61,629 research outputs found
Computing the complete CS decomposition
An algorithm is developed to compute the complete CS decomposition (CSD) of a
partitioned unitary matrix. Although the existence of the CSD has been
recognized since 1977, prior algorithms compute only a reduced version (the
2-by-1 CSD) that is equivalent to two simultaneous singular value
decompositions. The algorithm presented here computes the complete 2-by-2 CSD,
which requires the simultaneous diagonalization of all four blocks of a unitary
matrix partitioned into a 2-by-2 block structure. The algorithm appears to be
the only fully specified algorithm available. The computation occurs in two
phases. In the first phase, the unitary matrix is reduced to bidiagonal block
form, as described by Sutton and Edelman. In the second phase, the blocks are
simultaneously diagonalized using techniques from bidiagonal SVD algorithms of
Golub, Kahan, and Demmel. The algorithm has a number of desirable numerical
features.Comment: New in v3: additional discussion on efficiency, Wilkinson shifts,
connection with tridiagonal QR iteration. New in v2: additional figures and a
reorganization of the tex
Truth Table Invariant Cylindrical Algebraic Decomposition by Regular Chains
A new algorithm to compute cylindrical algebraic decompositions (CADs) is
presented, building on two recent advances. Firstly, the output is truth table
invariant (a TTICAD) meaning given formulae have constant truth value on each
cell of the decomposition. Secondly, the computation uses regular chains theory
to first build a cylindrical decomposition of complex space (CCD) incrementally
by polynomial. Significant modification of the regular chains technology was
used to achieve the more sophisticated invariance criteria. Experimental
results on an implementation in the RegularChains Library for Maple verify that
combining these advances gives an algorithm superior to its individual
components and competitive with the state of the art
A Backward Stable Algorithm for Computing the CS Decomposition via the Polar Decomposition
We introduce a backward stable algorithm for computing the CS decomposition
of a partitioned matrix with orthonormal columns, or a
rank-deficient partial isometry. The algorithm computes two polar
decompositions (which can be carried out in parallel) followed by an
eigendecomposition of a judiciously crafted Hermitian matrix. We
prove that the algorithm is backward stable whenever the aforementioned
decompositions are computed in a backward stable way. Since the polar
decomposition and the symmetric eigendecomposition are highly amenable to
parallelization, the algorithm inherits this feature. We illustrate this fact
by invoking recently developed algorithms for the polar decomposition and
symmetric eigendecomposition that leverage Zolotarev's best rational
approximations of the sign function. Numerical examples demonstrate that the
resulting algorithm for computing the CS decomposition enjoys excellent
numerical stability
Computational Complexity of Atomic Chemical Reaction Networks
Informally, a chemical reaction network is "atomic" if each reaction may be
interpreted as the rearrangement of indivisible units of matter. There are
several reasonable definitions formalizing this idea. We investigate the
computational complexity of deciding whether a given network is atomic
according to each of these definitions.
Our first definition, primitive atomic, which requires each reaction to
preserve the total number of atoms, is to shown to be equivalent to mass
conservation. Since it is known that it can be decided in polynomial time
whether a given chemical reaction network is mass-conserving, the equivalence
gives an efficient algorithm to decide primitive atomicity.
Another definition, subset atomic, further requires that all atoms are
species. We show that deciding whether a given network is subset atomic is in
, and the problem "is a network subset atomic with respect to a
given atom set" is strongly -.
A third definition, reachably atomic, studied by Adleman, Gopalkrishnan et
al., further requires that each species has a sequence of reactions splitting
it into its constituent atoms. We show that there is a to decide whether a given network is reachably atomic, improving
upon the result of Adleman et al. that the problem is . We
show that the reachability problem for reachably atomic networks is
-.
Finally, we demonstrate equivalence relationships between our definitions and
some special cases of another existing definition of atomicity due to Gnacadja
Generic Regular Decompositions for Parametric Polynomial Systems
This paper presents a generalization of our earlier work in [19]. In this
paper, the two concepts, generic regular decomposition (GRD) and
regular-decomposition-unstable (RDU) variety introduced in [19] for generic
zero-dimensional systems, are extended to the case where the parametric systems
are not necessarily zero-dimensional. An algorithm is provided to compute GRDs
and the associated RDU varieties of parametric systems simultaneously on the
basis of the algorithm for generic zero-dimensional systems proposed in [19].
Then the solutions of any parametric system can be represented by the solutions
of finitely many regular systems and the decomposition is stable at any
parameter value in the complement of the associated RDU variety of the
parameter space. The related definitions and the results presented in [19] are
also generalized and a further discussion on RDU varieties is given from an
experimental point of view. The new algorithm has been implemented on the basis
of DISCOVERER with Maple 16 and experimented with a number of benchmarks from
the literature.Comment: It is the latest version. arXiv admin note: text overlap with
arXiv:1208.611
A subtraction scheme for computing QCD jet cross sections at NNLO: integrating the doubly unresolved subtraction terms
We finish the definition of a subtraction scheme for computing NNLO
corrections to QCD jet cross sections. In particular, we perform the
integration of the soft-type contributions to the doubly unresolved
counterterms via the method of Mellin-Barnes representations. With these final
ingredients in place, the definition of the scheme is complete and the
computation of fully differential rates for electron-positron annihilation into
two and three jets at NNLO accuracy becomes feasible.Comment: 33 pages, references added, exposition expanded, minor typos
corrected. Version published in JHE
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