#### Volume 40, pp. 414-435, 2013.

## Computing approximate extended Krylov subspaces without explicit inversion

Thomas Mach, Miroslav S. Pranić, and Raf Vandebril

### Abstract

It is shown that extended Krylov subspaces–-under some assumptions–-can
be computed approximately without any explicit inversion or
system solves involved. Instead, the necessary computations are done in an implicit
way using the information from an enlarged standard Krylov subspace.

For both the classical and extended Krylov spaces, the matrices capturing the
recurrence coefficients can be retrieved by projecting the original matrix on
a particular orthogonal basis of the associated (extended) Krylov space. It is
also well-known that for (extended) Krylov spaces of full dimension,
i.e., equal to the matrix size, the matrix of recurrences can be obtained
directly by executing similarity transformations on the original matrix. In
practice, however, for large dimensions, computing time is saved by making use
of iterative procedures to gradually gather the recurrences in a matrix.
Unfortunately, for extended Krylov spaces, one is obliged to frequently solve
systems of equations.

In this paper the iterative and the direct similarity approach are integrated,
thereby avoiding system solves. At first, an orthogonal basis of a standard Krylov
subspace of dimension $m_\ell+m_r+p$ and the matrix of recurrences are
constructed iteratively. After that, cleverly chosen unitary similarity
transformations are executed to alter the matrix of recurrences, thereby also
changing the orthogonal basis vectors spanning the large Krylov
space. Finally, only the first $m_\ell+m_r-1$ new basis vectors are retained
resulting in an orthogonal basis approximately spanning the extended Krylov
subspace
\[
\mathcal{K}_{m_\ell,m_r}(A,v) = \mathop{\mathrm{span}}\left\lbrace{A^{-m_r+1}v, \dotsc, A^{-1}v, v, Av, A^{2}v, \dotsc, A^{m_\ell-1}v}\right\rbrace.
\]
Numerical experiments support the claim that this approximation is very good
if the large Krylov subspace approximately contains $\mathop{\mathrm{span}}\left\lbrace{A^{-m_r+1}v,
\dotsc, A^{-1}v}\right\rbrace$. This can culminate in significant dimensionality
reduction and as such can also lead to time savings when approximating or
solving, e.g., matrix functions or equations.

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### Key words

Krylov, extended Krylov, iterative methods, Ritz values, polynomial approximation, rotations, QR factorization

### AMS subject classifications

65F60, 65F10, 47J25, 15A16

### ETNA articles which cite this article

Vol. 43 (2014-2015), pp. 100-124 Thomas Mach, Miroslav S. Pranić, and Raf Vandebril: Computing approximate (block) rational Krylov subspaces without explicit inversion with extensions to symmetric matrices |

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