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Initial values of $ \lambda _L$ and $ \lambda _U$

Using the well-known Gershgorin bound:

$\displaystyle \min_i \Bigg(
[H]_{i,i} - \sum_{i\neq j } \vert [H]_{i,j} \vert ...
...] \leq \max_i \Bigg(
[H]_{i,i} + \sum_{i\neq j } \vert [H]_{i,j} \vert \Bigg) $

The frobenius or Euclidean norm:

$\displaystyle \Vert H \Vert _F =
\sqrt{\sum_{i=1}^m \sum_{j=1}^n [H]_{i,j}^2} $

The infinitum norm:

$\displaystyle \Vert H \Vert _{\infty} = \max_{1 \leq i \leq
\text{nLine}} \Vert H^T e_i\Vert _1 $

We finally obtain:

$\displaystyle \lambda_L := \max \Bigg[ 0, - \min_i
[H]_{i,i}, \frac{\Vert g\Ve...
... [H]_{i,j} \vert \Big], \Vert H\Vert _F,
\Vert H\Vert _{\infty} \bigg] \Bigg] $

$\displaystyle \lambda_U := \max \Bigg[ 0,
\frac{\Vert g\Vert _2}{\Delta} + \mi...
... [H]_{i,j} \vert \Big], \Vert H\Vert _F, \Vert H\Vert _{\infty}
\bigg] \Bigg] $

Frank Vanden Berghen 2004-04-19