# Shifted Legendre Polynomials

### Description

Computes shifted Legendre polynomials.

### Usage

1 |

### Arguments

`p` |
the variable for which to compute the polynomials. Must be |

`k` |
the degree of the polynomial. |

`intercept` |
logical. If |

### 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 `iqr`

.
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)`

.

### 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.

### Note

The estimation algorithm of `iqr`

is optimized for objects of class “slp”,
which means that using `formula.p = ~ slp(p, k)`

instead of
`formula.p = ~ p + I(p^2) + ... + I(p^k)`

will result in a quicker
computation, even with `k = 1`, with equivalent results.
The default for `iqr`

is `formula.p = ~ slp(p, k = 3)`

.

### Author(s)

Paolo Frumento paolo.frumento@ki.se

### 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

1 2 3 4 |