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    <p>Here is a solution based on <code>RcppArmadillo</code>. It is over 100 times faster than the R implementation. First, the c++ implementation, which relies on <a href="http://gallery.rcpp.org/articles/dmvnorm_arma/" rel="noreferrer">this rcpp gallery example</a>.</p> <pre><code>// [[Rcpp::export]] arma::mat dmvnormderiv_arma(arma::mat x, arma::rowvec mean, arma::mat sigma, bool log = false) { // get result for mv normal arma::vec distval = Mahalanobis(x, mean, sigma); double logdet = sum(arma::log(arma::eig_sym(sigma))); double log2pi = std::log(2.0 * M_PI); arma::vec mvnorm = exp(-( (x.n_cols * log2pi + logdet + distval)/2)); // create output matrix with one column for each derivative int n = x.n_rows; arma::mat deriv; deriv.copy_size(x); for (int i=0; i &lt; n; i++) { deriv.row(i) = -1 * mvnorm(i) * trans(solve(sigma, trans(x.row(i) - mean))); } return(deriv); } </code></pre> <p>And two R implementations. One is pure R and one is based on <code>dmvnorm</code> in the package <code>mvtnorm</code>.</p> <pre><code>library('RcppArmadillo') library('mvtnorm') library('rbenchmark') sourceCpp('mvnorm.cpp') mvnormDeriv = function(X, mu=rep(0,ncol(X)), sigma=diag(ncol(X))) { fn = function(x) -1 * c((1/sqrt(det(2*pi*sigma))) * exp(-0.5*t(x-mu)%*%solve(sigma)%*%(x-mu))) * solve(sigma,(x-mu)) out = t(apply(X,1,fn)) return(out) } dmvnormDeriv = function(X, mean, sigma) { if (is.vector(X)) X &lt;- matrix(X, ncol = length(X)) if (missing(mean)) mean &lt;- rep(0, length = ncol(X)) if (missing(sigma)) sigma &lt;- diag(ncol(X)) n = nrow(X) mvnorm = dmvnorm(X, mean = mean, sigma = sigma) deriv = array(NA,c(n,ncol(X))) for (i in 1:n) deriv[i,] = -mvnorm[i] * solve(sigma,(X[i,]-mean)) return(deriv) } </code></pre> <p>Finally some benchmarks:</p> <pre><code>set.seed(123456789) sigma = rWishart(1, 2, diag(2))[,,1] means = rnorm(2) X = rmvnorm(10000, means, sigma) benchmark(dmvnormderiv_arma(X,means,sigma), mvnormDeriv(X,mu=means,sigma=sigma), dmvnormDeriv(X,mean=means,sigma=sigma), order="relative", replications=5)[,1:4] test replications elapsed 1 dmvnormderiv_arma(X, means, sigma) 5 0.016 3 dmvnormDeriv(X, mean = means, sigma = sigma) 5 2.118 2 mvnormDeriv(X, mu = means, sigma = sigma) 5 5.939 relative 1 1.000 3 132.375 2 371.187 </code></pre>
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    1. POfirst derivative of multivariate normal/gaussian with rcpp and RcppArmadillo
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    1. CONicely done. That looks pretty nice, and almost ready. Take a look at the existing `.Rmd` (or, if you prefer, `.cpp`) files there and either send me something by mail, or fork and do the pull request thing common on Github.
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      1. USDirk Eddelbuettel
    2. COInstead of doing sum(arma::log(arma::eig_sym(sigma))), the [log_det()](http://arma.sourceforge.net/docs.html#log_det) might be more efficient
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    3. CODirk, [here](https://gist.github.com/jlegewie/6485612) is a gist with the Rmd file. I will add an issue to the gallery github page so the discussion can continue there.
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