Evaluates the falling factorial basis of a given order, with respect to given design points, at arbitrary query points.
Arguments
- k
Order for the falling factorial basis. Must be >= 0.
- xd
Design points. Must be sorted in increasing order, and have length at least
k+1
.- x
Query points. Must be sorted in increasing order.
- col_idx
Vector of indices, a subset of
1:n
wheren = length(xd)
, that indicates which columns of the constructed matrix should be returned. The default isNULL
, which is taken to mean1:n
.
Details
The falling factorial basis functions of order \(k\), defined with
respect to design points \(x_1 < \ldots < x_n\), are denoted \(h^k_1,
\ldots, h^k_n\). For their precise definition and further references, see
the help file for h_mat()
. The current function produces a matrix of
evaluations of the falling factorial basis at an arbitrary sequence of
query points. For each query point \(x\), this matrix has a corresponding
row with entries:
$$
h^k_j(x), \; j = 1, \ldots, n.
$$
See also
h_mat()
for constructing evaluations of the falling factorial
basis at the design points.
Examples
xd = 1:10 / 10
x = 1:9 / 10 + 0.05
h_mat(2, xd)
#> 10 x 10 sparse Matrix of class "dgCMatrix"
#>
#> [1,] 1 . . . . . . . . .
#> [2,] 1 0.1 . . . . . . . .
#> [3,] 1 0.2 0.01 . . . . . . .
#> [4,] 1 0.3 0.03 0.01 . . . . . .
#> [5,] 1 0.4 0.06 0.03 0.01 . . . . .
#> [6,] 1 0.5 0.10 0.06 0.03 0.01 . . . .
#> [7,] 1 0.6 0.15 0.10 0.06 0.03 0.01 . . .
#> [8,] 1 0.7 0.21 0.15 0.10 0.06 0.03 0.01 . .
#> [9,] 1 0.8 0.28 0.21 0.15 0.10 0.06 0.03 0.01 .
#> [10,] 1 0.9 0.36 0.28 0.21 0.15 0.10 0.06 0.03 0.01
h_eval(2, xd, x)
#> 9 x 10 sparse Matrix of class "dgCMatrix"
#>
#> [1,] 1 0.05 -0.00125 . . . . . . .
#> [2,] 1 0.15 0.00375 . . . . . . .
#> [3,] 1 0.25 0.01875 0.00375 . . . . . .
#> [4,] 1 0.35 0.04375 0.01875 0.00375 . . . . .
#> [5,] 1 0.45 0.07875 0.04375 0.01875 0.00375 . . . .
#> [6,] 1 0.55 0.12375 0.07875 0.04375 0.01875 0.00375 . . .
#> [7,] 1 0.65 0.17875 0.12375 0.07875 0.04375 0.01875 0.00375 . .
#> [8,] 1 0.75 0.24375 0.17875 0.12375 0.07875 0.04375 0.01875 0.00375 .
#> [9,] 1 0.85 0.31875 0.24375 0.17875 0.12375 0.07875 0.04375 0.01875 0.00375