chol0: Cholesky decomposition

In ADtools: Automatic Differentiation Toolbox

Description

Cholesky decomposition

Usage

chol0(x)

Arguments

x	A numeric matrix.

Details

chol0 is implemented as (t o chol o t) becauase the Cholesky decomposition in R (chol)

1. returns an upper-triangle matrix;

2. uses only the upper half of the matrix when the matrix is real.

This does not match with the usual math notation of A = LL^T, where L is a lower triangular matrix. (Note that the Cholesky-Banachiewicz algorithm for example only uses the lower triangular part of A to compute L, so one should expect d L / d A to look like something you get by differentiating a lower triangular matrix w.r.t. a lower triangular matrix, i.e. the resulting Jacobian matrix is also lower triangular.)

To convert the R version of chol to the usual math version of chol, we define chol0 <- function(x) { t(chol(t(x))) } The first transpose ensures the output is lower-triangular, and the second ensures the lower-triangular part of the input is used. Now, chol0

1. returns a lower-trianglar matrix

2. uses only the lower half of the matrix when the matrix is real

and this is what we want. (Additional note: finite-differencing with Cholesky is not too accurate due to the many floating point operations involved.)

Note

This function uses only the lower-triangular part of the input and returns a lower-triangular matrix.

ADtools documentation built on Nov. 9, 2020, 5:09 p.m.