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)
x |
a numeric vector of abscissae. When evaluated at |
degree |
maximum degree required. The default is one fewer than
the number of distinct values in |
norm |
a logical indicating whether the polynomials should be normalized. |
A list of class "polylist" of objects of class
"polynomial" of degree 1, 2, ..., degree.
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
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