How should I find the Cholesky factorization of the gramian in MATLAB?

I used two methods to find the Cholesky factorization of a gramian. These methods yield different results for certain systems.

The first method is to use command GRAM with option 'of'

rq = gram(SYS, 'of')

The second method is to find the gramian first, then take the Cholesky factorization.

q = gram(SYS, 'o');

rq = chol (q);

Answer 1

[MathWorks Support Team]

The command GRAM calculates the Cholesky factorization of the gramian directly from a Cholesky Lyapunov solver. If there is no specified factorization, GRAM explicitly forms the product Wo = R' * R. In other words,

q = gram(SYS, 'o');

is equivalent to:

r = gram(SYS, 'of');

q = r' * r;

To calculate the Cholesky factorization of the gramian,

rq = gram(SYS, 'of');

should be used. Using:

q = gram(SYS, 'o');

rq = chol(q);

may cause round-off errors and scaling issues. This is so because it performs the product q = r' * r first in the call to GRAM and then refactorizes the product q using CHOL.