% Generated by roxygen2: do not edit by hand % Please edit documentation in R/NICER.R name{NICER} alias{NICER} title{New Incremental Cell Enumeration (in) R} usage{ NICER(A, b, initial = c(0, 0), verb = TRUE, epsbound = 1,

epstol = 1e-07)

} description{ Modified version of the algorithm of Rada and Cerny (2018).

The main modifications include preprocessing as hyperplanes are added
to determine which new cells are created, thereby reducing the number of
calls to the witness function to solve LPs, and treatment of degenerate
configurations as well as those in "general position."  When the hyperplanes
are in general position the number of cells (polytopes) is determined by the
elegant formula of Zazlavsky (1975) \deqn{m = {n \choose d} - n + 1}.  In
degenerate cases, i.e. when hyperplanes are not in general position, the
number of cells is more complicated as considered by Alexanderson and Wetzel (1981).
The function \code{polycount} is provided to check agreement with their results
in an effort to aid in the selection of tolerances for the \code{witness} function.
Current version is intended for use with \eqn{d = 2}, but the algorithm is adaptable to
\eqn{d > 2}.

} details{ @param A is a n by m matrix of hyperplane slope coefficients

@param  b is an n vector of hyperplane intercept coefficients
@param  initial origin for the interior point vectors \code{w}
@param  epsbound is a scalar tolerance controlling how close the witness point

can be to an edge of the polytope

@param  epstol is a scalar tolerance for the LP convergence
@param  verb controls verbosity of Mosek solution

@return A list with components:

\itemize{

item SignVector a n by m matrix of signs determining position of cell relative

to each hyperplane.

item w a d by m matrix of interior points for the m cells }

@references
Alexanderson, G.L and J.E. Wetzel, (1981) Arrangements of planes in space,

Discrete Math, 34, 219–240.

Rada, M. and M. Cerny (2018) A new algorithm for the enumeration of cells

of hyperplane arrangements and a comparison with Avis and Fukada's reverse search, SIAM J. of Discrete Math, 32, 455-473.

Zaslavsky, T. (1975) Facing up to arrangements:  Face-Count Formulas for

Partitions of Space by Hyperplanes, Memoirs of the AMS, Number 154.

@export

}