Mqrcm (version 1.0)

slp: Shifted Legendre Polynomials

Description

Computes shifted Legendre polynomials.

Usage

slp(p, k = 3, intercept = FALSE)

Arguments

p

the variable for which to compute the polynomials. Must be 0 <= p <= 1.

k

the degree of the polynomial.

intercept

logical. If TRUE, the polynomials include the constant term.

Value

An object of class “slp”, i.e., a matrix with the same number of rows as p, and with k columns named slp1, slp2, … containing the SLP of the corresponding orders. The value of k is reported as attribute.

Details

Shifted Legendre polynomials (SLP) are orthogonal polynomial functions in (0,1) that can be used to build a spline basis, typically within a call to iMqr. The constant term is omitted unless intercept = TRUE: for example, the first two SLP are (2*p - 1, 6*p^2 - 6*p + 1), but slp(p, k = 2) will only return (2*p, 6*p^2 - 6*p).

References

Refaat El Attar (2009), Legendre Polynomials and Functions, CreateSpace, ISBN 978-1-4414-9012-4.

See Also

plf, for piecewise linear functions in the unit interval.

Examples

# NOT RUN {
  p <- seq(0,1,0.1)
  slp(p, k = 1) # = 2*p
  slp(p, k = 1, intercept = TRUE) # = 2*p - 1 (this is the true SLP of order 1)
  slp(p, k = 3) # a linear combination of (p, p^2, p^3), with slp(0,k) = 0
# }