{i}" />'s ranging from <img src="https://latex.codecogs.com/svg.latex?a_{1}" title="a_{1}" /> to <img src="https://latex.codecogs.com/svg.latex?a_{n-1}" title="a_{n-1}" />; this makes the algorithm slightly more straightforward to implement.

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