Create Test Rational Matrix in LMFD Form — test

This simple tool may be used to create a random, $(m,n)$-dimensional, rational matrix in LMFD form $$x(z)=a^{-1}(z) b(z)$$ The degrees of the polynomials $a(z), b(z)$ is denoted with $p$ and $q$ respectively.

Usage

test_lmfd(
  dim = c(1, 1),
  degrees = c(1, 1),
  digits = NULL,
  bpoles = NULL,
  bzeroes = NULL,
  n.trials = 100
)

Arguments

dim: integer vector c(m,n).
degrees: integer vector c(p,q).
digits: integer, if non NULL then the randomly generated numbers are rounded to "digits" number of decimal places.
bpoles: lower bound for the moduli of the poles of the rational matrix (or NULL).
bzeroes: lower bound for the moduli of the zeroes of the rational matrix (or NULL). This parameter is ignored for non-square matrices (m != n).
n.trials: maximum number of trials.

Value

lmfd object, which represents the generated rational matrix $x(z)$ in LMFD form.

Details

We require $m>0$ and $p\geq 0$. The left factor $a(z)$ is normalized as $a(0)=I_m$ where $I_m$ denotes the $(m,m)$-dimensional identity matrix.

The user may prescribe lower bounds for the moduli of the zeroes and/or poles of the rational matrix. In this case the procedure simply generates (up to n.trials) random matrices until a matrix is found which satisfies the constraint. The standard deviation of the normal distribution, which is used to generate the random entries, is decreased in each step. Of course this is a very crude method and it may fail or need a very large number of randomly generated matrices.

Examples

### generate a random (2 x 2) rational matrix in LMFD form with degrees p=1 and q =1
### we require that the matrix has no poles and no zeroes within the unit circle!
x = try(test_lmfd(dim = c(2,2), degrees = c(1,1), digits = 2, bpoles = 1, bzeroes = 1))
if (!inherits(x, 'try-error')) {
   print(x)
   print(abs(poles(x)))
   print(abs(zeroes(x)))
}
#> ( 2 x 2 ) left matrix fraction description a^(-1)(z) b(z) with degrees (p = 1, q = 1)
#> left factor a(z):
#>      z^0 [,1]  [,2] z^1 [,1]  [,2]
#> [1,]        1     0     0.84  1.99
#> [2,]        0     1    -0.34 -0.28
#> right factor b(z):
#>      z^0 [,1]  [,2] z^1 [,1]  [,2]
#> [1,]    -0.98  1.72     0.71 -0.63
#> [2,]    -2.49  0.21     0.51 -0.82
#> [1] 1.505164 1.505164
#> [1] 2.018492 7.741758