Construct Orthogonal Polynomials

Construct the orthogonal polynomials on a given vector, up to a specified degree.

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 required. 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.

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
## [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 
#> 
## 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
#>      [,1] [,2] [,3] [,4]
#> [1,]    1    0    0    0
#> [2,]    0    1    0    0
#> [3,]    0    0    1    0
#> [4,]    0    0    0    1