Multiplication bloards after factorization

[yiorgos]

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:

$\epsilon = \left\| \textbf{A}-\textbf{PQ} \right\| / \left\| \textbf{A} \right\|$

which gives

$\epsilon <1\text{e}-16$

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

$r = \left\| \textbf{AB}-\textbf{PQB} \right\| / \left\| \textbf{AB} \right\|$

now bloats, i.e. I get
$r>0.1.$

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)

 

[yiorgos]

Modification: I mistakenly added also "identity matrix" to previous post. Please ignore this from the list of matrices I have tried.

 

[AlephZero]

It's hard to know exactly what you are doing since you won't tell us all the facts, but I think the basic issue is the difference between "forward" and "backward" error analysis.

If you are trying to solve $Ax = b$ numerically, you can estimate the error two different ways.

Forward error analysis: try to estimate the error in $x$, i.e assume the exact solution is $A(x + e) = b$ and find an estimate for the vector $e$.
Backward error analysis: consider you have the exact solution to the "wrong" equation, i.e. estimate the size of a matrix $e$ such that $(A+e)x = b$.

Backward error analysis (first proposed by Wilkinson in the 1960s) is generally more useful than the more "obvious" forward analysis.