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Fixed the formula as for the second reference of the article. I don't have access to the first reference to double check Tags: Visual edit Mobile edit Mobile web edit Advanced mobile edit |
→Inversion: symmetric tridiagonal inverse is single-pair Tag: possible vandalism |
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Closed form solutions can be computed for special cases such as [[symmetric matrix|symmetric matrices]] with all diagonal and off-diagonal elements equal<ref>{{Cite journal | last1 = Hu | first1 = G. Y. | last2 = O'Connell | first2 = R. F. | doi = 10.1088/0305-4470/29/7/020 | title = Analytical inversion of symmetric tridiagonal matrices | journal = Journal of Physics A: Mathematical and General | volume = 29 | issue = 7 | pages = 1511 | year = 1996 }}</ref> or [[Toeplitz matrices]]<ref>{{Cite journal | last1 = Huang | first1 = Y. | last2 = McColl | first2 = W. F. | doi = 10.1088/0305-4470/30/22/026 | title = Analytical inversion of general tridiagonal matrices | journal = Journal of Physics A: Mathematical and General | volume = 30 | issue = 22 | pages = 7919 | year = 1997 }}</ref> and for the general case as well.<ref>{{Cite journal | last1 = Mallik | first1 = R. K. | doi = 10.1016/S0024-3795(00)00262-7 | title = The inverse of a tridiagonal matrix | journal = Linear Algebra and its Applications | volume = 325 | pages = 109–139 | year = 2001 | doi-access = free }}</ref><ref>{{Cite journal | last1 = Kılıç | first1 = E. | doi = 10.1016/j.amc.2007.07.046 | title = Explicit formula for the inverse of a tridiagonal matrix by backward continued fractions | journal = Applied Mathematics and Computation | volume = 197 | pages = 345–357 | year = 2008 }}</ref>
In general, the inverse of a tridiagonal matrix is a [[semiseparable matrix]] and vice versa.<ref name="VandebrilBarel2008">{{cite book|author1=Raf Vandebril|author2=Marc Van Barel|author3=Nicola Mastronardi|title=Matrix Computations and Semiseparable Matrices. Volume I: Linear Systems|year=2008|publisher=JHU Press|isbn=978-0-8018-8714-7|at=Theorem 1.38, p. 41}}</ref> The inverse of a symmetric tridiagonal matrix can be written as a [[single-pair matrix]] (a.k.a. ''generator-representable semiseparable matrix'') of the form<ref name="Meurant1992">{{cite journal |last1=Meurant |first1=Gerard |title=A review on the inverse of symmetric tridiagonal and block tridiagonal matrices |journal=SIAM Journal on Matrix Analysis and Applications |date=1992 |volume=13 |issue=3 |pages=707-728 |doi=10.1137/0613045 |url=https://doi.org/10.1137/0613045}}</ref><ref>{{cite journal |last1=Bossu |first1=Sebastien |title=Tridiagonal and single-pair matrices and the inverse sum of two single-pair matrices |journal=Linear Algebra and its Applications |date=2024 |doi=10.1016/j.laa.2024.06.018 |url=https://authors.elsevier.com/a/1jOTP5YnCtZEc}}</ref>
<math>\begin{pmatrix}
\alpha_1 & -\beta_1 \\
-\beta_1 & \alpha_2 & -\beta_2 \\
& \ddots & \ddots & \ddots & \\
& & \ddots & \ddots & -\beta_{n-1} \\
& & & -\beta_{n-1} & \beta_n
\end{pmatrix}^{-1} =
\begin{pmatrix}
a_1 b_1 & a_1 b_2 & \cdots & a_1 b_n \\
a_1 b_2 & a_2 b_2 & \cdots & a_2 b_n \\
\vdots & \vdots & \ddots & \vdots \\
a_1 b_n & a_2 b_n & \cdots & a_n b_n
\end{pmatrix}
= \left( a_{\min(i,j)} b_{\max(i,j)} \right)
</math>
where <math>\begin{cases} \displaystyle a_i = \frac{\beta_{i} \cdots \beta_{n-1}}{\delta_i \cdots \delta_n\,b_n}
\\ \displaystyle
b_i = \frac{\beta_1 \cdots \beta_{i-1}}{d_1 \cdots d_i}\end{cases}</math>
with <math>\begin{cases}
d_n = \alpha_n,\quad d_{i-1} = \alpha_{i-1} - \frac{\beta_{i-1}^2}{d_{i}}, & i = n, n-1, \cdots, 2,
\\
\delta_1 = \alpha_1, \quad
\delta_{i+1} = \alpha_{i+1} - \frac{\beta_{i}^2}{\delta_{i}}, & i = 1, 2, \cdots, n-1.
\end{cases}
</math>
===Solution of linear system===
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