Naslov
On Principal Angles Between Subspaces of Euclidean Space

Autori
Drmač, Zlatko

Izvornik
SIAM journal on matrix analysis and applications (0895-4798) 22 (1999), 1;

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
principal angles; singular value decomposition; mixed stability

Sažetak
The cosines of the principal angles between the column spaces of full column rank matrices X\in\Rmp$ and $Y\in\Rmq$ are efficiently computed, using the Bj\"{o}rck--Golub algorithm, as the singular values of $Q_x^{T}Q_y$, where $Q_x$ and $Q_y$ are orthonormal matrices computed by the QR factorizations of $X$ and $Y$, respectively. This paper shows that the Bj\"{o}rck--Golub algorithm is mixed stable in the following sense: the computed singular values approximate with small relative error the exact cosines of the principal angles between the column spaces of $X+\Delta X$ and $Y+\Delta Y$, where $\Delta X$, $\Delta Y$ are small backward errors. Further, theoretical analysis and numerical evidence show that the algorithm becomes more robust if the QR factorizations are computed with the complete pivoting scheme of Powell and Reid. Moreover, it is shown that Gaussian elimination with complete pivoting can be used as an efficient preconditioner in computation and as a useful tool in analysis of the ensitivity of the QR factorization.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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