40 research outputs found
Selecting Algorithms for Black Box Matrices: Checking for Matrix Properties That Can Simplify Computations
Processes to automate the selection of appropriate algorithms for various
matrix computations are described. In particular, processes to check for, and
certify, various matrix properties of black box matrices are presented. These
include sparsity patterns and structural properties that allow "superfast"
algorithms to be used in place of black-box algorithms. Matrix properties that
hold generically, and allow the use of matrix preconditioning to be reduced or
eliminated, can also be checked for and certified - notably including in the
small-field case, where this presently has the greatest impact on the
efficiency of the computation.Comment: Department of Computer Science Technical Report 2016-1085-0
Solving Sparse Integer Linear Systems
We propose a new algorithm to solve sparse linear systems of equations over
the integers. This algorithm is based on a -adic lifting technique combined
with the use of block matrices with structured blocks. It achieves a sub-cubic
complexity in terms of machine operations subject to a conjecture on the
effectiveness of certain sparse projections. A LinBox-based implementation of
this algorithm is demonstrated, and emphasizes the practical benefits of this
new method over the previous state of the art
Faster Inversion and Other Black Box Matrix Computations Using Efficient Block Projections
Block projections have been used, in [Eberly et al. 2006], to obtain an
efficient algorithm to find solutions for sparse systems of linear equations. A
bound of softO(n^(2.5)) machine operations is obtained assuming that the input
matrix can be multiplied by a vector with constant-sized entries in softO(n)
machine operations. Unfortunately, the correctness of this algorithm depends on
the existence of efficient block projections, and this has been conjectured. In
this paper we establish the correctness of the algorithm from [Eberly et al.
2006] by proving the existence of efficient block projections over sufficiently
large fields. We demonstrate the usefulness of these projections by deriving
improved bounds for the cost of several matrix problems, considering, in
particular, ``sparse'' matrices that can be be multiplied by a vector using
softO(n) field operations. We show how to compute the inverse of a sparse
matrix over a field F using an expected number of softO(n^(2.27)) operations in
F. A basis for the null space of a sparse matrix, and a certification of its
rank, are obtained at the same cost. An application to Kaltofen and Villard's
Baby-Steps/Giant-Steps algorithms for the determinant and Smith Form of an
integer matrix yields algorithms requiring softO(n^(2.66)) machine operations.
The derived algorithms are all probabilistic of the Las Vegas type
Computational Arithmetic of Modular Forms
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted student to implement it over any ring (such that a sufficient linear algebra theory is available in the chosen computer algebra system). The chosen approach is based on group cohomology and along the way the needed tools from homological algebra are provided
Omecamtiv mecarbil in chronic heart failure with reduced ejection fraction, GALACTICâHF: baseline characteristics and comparison with contemporary clinical trials
Aims:
The safety and efficacy of the novel selective cardiac myosin activator, omecamtiv mecarbil, in patients with heart failure with reduced ejection fraction (HFrEF) is tested in the Global Approach to Lowering Adverse Cardiac outcomes Through Improving Contractility in Heart Failure (GALACTICâHF) trial. Here we describe the baseline characteristics of participants in GALACTICâHF and how these compare with other contemporary trials.
Methods and Results:
Adults with established HFrEF, New York Heart Association functional class (NYHA)ââ„âII, EF â€35%, elevated natriuretic peptides and either current hospitalization for HF or history of hospitalization/ emergency department visit for HF within a year were randomized to either placebo or omecamtiv mecarbil (pharmacokineticâguided dosing: 25, 37.5 or 50âmg bid). 8256 patients [male (79%), nonâwhite (22%), mean age 65âyears] were enrolled with a mean EF 27%, ischemic etiology in 54%, NYHA II 53% and III/IV 47%, and median NTâproBNP 1971âpg/mL. HF therapies at baseline were among the most effectively employed in contemporary HF trials. GALACTICâHF randomized patients representative of recent HF registries and trials with substantial numbers of patients also having characteristics understudied in previous trials including more from North America (n = 1386), enrolled as inpatients (n = 2084), systolic blood pressureâ<â100âmmHg (n = 1127), estimated glomerular filtration rate <â30âmL/min/1.73 m2 (n = 528), and treated with sacubitrilâvalsartan at baseline (n = 1594).
Conclusions:
GALACTICâHF enrolled a wellâtreated, highârisk population from both inpatient and outpatient settings, which will provide a definitive evaluation of the efficacy and safety of this novel therapy, as well as informing its potential future implementation
LOGARITHMIC DEPTH CIRCUITS FOR HERMITE INTERPOLATION
We present a new parallel algorithm for Hermite interpolation. The
algorithm can be implemented using arithmetic-boolean circuits of
depth logarithmic and size polynomial in the input size. A
corresponding Boolean algorithm can be used to compute the coefficients
of the Hermite interpolating polynomial from binary representations of
evaluation points and derivatives over finite fields and number fields,
using a P-uniform family of circuits of depth logarithmic in the
input size and of polynomial size.We are currently acquiring citations for the work deposited into this collection. We recognize the distribution rights of this item may have been assigned to another entity, other than the author(s) of the work.If you can provide the citation for this work or you think you own the distribution rights to this work please contact the Institutional Repository Administrator at [email protected]
Asymptotically efficient algorithms for the Frobenius form
A new randomized algorithm is presented for computation of the Frobenius form of an nn matrix over a field. A version of the algorithm is presented that uses standard arithmetic whose asymptotic expected complexity matches the worst case complexity of the best known deterministic algorithm for this problem, recently given by Storjohann and Villard [25], and that seems to be superior when applied to sparse or structured matrices with a small number of invariant factors. A version that uses asymptotically fast matrix multiplication is also presented. This is the first known algorithm for this computation over small fields whose asymptotic complexity matches that of the best algorithm for computations over large fields and that also provides a Frobenius transition matrix over the ground field. As an application, it is shown that a "rational Jordan form" of an nn matrix over a finite field can also be computed asymptotically efficiently