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\name{poly.orth}
\alias{poly.orth}
\title{Construct Orthogonal Polynomials}
\description{
Construct the orthogonal polynomials on a given vector, up to a
specified degree.
}
\usage{
poly.orth(x, degree = length(unique(x)) - 1, norm = TRUE)
}
\arguments{
\item{x}{a numeric vector of abscissae. When evaluated at \code{x}
the polynomials will generate an orthonormal set.}
\item{degree}{maximum degree required. The default is one fewer than
the number of distinct values in \code{x}, which is maximum
possible.}
\item{norm}{a logical indicating whether the polynomials should be
normalized.}
}
\value{
A list of class \code{"polylist"} of objects of class
\code{"polynomial"} of degree 1, 2, \dots, \code{degree}.
}
\examples{
x <- rep(1:4, 1:4) # x with repetitions for weighting
x
## [1] 1 2 2 3 3 3 4 4 4 4
polx <- poly.orth(x, 3) # calculate orthogonal polynomials
polx
## List of polynomials:
## [[1]]
## 0.3162278
##
## [[2]]
## -0.9486833 + 0.3162278*x
##
## [[3]]
## 2.139203 - 1.863177*x + 0.3450328*x^2
##
## [[4]]
## -5.831564 + 8.80369*x - 3.803194*x^2 + 0.4930066*x^3
v <- sapply(polx, predict, x) # orthonormal basis
round(crossprod(v), 10) # check orthonormality
}
\keyword{symbolmath}
|
