Block Hankel matrix

Construct a block Hankel matrix with (block) entries from a 3-dimensional array R. The \((i,j)\)-th block of the Hankel matrix is R[,,i+j-1].

Usage

bhankel(R, d = NULL)

Arguments

R

3-dimensional array, matrix or vector. A vector of length \(k\) is coerced to a \((1,1,k)\)-dimensional array and a \((p,q)\) matrix is treated as an array of dimension \((p,q,1)\).

d

determines the number of block rows and columns. Suppose that R is an array of size \((p,q,k)\). If \(d=(m,n)\) then bhankel returns a block Hankel matrix with \(m\) block rows and \(n\) block columns. If \(d=m\) then a Hankel matrix with \(m\) block rows and \(n=\max(k+1-m,1)\) block columns is returned. In the default case d = NULL the number of block rows is \(m=(k+1)/2\) for odd \(k\) and \(m=(k/2+1)\) else and the number of block columns is set to \(n=\max(k+1-m,1)\). In the case \((m+n-1)>k\) the array R is padded with zeros.

Value

Matrix of size \((pm\times qn)\).

Examples

bhankel(1:6)
#>      [,1] [,2] [,3]
#> [1,]    1    2    3
#> [2,]    2    3    4
#> [3,]    3    4    5
#> [4,]    4    5    6
bhankel(1:6, d = 3)
#>      [,1] [,2] [,3] [,4]
#> [1,]    1    2    3    4
#> [2,]    2    3    4    5
#> [3,]    3    4    5    6
bhankel(letters[1:6], d = c(3,7))  # note the "zero padding"
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] "a"  "b"  "c"  "d"  "e"  "f"  "0" 
#> [2,] "b"  "c"  "d"  "e"  "f"  "0"  "0" 
#> [3,] "c"  "d"  "e"  "f"  "0"  "0"  "0" 
bhankel(test_array(dim = c(2,2,6)))
#>      [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,]  111  121  112  122  113  123
#> [2,]  211  221  212  222  213  223
#> [3,]  112  122  113  123  114  124
#> [4,]  212  222  213  223  214  224
#> [5,]  113  123  114  124  115  125
#> [6,]  213  223  214  224  215  225
#> [7,]  114  124  115  125  116  126
#> [8,]  214  224  215  225  216  226
bhankel(test_array(dim = c(3,2,6)), d = 3)
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
#>  [1,]  111  121  112  122  113  123  114  124
#>  [2,]  211  221  212  222  213  223  214  224
#>  [3,]  311  321  312  322  313  323  314  324
#>  [4,]  112  122  113  123  114  124  115  125
#>  [5,]  212  222  213  223  214  224  215  225
#>  [6,]  312  322  313  323  314  324  315  325
#>  [7,]  113  123  114  124  115  125  116  126
#>  [8,]  213  223  214  224  215  225  216  226
#>  [9,]  313  323  314  324  315  325  316  326
bhankel(test_array(dim = c(2,2,6)), d = c(3,7))
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
#> [1,]  111  121  112  122  113  123  114  124  115   125   116   126     0     0
#> [2,]  211  221  212  222  213  223  214  224  215   225   216   226     0     0
#> [3,]  112  122  113  123  114  124  115  125  116   126     0     0     0     0
#> [4,]  212  222  213  223  214  224  215  225  216   226     0     0     0     0
#> [5,]  113  123  114  124  115  125  116  126    0     0     0     0     0     0
#> [6,]  213  223  214  224  215  225  216  226    0     0     0     0     0     0
bhankel(test_array(dim = c(1,2,6)), d = c(1,2))
#>      [,1] [,2] [,3] [,4]
#> [1,]  111  121  112  122
bhankel(test_array(dim = c(2,3)), d = c(2,2))
#>      [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,]   11   12   13    0    0    0
#> [2,]   21   22   23    0    0    0
#> [3,]    0    0    0    0    0    0
#> [4,]    0    0    0    0    0    0
bhankel(test_array(dim = c(2,2,6)), d = c(3,0))
#>     
#> [1,]
#> [2,]
#> [3,]
#> [4,]
#> [5,]
#> [6,]
bhankel(test_array(dim = c(2,2,0)), d = c(2,2))
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0    0
#> [2,]    0    0    0    0
#> [3,]    0    0    0    0
#> [4,]    0    0    0    0

if (FALSE) {
# the following examples throw an error
bhankel(1:5, d = c(-1,1))
bhankel(test_array(dim = c(2,3,2,1)))
}