samples/texts/227520/page_7.md

# 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

$$ \hat{B}_i = Q_i\Sigma_iQ_i, \quad i = 1, 2, \dots, p. \qquad (2.1) $$