Certain choices of x we have 1 and consequently a a1

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Unformatted text preview: der A ∈ Cm×n , determine a condition number corresponding to perturbations of x κ= δx A(x + δ x ) − Ax Ax that is κ= A x Ax δx x (κ = = sup δx Aδ x δx Ax , x J (x) ) f (x) / x which is an exact formula for κ, dependent on both A and x If A happens to be square and non-singular, we can use the fact that x / Ax ≤ A−1 to loosen the bound κ ≤ A A−1 , or κ = α A A−1 , α= x Ax A−1 15 / 21 Condition of matrix-vector multiplication (cont’d) For certain choices of x, we have α = 1, and consequently κ = A A−1 If we use 2-norm, this will occur when x is a multiple of a minimal right singular vector of A If A ∈ Cm×n with m ≥ n has full rank, the above equations hold with A−1 replaced by the pseudo-inverse A† = (AH A)−1 AH ∈ Cn×m Given A, compute A−1 b from input b? Mathematically, identical to the problem just considered except that A is repalced by A−1 16 / 21 Condition of matrix-vector multiplication (cont’d) Theorem Let A ∈ Cm×n be nonsingular and consider the equation Ax = b. The problem of computing b given x, has condition number κ= A x ≤ A A−1 b (1) with respect to perturbation of x. The problem of computing x given b, has condition number κ = A−1 b ≤ A A−1 x (2) with respect to perturbation of b. If we use 2-norm, then the equality hold...
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This document was uploaded on 02/10/2014.

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