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:

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

$${\widehat{B}}_i=Q_i{\mathrm{\Sigma }}_i{Q}_i,i=1,2,\dots ,p\mathrm{.}(2.1)\backslash hat\{B\}\_i\; =\; Q\_i\backslash Sigma\_iQ\_i,\; \backslash quad\; i\; =\; 1,\; 2,\; \backslash dots,\; p.\; \backslash qquad\; (2.1)$$