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Let's take a tridiagonal matrix (in general not Toeplitz, nor symmetric)

$$L=\begin{pmatrix}a_1 & -b_1 & & & \\ -c_1 & a_2 & -b_2 \\ & -c_2 & \ddots & \ddots\\ & & \ddots\end{pmatrix}$$

and suppose that it's an M-matrix, diagonally dominant, with $a_i,b_i,c_i>0$, $a_i\ge c_{i-1}+ b_i$, ($c_0=b_n=0$) and such that there exists at least an index $i$ with $a_i> c_{i-1}+ b_i$.

I'm interested in an upper bound for the infinity norm of the inverse.

I already know that if $0<\gamma\le a_i-c_{i-1}- b_i$ for all indexes, then $$\|L^{-1}\|_{\infty}\le \gamma^{-1}$$ and, more in general, if $w\ge 0$ and $Lw\ge \gamma e$, then $$\|L^{-1}\|_{\infty}\le \gamma^{-1}\|w\|_{\infty}$$

I also know that for the particular matrix with $a_i=2, b_i=c_i=1$, we have $$\|L^{-1}\|_{\infty}\le \frac{(n+1)^2}{8}$$

Is there some general result of this kind? And if the matrix is Toeplitz, there's a rule for the dependence of such a bound to $n^k$ for some $k$?

Note: This Link may be useful.

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  • $\begingroup$ @epimorphic I hope you get to read this. I was referring to the discrete version of the maximum principle, applicable on matrices, that gives indeed useful bounds to the norm. $\endgroup$
    – Exodd
    Jun 7, 2015 at 8:38
  • $\begingroup$ Apologies if that's the case. I'm trying to gradually clean up the tag since it is so often mistakenly used. The tag definition specifies PDE and complex analysis, though I do try to make exceptions for similarly-named theorems in other fields to the extent that I'm aware. Could you provide a reference for this discrete maximum principle? $\endgroup$
    – epimorphic
    Jun 7, 2015 at 14:32
  • $\begingroup$ @epimorphic It's often used to determine the stability of numerical methods for computing PDE. Here an example with the standard Laplacian Matrix. books.google.it/… When I find some other reference, I'll let you know $\endgroup$
    – Exodd
    Jun 7, 2015 at 19:00

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