Evaluate a Basis Penalty Matrix
A basis roughness penalty matrix is the matrix containing the possible inner products of pairs of basis functions. These inner products are typically defined in terms of the value of a derivative or of a linear differential operator applied to the basis function. The basis penalty matrix plays an important role in the computation of functions whose roughness is controlled by a roughness penalty.
eval.penalty(basisobj, Lfdobj=int2Lfd(0), rng=rangeval)
a basis object.
either a nonnegative integer defining an order of a derivative or a linear differential operator.
a vector of length 2 defining a restricted range. Optionally, the inner products can be computed over a range of argument values that lies within the interval covered by the basis function definition.
The inner product can be computed exactly for many types of bases if $m$ is an integer. These include B-spline, fourier, exponential, monomial, polynomial and power bases. In other cases, and for noninteger operators, the inner products are computed by an iterative numerical integration method called Richard extrapolation using the trapezoidal rule.
If the penalty matrix must be evaluated repeatedly, computation can be greatly speeded up by avoiding the use of this function, and instead using quadrature points and weights defined by Simpson's rule.
a square symmetric matrix whose order is equal
to the number of basis functions defined by
the basis function object
Lfdobj is $m$ or a linear
differential operator of order $m$, the rank
of the matrix should be at least approximately equal to
its order minus $m$.