LDL' Decomposition of a Symmetric Positive-Definite Matrix

Performs an LDL' factorization of a symmetric positive-definite matrix $X$, such that $$X = L D L^\top,$$ where $L$ is unit lower-triangular (ones on the diagonal) and $D$ is diagonal.

Usage

LDL(x)

Arguments

x: Numeric matrix. Must be symmetric positive-definite.

Value

A list with components:

l_mat_unit: Unit lower-triangular matrix $L$.
l_mat_strict: Strictly lower-triangular part of $L$.
d_mat: Diagonal matrix $D$.
d_vec: Vector of diagonal entries of $D$.
d_uc: Unconstrained vector with $\mathrm{softplus}(d\_uc) = d\_vec$.
x: Original input matrix.
y: Reconstructed matrix $L D L^\top$.
diff: Difference x - y.

Details

This function returns both the unit lower-triangular factor $L$ and the diagonal factor $D$. The strictly lower-triangular part of $L$ is also provided for convenience. The function additionally computes an unconstrained vector d_uc such that softplus(d_uc) = d_vec, using $\mathrm{softplus}^{-1}(y) = \log(\exp(y) - 1)$ for stable back-transformation.

Examples

set.seed(123)
A <- matrix(rnorm(16), 4, 4)
S <- crossprod(A) + diag(1e-6, 4) # SPD
out <- LDL(S)
max(abs(out$diff))
#> [1] 4.440892e-16