11,751 research outputs found
A computer algebra user interface manifesto
Many computer algebra systems have more than 1000 built-in functions, making
expertise difficult. Using mock dialog boxes, this article describes a proposed
interactive general-purpose wizard for organizing optional transformations and
allowing easy fine grain control over the form of the result even by amateurs.
This wizard integrates ideas including:
* flexible subexpression selection;
* complete control over the ordering of variables and commutative operands,
with well-chosen defaults;
* interleaving the choice of successively less main variables with applicable
function choices to provide detailed control without incurring a combinatorial
number of applicable alternatives at any one level;
* quick applicability tests to reduce the listing of inapplicable
transformations;
* using an organizing principle to order the alternatives in a helpful
manner;
* labeling quickly-computed alternatives in dialog boxes with a preview of
their results,
* using ellipsis elisions if necessary or helpful;
* allowing the user to retreat from a sequence of choices to explore other
branches of the tree of alternatives or to return quickly to branches already
visited;
* allowing the user to accumulate more than one of the alternative forms;
* integrating direct manipulation into the wizard; and
* supporting not only the usual input-result pair mode, but also the useful
alternative derivational and in situ replacement modes in a unified window.Comment: 38 pages, 12 figures, to be published in Communications in Computer
Algebr
Computing Real Roots of Real Polynomials
Computing the roots of a univariate polynomial is a fundamental and
long-studied problem of computational algebra with applications in mathematics,
engineering, computer science, and the natural sciences. For isolating as well
as for approximating all complex roots, the best algorithm known is based on an
almost optimal method for approximate polynomial factorization, introduced by
Pan in 2002. Pan's factorization algorithm goes back to the splitting circle
method from Schoenhage in 1982. The main drawbacks of Pan's method are that it
is quite involved and that all roots have to be computed at the same time. For
the important special case, where only the real roots have to be computed, much
simpler methods are used in practice; however, they considerably lag behind
Pan's method with respect to complexity.
In this paper, we resolve this discrepancy by introducing a hybrid of the
Descartes method and Newton iteration, denoted ANEWDSC, which is simpler than
Pan's method, but achieves a run-time comparable to it. Our algorithm computes
isolating intervals for the real roots of any real square-free polynomial,
given by an oracle that provides arbitrary good approximations of the
polynomial's coefficients. ANEWDSC can also be used to only isolate the roots
in a given interval and to refine the isolating intervals to an arbitrary small
size; it achieves near optimal complexity for the latter task.Comment: to appear in the Journal of Symbolic Computatio
Computing Real Roots of Real Polynomials ... and now For Real!
Very recent work introduces an asymptotically fast subdivision algorithm,
denoted ANewDsc, for isolating the real roots of a univariate real polynomial.
The method combines Descartes' Rule of Signs to test intervals for the
existence of roots, Newton iteration to speed up convergence against clusters
of roots, and approximate computation to decrease the required precision. It
achieves record bounds on the worst-case complexity for the considered problem,
matching the complexity of Pan's method for computing all complex roots and
improving upon the complexity of other subdivision methods by several
magnitudes.
In the article at hand, we report on an implementation of ANewDsc on top of
the RS root isolator. RS is a highly efficient realization of the classical
Descartes method and currently serves as the default real root solver in Maple.
We describe crucial design changes within ANewDsc and RS that led to a
high-performance implementation without harming the theoretical complexity of
the underlying algorithm.
With an excerpt of our extensive collection of benchmarks, available online
at http://anewdsc.mpi-inf.mpg.de/, we illustrate that the theoretical gain in
performance of ANewDsc over other subdivision methods also transfers into
practice. These experiments also show that our new implementation outperforms
both RS and mature competitors by magnitudes for notoriously hard instances
with clustered roots. For all other instances, we avoid almost any overhead by
integrating additional optimizations and heuristics.Comment: Accepted for presentation at the 41st International Symposium on
Symbolic and Algebraic Computation (ISSAC), July 19--22, 2016, Waterloo,
Ontario, Canad
Tracking p-adic precision
We present a new method to propagate -adic precision in computations,
which also applies to other ultrametric fields. We illustrate it with many
examples and give a toy application to the stable computation of the SOMOS 4
sequence
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