Obtain the value of the m-th order basis functions (all) at time 't'. Or alternatively, obtain the basis function by calling bf(m)(j) only. Ex: val x = bf(m)(t) retrieves the value of all the basis functions at 't'. val f = bf(m) retrieves all the basis functions.

Evaluate each of the τ.dim-1-many order m B-spline basis functions at t non-recursively, using an efficient dynamic programming approach.

Example

val m = 4                       // order m (degree m-1)
val t = VectorD (1, 2, 3, 4)    // original time points
val N = B_Spline (t, k)         // B_Spline instance
val z = N.abf_ (m)(t)           // vector of evaluations
println ("order k basis functions at t = z")

Implementation Details

Let N(m)(i) denote the i-th order k basis function evaluated at t. Then each N(m)(i) depends on the evaluation of N(m-1)(i) and N(m-1)(i+1). Here is an example, given a set of order m=4 B-spline basis functions and knot vector τ of length n:

N(1)(0), N(1)(1), N(1)(2), N(1)(3), ..., N(1)(n-1)
      |/       |/       |/            |/
N(2)(0), N(2)(1), N(2)(2), ..., N(2)(n-2)
      |/       |/            |/
N(3)(0), N(3)(1), ..., N(3)(n-3)
      |/            |/
N(4)(0), ..., N(4)(n-4)

This algorithm applies the procedure described above using O(n) storage and O(k*n) floating point operations.

Obtain the value of the m-th order 'j'-th basis function at time 't'. Or alternatively, obtain the basis function by calling bf(m)(j) only. Ex: val x = bf(m)(j)(t) retrieves the value of the j-th basis function at 't'. val f = bf(m)(j) retrieves the j-th basis function.

Order 'm' B-Spline basis functions (general recurrence).

Evaluate the 'j+1'-th order m B-spline basis function at t non-recursively, using an efficient dynamic programming approach.

Example

val m = 4                       // order m (degree m-1)
val t = VectorD (1, 2, 3, 4)    // original time points
val N = B_Spline (t, k)         // B_Spline instance
val j = 0                       // spline j in N.range(m)
val z = N.bf (m)(j)(t(2))       // evaluate at t(2)
println ("order k basis function j at t(2) = z")

Implementation Details

Let N(m)(i) denote the i-th order k basis function evaluated at t. Then each N(m)(i) depends on the evaluation of N(m-1)(i) and N(m-1)(i+1). Here is an example, given a set of order m=4 B-spline basis functions and knot vector τ of length n:

N(1)(0), N(1)(1), N(1)(2), N(1)(3)
      |/       |/       |/
N(2)(0), N(2)(1), N(2)(2)
      |/       |/
N(3)(0), N(3)(1)
      |/
N(4)(0)

This algorithm applies the procedure described above using O(k) storage and O(k*k) floating point operations.

Obtain the value of the 1st order 'j'-th basis function at time 't'. Or alternatively, obtain the basis function by calling bf1(j) only. Ex: val x = bf1(j)(t) retrieves the value of the j-th basis function at 't'. val f = bf1(j) retrieves the j-th basis function.

Obtain the value of the 2nd order 'j'-th basis function at time 't'. Or alternatively, obtain the basis function by calling bf2(j) only. Ex: val x = bf2(j)(t) retrieves the value of the j-th basis function at 't'. val f = bf2(j) retrieves the j-th basis function.

Obtain the value of the 3rd order 'j'-th basis function at time 't'. Or alternatively, obtain the basis function by calling bf3(j) only. Ex: val x = bf3(j)(t) retrieves the value of the j-th basis function at 't'. val f = bf3(j) retrieves the j-th basis function.

Obtain the value of the 4th order 'j'-th basis function at time 't'. Or alternatively, obtain the basis function by calling bf4(j) only. Ex: val x = bf4(j)(t) retrieves the value of the j-th basis function at 't'. val f = bf4(j) retrieves the j-th basis function.

Adjusted order 'm' B-Spline basis functions (general recurrence). These are adjusted so that the first "usable" spline is at j = 0. The valid range of usable splines is defined in range.

The number of basis functions for a specified order.

Compute the dot/inner product of 'this' basis function object and basis function 'g'.

Retrieves the cached design matrices

Retrieve the order of the this B_Spline.

Range of "usable" splines when using the bs function. This is needed, since extra splines may be generated by the general B-spline recurrence.

Recompute cached matrices

The number of usable spline basis functions for a specified order, given the configured knot vector.