Forward and Backward Bracket

For an lpolm object as input, get_bwd discards all coefficient matrices pertaining to negative powers and returns a lpolm object with min_deg = 0. Similarly, get_fwd discards all coefficient matrices pertaining to non-negative powers, and also returns an lpolm object.

Usage

get_fwd(lpolm_obj)

get_bwd(lpolm_obj)

Arguments

lpolm_obj

Laurent polynomial object lpolm

Value

Laurent polynomial object lpolm without non-negative coefficients or without negative

Details

Obtain the forward or backward part of a Laurent polynomial, i.e. apply \([.]_-\) or \([.]_+\) to $$a(z) = a_{-q} z^{-q} + \cdots + a_{-1} z^{-1} + a_0 + a_1 z^1 + \cdots + a_p z^p$$ and obtain for get_fwd $$[a(z)]_- = a_{-q} z^{-q} + \cdots + a_{-1} z^{-1}$$ or for get_bwd $$[a(z)]_+ = a_0 + a_1 z^1 + \cdots + a_p z^p$$

Examples

(lp = test_lpolm(degree_max = 2, degree_min = -2))
#> ( 1 x 1 ) Laurent polynomial matrix with degree <= 2, and minimal degree >= -2
#>       z^-2 [,1] z^-1 [,1]   z^0 [,1]  z^1 [,1] z^2 [,1]
#> [1,] 0.03531025 0.4941645 -0.6351894 -1.043395 2.096363
get_fwd(lp)
#> ( 1 x 1 ) Laurent polynomial matrix with degree <= -1, and minimal degree >= -2
#>       z^-2 [,1] z^-1 [,1]
#> [1,] 0.03531025 0.4941645
(lp = test_lpolm(degree_max = 2, degree_min = -2))
#> ( 1 x 1 ) Laurent polynomial matrix with degree <= 2, and minimal degree >= -2
#>      z^-2 [,1]  z^-1 [,1] z^0 [,1]  z^1 [,1]  z^2 [,1]
#> [1,] 0.5202974 -0.3023961 1.245488 -1.409315 0.1863452
get_bwd(lp)
#> ( 1 x 1 ) Laurent polynomial matrix with degree <= 2, and minimal degree >= 0
#>      z^0 [,1]  z^1 [,1]  z^2 [,1]
#> [1,] 1.245488 -1.409315 0.1863452

(lp = lpolm(1:3, min_deg = 2))
#> ( 1 x 1 ) Laurent polynomial matrix with degree <= 4, and minimal degree >= 2
#>      z^2 [,1] z^3 [,1] z^4 [,1]
#> [1,]        1        2        3
get_bwd(lp)
#> ( 1 x 1 ) Laurent polynomial matrix with degree <= 4, and minimal degree >= 0
#>      z^0 [,1] z^1 [,1] z^2 [,1] z^3 [,1] z^4 [,1]
#> [1,]        0        0        1        2        3

(lp = lpolm(1:3, min_deg = -1))
#> ( 1 x 1 ) Laurent polynomial matrix with degree <= 1, and minimal degree >= -1
#>      z^-1 [,1] z^0 [,1] z^1 [,1]
#> [1,]         1        2        3
get_bwd(lp)
#> ( 1 x 1 ) Laurent polynomial matrix with degree <= 1, and minimal degree >= 0
#>      z^0 [,1] z^1 [,1]
#> [1,]        2        3

(lp = lpolm(1:3, min_deg = -5))
#> ( 1 x 1 ) Laurent polynomial matrix with degree <= -3, and minimal degree >= -5
#>      z^-5 [,1] z^-4 [,1] z^-3 [,1]
#> [1,]         1         2         3
get_bwd(lp)
#> ( 1 x 1 ) Laurent polynomial matrix with degree <= -1, and minimal degree >= 0