### Using the Cholesky Decomposition of .

Instead of computing directly, we store the Cholesky decomposition of computed using the QR decomposition. Consider a matrix of column vectors . These columns correspond to the vectors     in Table 13.1. A covariance estimate is obtained by forming the matrix

where is the diagonal matrix formed from data weights , , and . It may be verified that this is the same as computing the elements of as follows:

But note that if you take the QR decomposition , that

Thus, we see that the QR decomposition of is related to the Cholesky factor of . There is no reason to ever compute explicitly. Computing requires twice the number of bits of precision as . A quadratic form can be computed using as follows:

where

This convention is used in the software ( software/gmix_step.m). More precisely, the matrix tmpidx stores where the rows of are    . The QR decomposition of tmpidx is , which is stored as a parameter. The subroutine for computing is lqr_eval.m. This routine inputs , , and . The mixture (13.1) is implemented by subroutine lqr_evp.m.

Baggenstoss 2017-05-19