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Publication# Iterative refinement of Schur decompositions

Abstract

The Schur decomposition of a square matrix A is an important intermediate step of state-of-the-art numerical algorithms for addressing eigenvalue problems, matrix functions, and matrix equations. This work is concerned with the following task: Compute a (more) accurate Schur decomposition of A from a given approximate Schur decomposition. This task arises, for example, in the context of parameter-dependent eigenvalue problems and mixed precision computations. We have developed a Newton-like algorithm that requires the solution of a triangular matrix equation and an approximate orthogonalization step in every iteration. We prove local quadratic convergence for matrices with mutually distinct eigenvalues and observe fast convergence in practice. In a mixed low-high precision environment, our algorithm essentially reduces to only four high-precision matrix-matrix multiplications per iteration. When refining double to quadruple precision, it often needs only 3-4 iterations, which reduces the time of computing a quadruple precision Schur decomposition by up to a factor of 10-20.

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In this thesis we address the computation of a spectral decomposition for symmetric
banded matrices. In light of dealing with large-scale matrices, where classical dense
linear algebra routines are not applicable, it is essential to design alternative techniques
that take advantage of data properties. Our approach is based upon exploiting the
underlying hierarchical low-rank structure in the intermediate results. Indeed, we study
the computation in the hierarchically off-diagonal low rank (HODLR) format of two
crucial tools: QR decomposition and spectral projectors, in order to devise a fast spectral
divide-and-conquer method.
In the first part we propose a new method for computing a QR decomposition of a
HODLR matrix, where the factor R is returned in the HODLR format, while Q is given
in a compact WY representation. The new algorithm enjoys linear-polylogarithmic complexity and is norm-wise accurate. Moreover, it maintains the orthogonality of the Q factor.
The approximate computation of spectral projectors is addressed in the second part.
This problem has raised some interest in the context of linear scaling electronic structure
methods. There the presence of small spectral gaps brings difficulties to existing algorithms based on approximate sparsity. We propose a fast method based on a variant of
the QDWH algorithm, and exploit that QDWH applied to a banded input generates a
sequence of matrices that can be efficiently represented in the HODLR format. From
the theoretical side, we provide an analysis of the structure preservation in the final
outcome. More specifically, we derive a priori decay bounds on the singular values in
the off-diagonal blocks of spectral projectors. Consequently, this shows that our method,
based on data-sparsity, brings benefits in terms of memory requirements in comparison to
approximate sparsity approaches, because of its logarithmic dependence on the spectral
gap. Numerical experiments conducted on tridiagonal and banded matrices demonstrate
that the proposed algorithm is robust with respect to the spectral gap and exhibits linear-polylogarithmic complexity. Furthermore, it renders very accurate approximations
to the spectral projectors even for very large matrices.
The last part of this thesis is concerned with developing a fast spectral divide-and-conquer
method in the HODLR format. The idea behind this technique is to recursively split the
spectrum, using invariant subspaces associated with its subsets. This allows to obtain
a complete spectral decomposition by solving the smaller-sized problems. Following
Nakatsukasa and Higham, we combine our method for the fast computation of spectral
projectors with a novel technique for finding a basis for the range of such a HODLR
matrix. The latter strongly relies on properties of spectral projectors, and it is analyzed
theoretically. Numerical results confirm that the method is applicable for large-scale
matrices, and exhibits linear-polylogarithmic complexity.