\(I - \rho W\) is singular exactly at \(\rho = 1/\omega\) for the
eigenvalues \(\omega\) of W; the interval containing 0 on which it
stays invertible is \((1/\omega_{min}, 1/\omega_{max})\) (Ord 1975;
spatialreg::eigenw), and the Neumann series converges when
\(|\rho| < 1/\)spectral radius. W must be symmetric or a
row-standardised symmetric neighbour matrix \(D^{-1}B\), whose
eigenvalues are those of \(D^{-1/2} B D^{-1/2}\).
Usage
RhoBounds(W, tol = 1e-10)
Arguments
- W
Spatial weights matrix.
- tol
Tolerance for the symmetry checks.
Value
List with lower, upper, spectral_radius,
neumann_radius and eigenvalues (increasing).
References
Ord, K. (1975). Estimation methods for models of spatial
interaction. Journal of the American Statistical Association 70,
120-126.
Examples
RhoBounds(matrix(c(0, 1, 0, .5, 0, .5, 0, 1, 0), 3, byrow = TRUE))[c("lower", "upper")]
#> $lower
#> [1] -1
#>
#> $upper
#> [1] 1
#>
