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pgs (version 0.4-0)

pgs-internal: Internal pgs functions

Description

Internal pgs functions. These functions should not be called by most users.

Usage

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)

Arguments

s
a list of vectors or a vector.
d
an integer, the space dimension. Default: 2.
x
an array, a matrix or a vector.
a
a numeric.
h
a vector or a matrix.
E
a matrix.
Einv
a matrix.
tol
a numeric.

Details

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.

Examples

Run this code
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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