Compute the Jacobian of the fixed-point map G

The function computes the Jacobian matrix of the fixed-point map with respect to the attractivenesses. The function returns NA when applied to spatial interaction models for which the map is not defined.

Usage

sim_fp_jacobian(sim, ...)

Arguments

sim: a spatial interaction model, an object of class sim
...: additional parameters

Value

a p x p matrix or NA

Details

The fixed point map $\mathbf{G}$ is given by

$$G_j=D_j-\kappa_j Z_j,$$

where $D_j$ is the destination flow (destination_flow()), $Z_j$ the attractiveness (attractiveness()) and $\kappa_j$ the conversion factor (sim_conversion()), all attached to destination $j$.

The derivatives of $\mathbf{G}$ w.r.t. $\mathbf{Z}$ can be used to analyse the stability of the solutions, among other applications.

Examples

distances <- french_cities_distances[1:10, 1:10] / 1000 ## convert to km
production <- log(french_cities$population[1:10])
attractiveness <- log(french_cities$area[1:10])
model <- static_blvim(distances, production, 1.5, 1 / 250, attractiveness)
sim_fp_jacobian(model) ## must NA
#> [1] NA