Construct Orthogonal Polynomials
Usage
poly.orth(x, degree = length(unique(x)) - 1, norm = TRUE)
Arguments
x
|
A numeric vector of abscissae. When evaluated at x
the polynomials will generate an orthonormal set.
|
degree
|
Maximum degree requried. The default is one fewer
than the number of distinct values in x , which is maximum
possible.
|
norm
|
a logical indicating whether the polynomials should be
normalized.
|
Description
Construct the orthogonal polynomials on a given vector, x
, up
to a specified degree.Value
A list of class "polylist"
of objects of class
"polynomial"
of degree 1, 2, ..., 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