Give the spectral parameters for zero eigenvalues of mc-matrices

Give the spectral parameters for zero eigenvalues of mc-matrices.

Usage

spec_root0(dim, mo, mo.col)

Arguments

dim

dimension of the matrix, a positive integer.

mo

multi-companion order, a positive integer.

mo.col

last non-zero column in the top of the mc-matrix, a non-negative integer.

Details

spec_root0 prepares a structure for the zero roots of an mc-matrix.

Value

a list with the following components:

mo

multi-companion order

ev.type

type of the eigenvalues

co.type

not used currently (:todo:)

n.root

number of non-zero roots

ev.abs

absolute values of roots

ev.arg

arguments of eigenvalues (0 for positive ev)

block.length

lengths of Jordan blocks

co.abs

absolute values of seed parameters

co.arg

arguments of seed parameters (Hz: 0 for positive; 1/2 for negative)

co0

redundant but keep it for now.

Author

Georgi N. Boshnakov

See also

spec_root1, mcSpec

Examples

spec_root0(4,2,3)
#> $mo
#> [1] 2
#> 
#> $ev.type
#> [1] "r"
#> 
#> $co.type
#> [1] "r"
#> 
#> $n.root
#> [1] 0
#> 
#> $ev.abs
#> [1] 0
#> 
#> $ev.arg
#> [1] 0
#> 
#> $block.length
#> [1] 1
#> 
#> $co.abs
#>      [,1]
#> [1,]    0
#> [2,]    1
#> 
#> $co.arg
#>      [,1]
#> [1,]    0
#> [2,]    0
#> 
#> $co0
#>      [,1]
#> [1,]    0
#> [2,]    1
#> 
spec_root0(4,2,2)
#> $mo
#> [1] 2
#> 
#> $ev.type
#> [1] "r" "r"
#> 
#> $co.type
#> [1] "r" "r"
#> 
#> $n.root
#> [1] 0
#> 
#> $ev.abs
#> [1] 0 0
#> 
#> $ev.arg
#> [1] 0 0
#> 
#> $block.length
#> [1] 1 1
#> 
#> $co.abs
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 
#> $co.arg
#>      [,1] [,2]
#> [1,]    0    0
#> [2,]    0    0
#> 
#> $co0
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 
spec_root0(4,2,1)
#> $mo
#> [1] 2
#> 
#> $ev.type
#> [1] "r" "r"
#> 
#> $co.type
#> [1] "r" "r"
#> 
#> $n.root
#> [1] 0
#> 
#> $ev.abs
#> [1] 0 0
#> 
#> $ev.arg
#> [1] 0 0
#> 
#> $block.length
#> [1] 2 1
#> 
#> $co.abs
#>      [,1] [,2] [,3]
#> [1,]    0    0    1
#> [2,]    1    0    0
#> 
#> $co.arg
#>      [,1] [,2] [,3]
#> [1,]    0    0    0
#> [2,]    0    0    0
#> 
#> $co0
#>      [,1] [,2] [,3]
#> [1,]    0    0    1
#> [2,]    1    0    0
#> 
spec_root0(5,2,3)
#> $mo
#> [1] 2
#> 
#> $ev.type
#> [1] "r" "r"
#> 
#> $co.type
#> [1] "r" "r"
#> 
#> $n.root
#> [1] 0
#> 
#> $ev.abs
#> [1] 0 0
#> 
#> $ev.arg
#> [1] 0 0
#> 
#> $block.length
#> [1] 1 1
#> 
#> $co.abs
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 
#> $co.arg
#>      [,1] [,2]
#> [1,]    0    0
#> [2,]    0    0
#> 
#> $co0
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 
spec_root1(4,2,2)
#> $mo
#> [1] 4
#> 
#> $ev.type
#> [1] "r" "r"
#> 
#> $co.type
#> [1] "r" "r"
#> 
#> $n.root
#> [1] 4
#> 
#> $ev.abs
#> [1] 1 1
#> 
#> $ev.arg
#> [1] 0 0
#> 
#> $block.length
#> [1] 2 2
#> 
#> $co.abs
#>      [,1] [,2] [,3] [,4]
#> [1,]   NA    0   NA    0
#> [2,]   NA    0   NA    0
#> [3,]   NA    0   NA    0
#> [4,]   NA    0   NA    0
#> 
#> $co.arg
#>      [,1] [,2] [,3] [,4]
#> [1,]   NA    0   NA    0
#> [2,]   NA    0   NA    0
#> [3,]   NA    0   NA    0
#> [4,]   NA    0   NA    0
#> 
#> $co1
#>      [,1] [,2] [,3] [,4]
#> [1,]   NA    0   NA    0
#> [2,]   NA    0   NA    0
#> [3,]   NA    0   NA    0
#> [4,]   NA    0   NA    0
#> 

