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Currently, I am working on solving a PDE using the finite element method (FEM) and facing the problem of finding the inverse of the following Toeplitz tridiagonal matrix

$$\mathbf{M} = \begin{pmatrix} a & b & & & & & 0 \\ b & a & b & & & & \\ & b & a & b & & & \\ & & b & a & & & \\ & & & & \ddots & & \\ & & & & & a & b \\ 0 & & & & & b & a \\ \end{pmatrix}$$

Basically, this matrix has diagonal elements are all $a \in \mathbb{R}$ and every elements adjacent to the diagonal elements are $b \in \mathbb{R}$, the rest are all zeros.

I tried using Newton's method (the inverse matrix of $\mathbf{M}$ is determined by the limit of $\mathbf{X}_n$ where $n$ to infinity of the sequence $\mathbf{X}_{n+1} = \mathbf{X}_n(2\mathbf{I} - \mathbf{M}\mathbf{X}_n)$) but this method is not very good with large matrix (or perhaps my coding skill is terrible). So I was wondering is there another method to find the inversion of this type of matrix?

For more detail about FEM (on 1D case), you can visit here

Bakkune
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    Do you need the inverse or are you trying to solve a system of equations? If you are solving a system of equations this may help: Tridiagonal matrix algorithm – GrapefruitIsAwesome Mar 12 '22 at 12:33
  • Sparsity is usually not upheld in an inverse. You could use iterative methods if you are trying to solve a linear equation. If you need some kind of inverse matrix, you could to implement an incomplete lu factorization (uphelds the sparsity) – Laray Mar 12 '22 at 12:36
  • @GrapefruitIsAwesome Cool, this is want I'm looking for. Thank you so much for your help! – Bakkune Mar 12 '22 at 12:43
  • How large is your matrix? Why do you need to invert it? – Rodrigo de Azevedo Mar 12 '22 at 17:23
  • @RodrigodeAzevedo On average, the size of this matrix is around $1000\times 1000$. And the reason I need to find the inverse of it is to solve the coefficients vector $\mathbf{c}$ of the linear system $\mathbf{M}\mathbf{c} = \mathbf{b}$, from there I can determine the approximated function $\tilde{u}(x) = \sum_{j} c_j\phi_j(x)$ of $u(x)$ in the boundary value problem $-\Delta u = f$ in $\Omega$, $u|_{\partial\Omega} = 0$. This is the main idea of Galerkin's method to numerically solve PDE.

    Edit: Fixed $u|{\partial\Omega} = g$ to $u|{\partial\Omega} = 0$

     – Bakkune Mar 13 '22 at 11:55
  • Please include the background and motivation in the question itself. – Rodrigo de Azevedo Mar 13 '22 at 12:22
  • Related – Rodrigo de Azevedo Mar 14 '25 at 14:57

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