# Precision matrix of stationary Gaussian Matern random fields with integer covariance exponent `rspde.matern.precision.integer.opt` is used for computing the precision matrix of stationary Gaussian random fields on $R^d$ with a Matern covariance function [REMOVE_ME]$$ C(h) = \frac{\sigma^2}{2^(\nu-1)\Gamma(\nu)}(\kappa h)^\nu K_\nu(\kappa h)C(h) =(\sigma^2/(2^(\nu-1)\Gamma(\nu))(\kappa h)^\nu K_\nu(\kappa h) [REMOVE_ME_2]$$, where $\alpha = \nu + d/2$ is a natural number. ```r rspde.matern.precision.integer( kappa, nu, tau = NULL, sigma = NULL, dim, fem_mesh_matrices ) ``` ## Arguments - `kappa`: Range parameter of the covariance function. - `nu`: Shape parameter of the covariance function. - `tau`: Scale parameter of the covariance function. - `sigma`: Standard deviation of the covariance function. If tau is not provided, sigma should be provided. - `dim`: The dimension of the domain - `fem_mesh_matrices`: A list containing the FEM-related matrices. The list should contain elements c0, g1, g2, g3, etc. ## Returns The precision matrix ## Description `rspde.matern.precision.integer.opt` is used for computing the precision matrix of stationary Gaussian random fields on $R^d$ with a Matern covariance function $$ C(h) = \frac{\sigma^2}{2^(\nu-1)\Gamma(\nu)}(\kappa h)^\nu K_\nu(\kappa h)C(h) =(\sigma^2/(2^(\nu-1)\Gamma(\nu))(\kappa h)^\nu K_\nu(\kappa h) $$ , where $\alpha = \nu + d/2$ is a natural number. ## Examples ```r set.seed(123) nobs <- 101 x <- seq(from = 0, to = 1, length.out = nobs) fem <- rSPDE.fem1d(x) kappa <- 40 sigma <- 1 d <- 1 nu <- 0.5 tau <- sqrt(gamma(nu) / (kappa^(2 * nu) * (4 * pi)^(d / 2) * gamma(nu + d / 2))) range <- sqrt(8 * nu) / kappa op_cov <- matern.operators( loc_mesh = x, nu = nu, range = range, sigma = sigma, d = 1, m = 2, parameterization = "matern" ) v <- t(rSPDE.A1d(x, 0.5)) c.true <- matern.covariance(abs(x - 0.5), kappa, nu, sigma) Q <- rspde.matern.precision.integer( kappa = kappa, nu = nu, tau = tau, d = 1, fem_mesh_matrices = op_cov$fem_mesh_matrices ) A <- Diagonal(nobs) c.approx_cov <- A %*% solve(Q, v) # plot the result and compare with the true Matern covariance plot(x, matern.covariance(abs(x - 0.5), kappa, nu, sigma), type = "l", ylab = "C(h)", xlab = "h", main = "Matern covariance and rational approximations" ) lines(x, c.approx_cov, col = 2) ```

rspde.matern.precision.integer function