spec_root0(6,4,2)
#> $mo
#> [1] 4
#> 
#> $ev.type
#> [1] "r" "r" "r" "r"
#> 
#> $co.type
#> [1] "r" "r" "r" "r"
#> 
#> $n.root
#> [1] 0
#> 
#> $ev.abs
#> [1] 0 0 0 0
#> 
#> $ev.arg
#> [1] 0 0 0 0
#> 
#> $block.length
#> [1] 1 1 1 1
#> 
#> $co.abs
#>      [,1] [,2] [,3] [,4]
#> [1,]    1    0    0    0
#> [2,]    0    1    0    0
#> [3,]    0    0    1    0
#> [4,]    0    0    0    1
#> 
#> $co.arg
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0    0
#> [2,]    0    0    0    0
#> [3,]    0    0    0    0
#> [4,]    0    0    0    0
#> 
#> $co0
#>      [,1] [,2] [,3] [,4]
#> [1,]    1    0    0    0
#> [2,]    0    1    0    0
#> [3,]    0    0    1    0
#> [4,]    0    0    0    1
#> 
spec_root0(6,4,4)
#> $mo
#> [1] 4
#> 
#> $ev.type
#> [1] "r" "r"
#> 
#> $co.type
#> [1] "r" "r"
#> 
#> $n.root
#> [1] 0
#> 
#> $ev.abs
#> [1] 0 0
#> 
#> $ev.arg
#> [1] 0 0
#> 
#> $block.length
#> [1] 1 1
#> 
#> $co.abs
#>      [,1] [,2]
#> [1,]    0    0
#> [2,]    0    0
#> [3,]    1    0
#> [4,]    0    1
#> 
#> $co.arg
#>      [,1] [,2]
#> [1,]    0    0
#> [2,]    0    0
#> [3,]    0    0
#> [4,]    0    0
#> 
#> $co0
#>      [,1] [,2]
#> [1,]    0    0
#> [2,]    0    0
#> [3,]    1    0
#> [4,]    0    1
#> 
spec_root0(10,4,8)
#> $mo
#> [1] 4
#> 
#> $ev.type
#> [1] "r" "r"
#> 
#> $co.type
#> [1] "r" "r"
#> 
#> $n.root
#> [1] 0
#> 
#> $ev.abs
#> [1] 0 0
#> 
#> $ev.arg
#> [1] 0 0
#> 
#> $block.length
#> [1] 1 1
#> 
#> $co.abs
#>      [,1] [,2]
#> [1,]    0    0
#> [2,]    0    0
#> [3,]    1    0
#> [4,]    0    1
#> 
#> $co.arg
#>      [,1] [,2]
#> [1,]    0    0
#> [2,]    0    0
#> [3,]    0    0
#> [4,]    0    0
#> 
#> $co0
#>      [,1] [,2]
#> [1,]    0    0
#> [2,]    0    0
#> [3,]    1    0
#> [4,]    0    1
#>