| # Chapter 2 | |
| ## The Divide-and-Conquer Process | |
| Like the tridiagonal divide-and-conquer method, a banded divide-and-conquer method can be structured into (i) a subdivision step, (ii) a solution step, and (iii) a synthesis step. | |
| ### 2.1 The Subdivision Step | |
| In analogy to the tridiagonal case, the band matrix $B$ is divided into $p$ smaller parts, each of them being a band matrix of size $n/p \times n/p$. For simplicity, only the case $p = 2$ is illustrated here. | |
| It should be noted that there are several possibilities for subdividing the original problem, a summary of which is given in Arbenz [2]. For the moment, the description will be restricted to the following decomposition: | |
| $$ | |
| \begin{aligned} | |
| B &= \begin{pmatrix} B_1 & R \\ R & B_2 \end{pmatrix} \\ | |
| &= \begin{pmatrix} B_1 - \begin{pmatrix} 0 \\ I_b \end{pmatrix} & 0 \\ 0 & B_2 - \begin{pmatrix} RR \\ 0 \end{pmatrix} \end{pmatrix} + \\ | |
| &\quad + \begin{pmatrix} 0 \\ I_b \\ R \\ 0 \end{pmatrix} (\begin{smallmatrix} 0 & | & I_b & | & R & | & 0 \end{smallmatrix}) \\ | |
| &= \begin{pmatrix} \hat{B}_1 & 0 \\ 0 & \hat{B}_2 \end{pmatrix} + WW^T. | |
| \end{aligned} | |
| $$ | |
| Here, $B_1, B_2, \hat{B}_1, \hat{B}_2$ are band matrices of size $n/2 \times n/2$, $R \in \mathbb{R}^{b \times b}$ is upper triangular and $W \in \mathbb{R}^{n \times b}$. | |
| ### 2.2 The Solution Step | |
| The $p$ smaller eigenproblems $\hat{B}_ix = \sigma x$ are solved independently, resulting in the factorizations | |
| $$ \hat{B}_i = Q_i\Sigma_iQ_i, \quad i = 1, 2, \dots, p. \qquad (2.1) $$ |