# F(X) = log(det(X))    if det(X) > 0
# dF(X) = tr(X^-1 dX)
# DF(X) = vec((X^-1)')'
test_that("Matrix Calculus 1", {
    # test data
    X <- tcrossprod(diag(5) + matrix(runif(5^2, -.1, .1), 5))

    num.grad <- num.deriv(function(X) log(det(X)), X)
    ana.grad <- c(solve(X))

    expect_equal(num.grad, ana.grad)
})

# F(mu) = <X - mu, (X - mu) x_{k in [r]} Delta_K^-1>    for Delta_k = Delta_k'
# DF(mu) = -2 vec((X - mu) x_{k in [r]} Delta_K^-1)'
test_that("Matrix Calculus 2", {
    p <- c(2, 3, 5)

    X <- array(rnorm(prod(p)), dim = p)
    mu <- array(rnorm(prod(p)), dim = p)
    Deltas <- Map(function(pj) tcrossprod(diag(pj) + runif(pj^2, -0.1, 0.1)), p)

    ana.grad <- -2 * c(mlm(X - mu, Map(solve, Deltas)))
    num.grad <- num.deriv(function(mu) sum((X - mu) * mlm(X - mu, Map(solve, Deltas))), mu)

    expect_equal(num.grad, ana.grad)
})

# F(Delta_j) = <X - mu, (X - mu) x_{k in [r]} Delta_K^-1>    for Delta_k = Delta_k'
# DF(Delta_j) = -((X - mu) x_{k in [r]} Delta_K^-1)_(j) (X - mu)_(j)' Delta_j^-1
test_that("Matrix Calculus 3", {
    # config
    p <- c(2, 3, 5)

    # generate test data
    X <- array(rnorm(prod(p)), dim = p)
    mu <- array(rnorm(prod(p)), dim = p)
    Deltas <- Map(function(pj) tcrossprod(diag(pj) + runif(pj^2, -0.1, 0.1)), p)

    # check analytic to numeric derivatives
    for (j in seq_along(Deltas)) {
        num.grad <- matrix(num.deriv(function(Delta_j) {
                Deltas[[j]] <- Delta_j
                sum((X - mu) * mlm(X - mu, Map(solve, Deltas)))
            }, Deltas[[j]]), p[j])
        ana.grad <- -mcrossprod(mlm(X - mu, Map(solve, Deltas)), X - mu, j) %*% solve(Deltas[[j]])

        expect_equal(num.grad, ana.grad)
    }
})