grandes-ecoles 2013 Q21

grandes-ecoles · France · x-ens-maths__pc Matrices Matrix Norm, Convergence, and Inequality

In the optimal version of the Jacobi algorithm, we choose for each $m$ a pair $(p_m, q_m)$ such that the absolute value of the coefficient $\sigma_{ij}^{(m)}$ is maximal precisely when $(i,j) = (p_m, q_m)$. In other words, $$\forall i < j, \quad \left|\sigma_{ij}^{(m)}\right| \leqslant \left|\sigma_{p_m q_m}^{(m)}\right|$$
Let $m \in \mathbb{N}$. Show that $$\left\|D - D^{(m)}\right\| \leqslant \frac{\rho^m}{1 - \rho} \left\|E^{(0)}\right\|$$

In the optimal version of the Jacobi algorithm, we choose for each $m$ a pair $(p_m, q_m)$ such that the absolute value of the coefficient $\sigma_{ij}^{(m)}$ is maximal precisely when $(i,j) = (p_m, q_m)$. In other words,
$$\forall i < j, \quad \left|\sigma_{ij}^{(m)}\right| \leqslant \left|\sigma_{p_m q_m}^{(m)}\right|$$

Let $m \in \mathbb{N}$. Show that
$$\left\|D - D^{(m)}\right\| \leqslant \frac{\rho^m}{1 - \rho} \left\|E^{(0)}\right\|$$

Paper Questions

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