linearity | R Documentation |
Given V-representation (convex hull of points and directions) or H-representation (intersection of half spaces) of convex polyhedron find non-linearity generators that can be made linearity without changing polyhedron
linearity(input, representation = c("H", "V"))
input |
either H-representation or V-representation of convex polyhedron (see details). |
representation |
if |
Interface to the function dd_ImpliedLinearityRows
of the
cddlib
library,
see cddlibman.pdf
in the doc
directory of this package,
especially Sections 1 and 2 and page 9.
See also scdd
for a description of the way this package
codes H-representations and V-representations as R matrices.
A row of a matrix that is an H-representation or V-representation is a linearity row if the first element of that row is 1. The row is an implied linearity row if the first element of that row is 0 but if it were 1 the convex polyhedron described would be unchanged.
The interpretation is as follows. For an H-representation, the linearity (given plus implied) determines the affine hull of the polyhedron (the smallest translate of a subspace containing it). For a V-representation, the linearity (given plus implied) determines the smallest affine set (translate of a subspace) contained in the polyhedron.
a numeric vector, the indices of the implied linearity rows. (Note: rows that are linearity rows in the input matrix are not contained in this vector.)
The input representation may
have type "character"
in which case its elements are interpreted
as unlimited precision rational numbers. They consist of an optional
minus sign, a string of digits of any length (the numerator),
a slash, and another string of digits of any length (the denominator).
The denominator must be positive. If the denominator is one, the
slash and the denominator may be omitted. This package
provides several functions (see ConvertGMP and ArithmeticGMP)
for conversion back and forth between R floating point numbers and rationals
and for arithmetic on GMP rationals.
If you want correct answers, use rational arithmetic. If you do not,
this function may (1) produce approximately correct answers, (2) fail with
an error, (3) give answers that are nowhere near correct with no error or
warning, or (4) crash R losing all work done to that point. In large
simulations (1) is most frequent, (2) occurs roughly one time in a thousand,
(3) occurs roughly one time in ten thousand, and (4) has only occurred once
and only with the redundant
function. So the R floating point
arithmetic version does mostly work, but you cannot trust its results unless
you can check them independently.
ArithmeticGMP
, ConvertGMP
,
validcdd
, makeH
### calculate affine hull
### determined by given + implied linearity rows
qux <- rbind(c(0, 2, 0, 0, 1),
c(0, 3, 1, 0, 0),
c(0, 4, 0, 1, 0),
c(0, -7, -1, -1, 0))
out <- linearity(qux, representation = "H")
print(out)
qux[out, 1] <- 1
redundant(qux, representation = "H")$output
### calculate minimal nonempty face of polyhedral convex cone
### determined by given + implied linearity rows
qux <- rbind(c(0, 0, 0, 0, 1),
c(0, 0, 1, 0, 0),
c(0, 0, 0, 1, 0),
c(0, 0, -1, -1, 0))
out <- linearity(qux, representation = "V")
print(out)
redundant(qux, representation = "V")$output
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