Solve RMFD system for given inputs — solve_RMFD

Compute the outputs $y_t$ of RMFD(p, q) systems of the form $$y_t = d_0 v_t + d_1 v_{t-1} + \cdots + d_q v_{t-q}$$ where $$v_t + c_1 v_{t-1} + \cdots + c_p v_{t-p} = u_t$$ for given inputs $u_t$ which are contained column-wis in the data matrix data_input, see below.

Usage

solve_RMFD_R(polm_c, polm_d, data_input, t0 = 1)

Arguments

polm_c, polm_d: polm objects. Describe jointly RMFD. $c(z)$ is square, has the identity as zero-lag coefficient, and its coefficients will be reversed in the procedure: $(c_p,...,c_1)$. $d(z)$ might be tall, and its zero-lag coefficient matrix is in general free.
data_input: $(n, N)$ matrix with the inputs $(u_1,...,u_N$.
t0: integer, start iteration at t=t0.

Value

The R implementation solve_RMFD_R() returns the matrix y with the computed outputs. $$y_t = d(z) c(z)^{-1} u_t$$

The RcppArmadillo implementation solve_rmfd_cpp() returns NULL but overwrites the input argument. Note that the RcppArmadillo implementation has a different user interface (it is intended for internal use only).

Details

Values $y_t$, $u_t$ for $t\leq 0$ are implicitely set to be zero. However, if we start the iteration with some $t_0>1$ we can enforce non-zero initial values.

The routines are used internally and hence do not check their arguments. We require the number of outputs to be positive $m > 0$, the number of inputs to be non-negative $n \geq 0$, the degree of $c(z)$ %>% to be non-negative $p \geq 0$, the degree of $d(z)$ is unrestricted, i.e. $(q+1) \geq 0$, and for the starting value and the sample size $1 \leq t_0 \leq N$ holds. Note also that the RcppArmadillo implementation overwrites the input argument y. Use this procedure with care!

Note the non standard arguments: The polynomial matrices $c(z)$ and $d(z)$ are saved as "wide" matrices. The order of the coefficients in $c(z)$ is reversed and there is no $c_0$ coefficient (because it is required to be the identity matrix). The order of the coefficients in $d(z)$ is as usual, $d_0$ is available too. The data matrices are organized columnwise (to avoid memory shuffling)!