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# This library is free software; you can redistribute it and/or
# modify it under the terms of the GNU Library General Public
# License as published by the Free Software Foundation; either
# version 2 of the License, or (at your option) any later version.
#
# This library is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Library General Public License for more details.
#
# You should have received a copy of the GNU Library General
# Public License along with this library; if not, write to the
# Free Foundation, Inc., 59 Temple Place, Suite 330, Boston,
# MA 02111-1307 USA
################################################################################
# FUNCTION: DESCRIPTION:
# pascal Creates a Pascal matrix
################################################################################
pascal <-
function(n)
{
# A function implemented by Diethelm Wuertz
# Description:
# Creates a Pascal matrix
# Arguments:
# n - the dimension of the square matrix
# Details:
# http://mathworld.wolfram.com/PascalMatrix.html
# Pascal matrices are symmetric and positive definite.
# The determinant of a Pascal matrix is 1.
# The inverse of a Pascal matrix has integer entries.
# If lambda is an eigenvalue of a Pascal matrix,
# then 1/lambda is also an eigenvalue of the matrix.
# The Cholesky factor of a Pascal matrix consists of
# the elements of Pascal's triangle
# FUNCTION:
# Pascal:
N = n-1
n.over.r = function(n, r) {
prod(1:n) / (prod(1:(n-r)) * prod(1:r) ) }
X = rep(1, N)
for ( i in 1:N )
for ( j in 1:N )
X = c(X, n.over.r(i+j, j))
X = cbind(rep(1, N+1), matrix(X, byrow = TRUE, ncol = N))
# Return Value:
X
}
################################################################################
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