- #1
yiorgos
- 18
- 0
Let a positive definite matrix A be factorized to P and Q, A=P*Q and let an arbitrary matrix B.
I am calculating the relative error of the factorization through the norm:
[itex]\epsilon = \left\| \textbf{A}-\textbf{PQ} \right\| / \left\| \textbf{A} \right\|[/itex]
which gives
[itex]\epsilon <1\text{e}-16[/itex]
so I assume factorization is correct.
But things go messy when I try to multiply the factorized form of A with B.
In particular, the relative error, r, of the product
[itex]r = \left\| \textbf{AB}-\textbf{PQB} \right\| / \left\| \textbf{AB} \right\|[/itex]
now bloats, i.e. I get
[itex]r>0.1.[/itex]
Note that B is arbitrary, in particular I have tried several different types: random, structured, all-ones matrix, even the identity matrix.
I'm confused. How come factorization is correct and then the multiplication bloats?
Has anything to do with condition number?
(Unfortunately I can't disclose the type of factorization but I can tell that P and Q are not triangular)
I am calculating the relative error of the factorization through the norm:
[itex]\epsilon = \left\| \textbf{A}-\textbf{PQ} \right\| / \left\| \textbf{A} \right\|[/itex]
which gives
[itex]\epsilon <1\text{e}-16[/itex]
so I assume factorization is correct.
But things go messy when I try to multiply the factorized form of A with B.
In particular, the relative error, r, of the product
[itex]r = \left\| \textbf{AB}-\textbf{PQB} \right\| / \left\| \textbf{AB} \right\|[/itex]
now bloats, i.e. I get
[itex]r>0.1.[/itex]
Note that B is arbitrary, in particular I have tried several different types: random, structured, all-ones matrix, even the identity matrix.
I'm confused. How come factorization is correct and then the multiplication bloats?
Has anything to do with condition number?
(Unfortunately I can't disclose the type of factorization but I can tell that P and Q are not triangular)