An Error Analysi S of Gaussian El Imination

ese notes were copied from Gene Golub’s CME 302 Matrix Computations class while David was a student at Stanford. WarningWe will adopt slightly non-standard notation. e quantity ∆ is a matrix, as are any symbols prexed with δ, such as δL or δU . Upper tildes, like L̃ will denote quantities represented on the computer. Our goal is to show that Gaussian Elimination is a backwards stable algorithm. If we show that Gaussian elimination is backwards stable, then we will show that we can compute x̃ such that (A+ ∆)x̃ = b. In words, we compute the solution to a perturbed system where the perturbation is the matrix ∆. If so, then, if Ax = b, we have:1 1 It’s a good exercise to stitch this bound togeter from x̃ + A−1∆x̃ = x. ∥x − x̃∥ ∥x∥ ≤ ρκ(A) 1 − ρκ(A) where ρ = ∥∆∥/∥A∥. us, if we can show that ∥∆∥ is small, we will have a nice bound on the error of the solution.