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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