cubicgrid(s,d=2)
dualmat(x)
gaic(a,x)
in.upper.halfspace(x)
normphase(h,E,Einv=solve(E))
pointcov(x,tol=.Machine$double.eps ^ 0.5)
sphereSurface(d=2)
ellipsoidSurface(a)cubicgrid computes the coordinates of the points of the
Cartesian product set s1 x...x sd where the si's are subsets of reals.
The si's may be provided as a list of vectors or a vector s. In
the latter case, the vector s is replicated d-times in the list
list(s,...,s).
dualmat computes the dual of a non-singular square matrix
x.
gaic computes the incomplete gamma function with parameter
a for a vector or an array of reals x.
in.upper.halfspace is used to test if the vector x is in
the upper half-space. Upper half-space: null vector or last non-zero
coordinate is greater than 0.
normphase normalizes the vector h
with respect to a vector lattice. The lattice is defined by its
generating matrix E. The normalized vector lies inside the
fundamental tile defined by E and differs from h by a
lattice vector. When h is a matrix, the transformation is applied
to each column vector.
pointcov computes the set S-S for a given finite set of
points S. The points are defined as the columns of a matrix
x. The result is given as a list with two components ud
and n. The component ud contains the points of S-S lying
in the upper half-space (S-S is symmetric). The component n
provides multiplicities: n[i] is the multiplicity of
ud[,i]. The parameter tol controls the precision level
in comparisons.
sphereSurface computes the surface area of the unit sphere in the
d-dimensional space. ellipsoidSurface computes the surface area of an ellipsoid in a
space of arbitrary dimension. The argument a is a vector containing
the semi-axis lengths. The algorithm is based on Garry Tee (2005) Surface area and capacity of
ellipsoids in n dimensions. New Zealand Journal of Mathematics,
34(2), 165--198. It involves numerical one-dimensional integration.
cubicgrid(c(1,2,5))
dualmat(diag(1:2))
gaic(1.2,seq(0.5,10,length=10))
in.upper.halfspace(c(-1,0))
in.upper.halfspace(c(1,0))
normphase(c(1.8,1.5),diag(1:2))
pointcov(matrix(c(0,0,1,0,1,1,0,1,1/2,1/2),nrow=2))
sphereSurface(2)
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