Computer we view a problem as a function f x y from a

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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...
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