This function calculates the B-Spline basis penalty. It follows the procedure outlined in the paper by Zheyuan Li, Jiguo Cao, 2022 "General P-Splines for Non-Uniform B-Splines" doi:10.48550/arXiv.2201.06808 . For equidistant knots it coincides with the usual penalty based on the identitiy. For non-equidistant knots it is a weighted penalty with respect to the knot distances. In addition to the above, we added the possibility to calculate periodic penalties which are based on the periodic differencing matrices.

## Arguments

- knots
Vector of knots.

- order
Order of the Basis (degree + 1).

- periodic
Whether the penalties should be periodic or not.

- max_diff
Maximum difference order to calculate.

## Examples

```
if (FALSE) {
# Equidistant knots with order 2
knots <- 1:10
P <- penalty(knots, order = 2)
print(P[[1]]) # First differences
# Non equidistant knots
knots <- c(0, 0, 0, 0, 1, 3, 4, 4, 4, 4)
P <- penalty(knots, order = 4)
print(P[[1]]) # First differences
print(P[[2]]) # Second differences
print(P[[3]]) # Third differences
# Periodic penalty for equidistant knots
oder <- 4
deg <- order - 1
knots <- 1:15
penalty(knots, order = order, periodic = TRUE)[[1]]
penalty(knots, order = order, periodic = TRUE)[[2]]
penalty(knots, order = order, periodic = TRUE)[[3]]
}
```