# llt

function llt(S:Real[_,_]) -> LLT

Cholesky decomposition of the symmetric positive definite matrix $S$.

• S: The symmetric positive definite matrix $S$.

Returns: an object representing the symmetric positive definite matrix $S$ in its decomposed form.

This differs from chol in that chol returns the lower-triangular Cholesky factor, while this returns the original matrix, but decomposed.

The object acts as the matrix $S$, defines conversion to and assignment from Real[_,_], and is intended as more or less a drop-in replacement for that type, albeit sharing, as usual for objects (i.e. copy-by-reference rather than copy-by-value semantics). That sharing permits, for example, multiple multivariate Gaussian distributions to share the same covariance or precision matrix with common posterior updates performed only once.

Various functions, such as solve, have overloads that make use of LLT objects for more efficient computation.

Attention

To emphasize, the matrix represented is $S$, not $L$, which is to say, code such as the following:

let A <- llt(S);
y <- solve(A, x);

computes the matrix-vector product $y = S^{^-1}x$, not $y = L^{-1}x$, however the Cholesky decomposition will be used to solve this more efficiently than a general matrix solve. The point of an LLT object is to maintain the original matrix in a decomposed form for more efficient computation.

function llt(S:LLT) -> LLT

Cholesky decomposition of the symmetric positive definite matrix $S$ (identity function).

function llt(y:Expression<Real[_,_]>) -> MatrixLLT

Lazy inv.