G 1 10 102 and ill conditioned if is large eg 106

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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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