Divide-and-conquer eigenvalue algorithm

Divide-and-conquer eigenvalue algorithms are a class of eigenvalue algorithms for Hermitian or real symmetric matrices that have recently (circa 1990s) become competitive in terms of stability and efficiency with more traditional algorithms such as the QR algorithm. The basic concept behind these algorithms is of course the famous divide-and-conquer approach from computer science. An eigenvalue problem is divided into two problems of roughly half the size, each of these are solved recursively, and the eigenvalues of the original problem are computed from the results of these smaller problems.

Here we present the simplest version of a divide-and-conquer algorithm, similar to the one originally proposed by Cuppen in 1981. Many details that lie outside the scope of this article will be omitted; it should be noted, however, that without considering these details, the algorithm is not fully stable.

Background

As with most eigenvalue algorithms for Hermitian matrices, divide-and-conquer begins with a reduction to tridiagonal form. For an $m\times m$ matrix, the standard method for this, Householder reduction, takes ${\frac {4}{3}}m^{3}$ flops, or ${\frac {8}{3}}m^{3}$ if eigenvectors are needed as well. There are other algorithms, such as the Arnoldi iteration, which may do better for certain classes of matrices; we will not consider this further here.

In certain cases, it is possible to deflate an eigenvalue problem into smaller problems. Consider a block diagonal matrix

$T={\begin{bmatrix}T_{1}&0\\0&T_{2}\end{bmatrix}}$

The eigenvalues and eigenvectors of $T$ are simply those of $T_{1}$ and $T_{2}$, and it will almost always be faster to solve these two smaller problems than to solve the original problem all at once. This technique can used to improve the efficiency of many eigenvalue algorithms, but it has special significance to divide-and-conquer.

For the rest of this article, we will assume the input to the divide-and-conquer algorithm is an $m\times m$ real symmetric tridiagonal matrix $T$. Although the algorithm can be modified for Hermitian matrices, we do not give the details here.

Divide

The divide part of the divide and conquer algorithm comes from the realization that a tridiagonal matrix is "almost" block diagonal.

$T={\begin{bmatrix}T_{1}&{\begin{matrix}&&\\&&\\\beta &&\end{matrix}}\\{\begin{matrix}&&\beta \\&&\\&&\end{matrix}}&T_{2}\end{bmatrix}}$

We write $T$ as a block diagonal matrix, plus a rank-1 correction:

$T={\begin{bmatrix}{\hat {T}}_{1}&0\\0&{\hat {T}}_{2}\end{bmatrix}}+{\begin{bmatrix}&&&&&\\&&&&&\\&&\beta &\beta &&\\&&\beta &\beta &&\\&&&&&\\&&&&&\end{bmatrix}}$