920 research outputs found
Subtropical Real Root Finding
We describe a new incomplete but terminating method for real root finding for
large multivariate polynomials. We take an abstract view of the polynomial as
the set of exponent vectors associated with sign information on the
coefficients. Then we employ linear programming to heuristically find roots.
There is a specialized variant for roots with exclusively positive coordinates,
which is of considerable interest for applications in chemistry and systems
biology. An implementation of our method combining the computer algebra system
Reduce with the linear programming solver Gurobi has been successfully applied
to input data originating from established mathematical models used in these
areas. We have solved several hundred problems with up to more than 800000
monomials in up to 10 variables with degrees up to 12. Our method has failed
due to its incompleteness in less than 8 percent of the cases
A Survey of Satisfiability Modulo Theory
Satisfiability modulo theory (SMT) consists in testing the satisfiability of
first-order formulas over linear integer or real arithmetic, or other theories.
In this survey, we explain the combination of propositional satisfiability and
decision procedures for conjunctions known as DPLL(T), and the alternative
"natural domain" approaches. We also cover quantifiers, Craig interpolants,
polynomial arithmetic, and how SMT solvers are used in automated software
analysis.Comment: Computer Algebra in Scientific Computing, Sep 2016, Bucharest,
Romania. 201
Smaller SDP for SOS Decomposition
A popular numerical method to compute SOS (sum of squares of polynomials)
decompositions for polynomials is to transform the problem into semi-definite
programming (SDP) problems and then solve them by SDP solvers. In this paper,
we focus on reducing the sizes of inputs to SDP solvers to improve the
efficiency and reliability of those SDP based methods. Two types of
polynomials, convex cover polynomials and split polynomials, are defined. A
convex cover polynomial or a split polynomial can be decomposed into several
smaller sub-polynomials such that the original polynomial is SOS if and only if
the sub-polynomials are all SOS. Thus the original SOS problem can be
decomposed equivalently into smaller sub-problems. It is proved that convex
cover polynomials are split polynomials and it is quite possible that sparse
polynomials with many variables are split polynomials, which can be efficiently
detected in practice. Some necessary conditions for polynomials to be SOS are
also given, which can help refute quickly those polynomials which have no SOS
representations so that SDP solvers are not called in this case. All the new
results lead to a new SDP based method to compute SOS decompositions, which
improves this kind of methods by passing smaller inputs to SDP solvers in some
cases. Experiments show that the number of monomials obtained by our program is
often smaller than that by other SDP based software, especially for polynomials
with many variables and high degrees. Numerical results on various tests are
reported to show the performance of our program.Comment: 18 page
- …