Precision of solutions given condition number of matrix (floating point) -

i have question on stuck. if point me in right direction, appreciate it.

i did poorly on last midterm question similar 1 i'd able understand concept in addition towards answer can better on final exam.

thanks in advance.

the standard approach error analysis of linear systems consider given system represents of systems

(a + Δa) * (x + Δx) = b + Δb

where Δa , Δb have entries of relative size μ = 5 * 10^-d,

||Δa|| ∼ μ * ||a|| , ||Δb|| ∼ μ * ||b||.

the idea being solution found represent exact solution of perturbed system perturbations in bounds given.

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by standard manipulations of truncated geometric or neumann series

(a + Δa) * Δx = Δb - Δa * x

and ignoring second order terms,

Δx ≃ a^-1 * Δb - a^-1 * Δa * x = a^-1 * Δb - a^-1 * Δa * a^-1 * b

so

||Δx|| ≃ ||a^-1|| * ||Δb|| + ||a^-1|| * ||Δa|| * ||x|| ≦ μ * (||a^-1|| * ||b|| + κ * ||x||)

||Δx|| ≦ 2 * μ * κ * ||x||

the relative error of x, ||Δx||/||x||, determine number of valid digits in x, or smaller 2 * μ * κ. per assignment, has smaller 5 * 10^-e, or

2 * κ ≦ 10^d-e.

and specific formula κ,

2 * λ * n^α ≦ 10^d-e.