531 research outputs found
Manifold interpolation and model reduction
One approach to parametric and adaptive model reduction is via the
interpolation of orthogonal bases, subspaces or positive definite system
matrices. In all these cases, the sampled inputs stem from matrix sets that
feature a geometric structure and thus form so-called matrix manifolds. This
work will be featured as a chapter in the upcoming Handbook on Model Order
Reduction (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A.
Schilders, L.M. Silveira, eds, to appear on DE GRUYTER) and reviews the
numerical treatment of the most important matrix manifolds that arise in the
context of model reduction. Moreover, the principal approaches to data
interpolation and Taylor-like extrapolation on matrix manifolds are outlined
and complemented by algorithms in pseudo-code.Comment: 37 pages, 4 figures, featured chapter of upcoming "Handbook on Model
Order Reduction
Rational minimax approximation via adaptive barycentric representations
Computing rational minimax approximations can be very challenging when there
are singularities on or near the interval of approximation - precisely the case
where rational functions outperform polynomials by a landslide. We show that
far more robust algorithms than previously available can be developed by making
use of rational barycentric representations whose support points are chosen in
an adaptive fashion as the approximant is computed. Three variants of this
barycentric strategy are all shown to be powerful: (1) a classical Remez
algorithm, (2) a "AAA-Lawson" method of iteratively reweighted least-squares,
and (3) a differential correction algorithm. Our preferred combination,
implemented in the Chebfun MINIMAX code, is to use (2) in an initial phase and
then switch to (1) for generically quadratic convergence. By such methods we
can calculate approximations up to type (80, 80) of on in
standard 16-digit floating point arithmetic, a problem for which Varga, Ruttan,
and Carpenter required 200-digit extended precision.Comment: 29 pages, 11 figure
Accurate and Efficient Expression Evaluation and Linear Algebra
We survey and unify recent results on the existence of accurate algorithms
for evaluating multivariate polynomials, and more generally for accurate
numerical linear algebra with structured matrices. By "accurate" we mean that
the computed answer has relative error less than 1, i.e., has some correct
leading digits. We also address efficiency, by which we mean algorithms that
run in polynomial time in the size of the input. Our results will depend
strongly on the model of arithmetic: Most of our results will use the so-called
Traditional Model (TM). We give a set of necessary and sufficient conditions to
decide whether a high accuracy algorithm exists in the TM, and describe
progress toward a decision procedure that will take any problem and provide
either a high accuracy algorithm or a proof that none exists. When no accurate
algorithm exists in the TM, it is natural to extend the set of available
accurate operations by a library of additional operations, such as , dot
products, or indeed any enumerable set which could then be used to build
further accurate algorithms. We show how our accurate algorithms and decision
procedure for finding them extend to this case. Finally, we address other
models of arithmetic, and the relationship between (im)possibility in the TM
and (in)efficient algorithms operating on numbers represented as bit strings.Comment: 49 pages, 6 figures, 1 tabl
Computing the singular value decomposition with high relative accuracy
AbstractWe analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy. This means that each computed singular value is guaranteed to have some correct digits, even if the singular values have widely varying magnitudes. This is in contrast to the absolute accuracy provided by conventional backward stable algorithms, which in general only guarantee correct digits in the singular values with large enough magnitudes. It is of interest to compute the tiniest singular values with several correct digits, because in some cases, such as finite element problems and quantum mechanics, it is the smallest singular values that have physical meaning, and should be determined accurately by the data. Many recent papers have identified special classes of matrices where high relative accuracy is possible, since it is not possible in general. The perturbation theory and algorithms for these matrix classes have been quite different, motivating us to seek a common perturbation theory and common algorithm. We provide these in this paper, and show that high relative accuracy is possible in many new cases as well. The briefest way to describe our results is that we can compute the SVD of G to high relative accuracy provided we can accurately factor G=XDYT where D is diagonal and X and Y are any well-conditioned matrices; furthermore, the LDU factorization frequently does the job. We provide many examples of matrix classes permitting such an LDU decomposition
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