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**Unformatted text preview: **condition number
Relative condition number:
κ = lim sup δ →0 δ x ≤δ δf
f (x) δx
x Again, assume δ x and δ f are inﬁnitesimal,
κ = sup
δx δf
f (x) δx
x If f is diﬀerentiable, we can express this in terms of Jacobian
κ= J (x)
f (x) / x Relative condition number is more important in numerical analysis
A problem is well-conditioned if κ is small (e.g., 1, 10, 102 ), and
ill-conditioned if κ is large (e.g., 106 , 1016 )
12 / 21 Well-conditioned and ill-conditioned problems
Consider the problem of obtaining the scalar x /2 from x ∈ C. The
Jacobian of the function f : x → x /2 is the derivative J = f = 1/2
κ= J
1/2
=
= 1.
f (x ) / x
(x /2)/x This problem is well-conditioned
Consider x 2 − 2x + 1 = (x − 1)2 , with a double root x = 1. A small
perturbation in the coeﬃcient may lead to large change in the roots
x 2 − 2x + 0.9999 = (x − 0.99)(x − 1.01).
In fact, the roots can change in proportion to the square root of the
change in the coeﬃcients, so in this case the Jacobian is inﬁnite (the
problem is not diﬀerentiable), and κ = ∞ 13 / 21 Computing eigenvalues of a non-symmetric matrix
Computing the eigenvalues of a non-symmetric matrix is often
ill-conditioned
Consider the two matrices
A= 1 1000
0
1 B= 1
1000
0.001
1 and the eigenvalues are {1, 1} and {0, 2} respectively
On the other hand, if a matrix A is symmetric (more generally, if it is
normal), then its eigenvalues are well-conditioned
It can be shown that if λ and λ + δλ are corresponding eigenvalues of
A and A + δ A, then |δλ| ≤ δ A 2 with equality if δ A is a multiple of
the identity
In that case, κ = 1 if perturbations are measured in the 2-norm and
ˆ
the relative condition number is κ = A 2 /|λ|
14 / 21 Condition of matrix-vector multiplication
Consi...

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