# Precision matrix of the covariance-based rational approximation of stationary Gaussian Matern random fields `rspde.matern.precision` is used for computing the precision matrix of the covariance-based rational SPDE approximation of a stationary Gaussian random fields on $R^d$ with a Matern covariance function [REMOVE_ME]$$ C(h) = \frac{\sigma^2}{2^(\nu-1)\Gamma(\nu)}(\kappa h)^\nuK_\nu(\kappa h)C(h) = (\sigma^2/(2^(\nu-1)\Gamma(\nu))(\kappa h)^\nuK_\nu(\kappa h) [REMOVE_ME_2]$$ ```r rspde.matern.precision( kappa, nu, tau = NULL, sigma = NULL, rspde.order, dim, fem_mesh_matrices, only_fractional = FALSE, return_block_list = FALSE, type_rational_approx = "brasil" ) ``` ## Arguments - `kappa`: Range parameter of the covariance function. - `nu`: Shape parameter of the covariance function. - `tau`: Scale parameter of the covariance function. If sigma is not provided, tau should be provided. - `sigma`: Standard deviation of the covariance function. If tau is not provided, sigma should be provided. - `rspde.order`: The order of the rational approximation - `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. - `only_fractional`: Logical. Should only the fractional-order part of the precision matrix be returned? - `return_block_list`: Logical. For `type = "covariance"`, should the block parts of the precision matrix be returned separately as a list? - `type_rational_approx`: Which type of rational approximation should be used? The current types are "brasil", "chebfun" or "chebfunLB". ## Returns The precision matrix ## Description `rspde.matern.precision` is used for computing the precision matrix of the covariance-based rational SPDE approximation of a stationary Gaussian random fields on $R^d$ with a Matern covariance function $$ C(h) = \frac{\sigma^2}{2^(\nu-1)\Gamma(\nu)}(\kappa h)^\nuK_\nu(\kappa h)C(h) = (\sigma^2/(2^(\nu-1)\Gamma(\nu))(\kappa h)^\nuK_\nu(\kappa h) $$ ## 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 <- 2.6 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, compute_higher_order = TRUE, parameterization = "matern" ) v <- t(rSPDE.A1d(x, 0.5)) c.true <- matern.covariance(abs(x - 0.5), kappa, nu, sigma) Q <- rspde.matern.precision( kappa = kappa, nu = nu, tau = tau, rspde.order = 2, d = 1, fem_mesh_matrices = op_cov$fem_mesh_matrices ) A <- Diagonal(nobs) Abar <- cbind(A, A, A) w <- rbind(v, v, v) c.approx_cov <- (Abar) %*% solve(Q, w) # 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 function