1,299 research outputs found
On Cayley's factorization with an application to the orthonormalization of noisy rotation matrices
The final publication is available at link.springer.comA real orthogonal matrix representing a rotation in four dimensions can be decomposed into the commutative product of a left- and a right-isoclinic rotation matrix. This operation, known as Cayley's factorization, directly provides the double quaternion representation of rotations in four dimensions. This factorization can be performed without divisions, thus avoiding the common numerical issues attributed to the computation of quaternions from rotation matrices. In this paper, it is shown how this result is particularly useful, when particularized to three dimensions, to re-orthonormalize a noisy rotation matrix by converting it to quaternion form and then obtaining back the corresponding proper rotation matrix. This re-orthonormalization method is commonly implemented using the Shepperd-Markley method, but the method derived here is shown to outperform it by returning results closer to those obtained using the Singular Value Decomposition which are known to be optimal in terms of the Frobenius norm.Peer ReviewedPostprint (author's final draft
Hierarchical interpolative factorization for elliptic operators: differential equations
This paper introduces the hierarchical interpolative factorization for
elliptic partial differential equations (HIF-DE) in two (2D) and three
dimensions (3D). This factorization takes the form of an approximate
generalized LU/LDL decomposition that facilitates the efficient inversion of
the discretized operator. HIF-DE is based on the multifrontal method but uses
skeletonization on the separator fronts to sparsify the dense frontal matrices
and thus reduce the cost. We conjecture that this strategy yields linear
complexity in 2D and quasilinear complexity in 3D. Estimated linear complexity
in 3D can be achieved by skeletonizing the compressed fronts themselves, which
amounts geometrically to a recursive dimensional reduction scheme. Numerical
experiments support our claims and further demonstrate the performance of our
algorithm as a fast direct solver and preconditioner. MATLAB codes are freely
available.Comment: 37 pages, 13 figures, 12 tables; to appear, Comm. Pure Appl. Math.
arXiv admin note: substantial text overlap with arXiv:1307.266
- …