This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: tional error oF ± M . In N operations, roundo² errors may accumulate ran-domly (error ∼ √ N± M ), systematically (error ∼ N± M ), or catastrophically (1 / ( b-a ) may → ± InF). (2) Truncation or discretization error: An infnite series has to be approxi-mated by a fnite number oF terms on the computer. ±or example, a contin-uous real interval [ a, b ] is approximated by a fnite set oF gridpoints x = a , x 1 = x + Δ x , x 2 = x + 2 Δ x , . . . , x n = x + n Δ x = b ; or, as another example, e x ≈ N ± n =0 x n n ! , | x | < 1 (3) Numerical instability: roundo² error introduced early in the computation can be exponentially amplifed and swamp the true solution. ±or example, the Forward Euler method For the heat equation u t = u xx is stable only iF Δ t ≤ Δ x 2 / 2. ±orward Euler computed solutions to the heat equation with 20Δ x . 2...
View Full Document
- Fall '11
- Numerical Analysis, IEEE Floating point, roundoff error, additional fractional error