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,]   68.63017  133.9147  10.11417  19.19290  2.182392  3.369295
#> [2,]  302.39270  590.3264  43.09670  76.68699  4.310962 13.347594
#> [3,] -214.09961 -410.3890 -27.34128 -56.95725 -3.011508 -6.213239
#> 
#> $rem
#> ( 3 x 2 ) matrix polynomial with degree <= 1 
#>        z^0 [,1]       [,2]   z^1 [,1]      [,2]
#> [1,]  -40.74383  -54.18968  -78.15372  240.8611
#> [2,] -182.98575 -241.99021 -348.40940 1067.6433
#> [3,]  127.80225  169.94282  245.16864 -743.3890
#> 

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