**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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