703,653 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
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
Real root finding for equivariant semi-algebraic systems
Let be a real closed field. We consider basic semi-algebraic sets defined
by -variate equations/inequalities of symmetric polynomials and an
equivariant family of polynomials, all of them of degree bounded by .
Such a semi-algebraic set is invariant by the action of the symmetric group. We
show that such a set is either empty or it contains a point with at most
distinct coordinates. Combining this geometric result with efficient algorithms
for real root finding (based on the critical point method), one can decide the
emptiness of basic semi-algebraic sets defined by polynomials of degree
in time . This improves the state-of-the-art which is exponential
in . When the variables are quantified and the
coefficients of the input system depend on parameters , one
also demonstrates that the corresponding one-block quantifier elimination
problem can be solved in time
New Structured Matrix Methods for Real and Complex Polynomial Root-finding
We combine the known methods for univariate polynomial root-finding and for
computations in the Frobenius matrix algebra with our novel techniques to
advance numerical solution of a univariate polynomial equation, and in
particular numerical approximation of the real roots of a polynomial. Our
analysis and experiments show efficiency of the resulting algorithms.Comment: 18 page
Novel Approach to Real Polynomial Root-finding and Matrix Eigen-solving
Univariate polynomial root-finding is both classical and important for modern
computing. Frequently one seeks just the real roots of a polynomial with real
coefficients. They can be approximated at a low computational cost if the
polynomial has no nonreal roots, but typically nonreal roots are much more
numerous than the real ones. We dramatically accelerate the known algorithms in
this case by exploiting the correlation between the computations with matrices
and polynomials, extending the techniques of the matrix sign iteration, and
exploiting the structure of the companion matrix of the input polynomial. We
extend some of the proposed techniques to the approximation of the real
eigenvalues of a real nonsymmetric matrix.Comment: 17 pages, added algorithm
Does Stationarity Characterize Real GDP Movements? Results from Non-Linear Unit Root Tests
Using non-linear unit root tests this paper investigates non- stationarity of real GDP per capita for seven OECD countries over the period 1900-2000. Non-linear unit root tests are more powerful than traditional ADF statistics in rejecting the null unit root hypothesis. To this end we adopt a first order Fourier approximation that may capture many features of non-linear adjustment. Empirical results show that, contrary to what the linear ADF statistics suggest, stationarity characterizes six out of the seven countries. This finding stands at variance with other recent studies which conclude that movements in real GDP per capita can be characterized as a non-stationary process.Unit root tests;non-linear model;real GDP
Some Multiple and Simple Real Root Finding Methods
Solving nonlinear equations with root finding is very common in science and engineering models. In particular, one applies it in mathematics, physics, electrical engineering and mechanical engineering. It is a researchable area in numerical analysis. This present work focuses on some iterative methods of higher order for multiple roots. New and existing novel multiple and simple root finding techniques are discussed. Methods independent of a multiplicity m of a root r, which function very well for both simple and multiple roots, are also presented. Error-correction and variatonal technique with some function estimations are used for the constructions. For the analysis of orders of convergence, some basic theorems are applied. Ample test examples are provided (in C++) for test of efficiencies with suitable initial guesses. And convergence of some methods to a root is shown graphically using matlab applications. Keywords:Iterative algorithms, error-correction, variational methods, multiple roots, applications
Is per capita GDP non-linear stationary in SAARC countries?
Using data for SAARC region, we found real GDP per capita is nonlinear stationary implying that shocks to economy by economic policies (external or internal) have permanent effects on real per capita GDP of SAARC countries. This finding reveals that classical growth model works better to boost economic growth in long run.GDP, Non-stationarity, panel unit root tets
- âŠ