Division Algorithm for Polynomial Matrices — polm

For given polynomial matrices $a(z), b(z)$ compute two matrices $c(z), d(z)$ such that $$a(z) = c(z) b(z) + d(z)$$ where the degree of $d(z)$ is smaller than the degree of $b(z)$. The matrix $b(z)$ must be square with a non singular leading coefficient matrix! The matrices must be compatible, i.e. the number of columns of $a(z)$ must equal the number of rows (and columns) of $b(z)$.

Usage

polm_div(a, b)

Arguments

a, b: Two compatible polynomial matrices.

Value

List with two slots

qucontains the polynomial $c(z)$
remcontains the polynomial $d(z)$.

Examples

a = test_polm(dim = c(3,2), degree = 4, random = TRUE)
b = test_polm(dim = c(2,2), degree = 2, random = TRUE)

(out = polm_div(a, b))
#> $qu
#> ( 3 x 2 ) matrix polynomial with degree <= 2 
#>        z^0 [,1]       [,2]    z^1 [,1]        [,2]  z^2 [,1]       [,2]
#> [1,] -1.1407040  1.3689253 -0.05007305 -0.98069292 0.1292603  0.9231022
#> [2,] -0.2694931 -0.2119719  0.59082032 -0.04523667 0.2019229  0.7190641
#> [3,]  4.2526376  1.2255002 -0.72896475  2.62352494 0.2758249 -1.1449855
#> 
#> $rem
#> ( 3 x 2 ) matrix polynomial with degree <= 1 
#>        z^0 [,1]       [,2]   z^1 [,1]      [,2]
#> [1,] -0.5390637  2.6952921  4.4251373  1.432490
#> [2,]  1.3147424 -0.1589452  0.7688883 -2.105304
#> [3,]  3.0560400 -3.3818161 -6.3485225  2.066395
#> 

all.equal(a, out$qu %r% b + out$rem)
#> [1] TRUE