Evaluate H basis

Evaluates the falling factorial basis of a given order, with respect to given design points, at arbitrary query points.

Usage

h_eval(k, xd, x, col_idx = NULL)

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 where n = length(xd), that indicates which columns of the constructed matrix should be returned. The default is NULL, which is taken to mean 1:n.

Value

Sparse matrix of dimension length(x) by length(col_idx).

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