**Unformatted text preview: **l structure, and this ﬁgure increases to O (m3 ) 9 / 21 Conditioning and condition numbers
Conditioning: pertain to the perturbation behavior of a mathematical
problem
Stability: pertain to the perturbation behavior of an algorithm used to
solve the problem on a computer
We view a problem as a function f : X → Y from a normed vector
space X of data to a normed vector space Y of solutions
f is usually nonlinear (even in linear algebra), but most of the time it
is at least continuous
A well-conditioned problem: all small perturbations of x lead to only
small changes in f (x)
An ill-conditioned problem: some small perturbation of x leads to a
large change in f (x) 10 / 21 Absolute condition number
Let δ x denote a small perturbation of x and write
δ f = f (x + δ x) − f (x)
Absolute condition number:
δf
κ = κ(x) = lim sup
ˆˆ
δ →0 δ x ≤δ δ x
For most problems, it can be interpreted as a supremum over all
inﬁnitesimal perturbations δ x, and can write simply as
δf
κ = sup
ˆ
δx
δx
If f is diﬀerentiable, we can evaluate the condition number by means
of the derivative of f
Let J (x) be the matrix where i , j entry is the partial derivative
∂ fi /∂ xj evaluated at x, known as the Jacobian of f at x
By the deﬁnition, δ f ≈ J (x)δ x with δ x → 0, and the absolute
condition number becomes
κ = J (x)
ˆ
where J (x) represents the norm of J (x) 11 / 21 Relative...

View
Full
Document