In this problem, we want to analyze numerical errors when calculating square roots. Specifically, for given...

 

Transcribed Image Text:

In this problem, we want to analyze numerical errors when calculating square roots. Specifically, for given b, c ER, we consider the function and the associated floating-point approximation y = f(b, c) = √b² - c := ŷ= f(b, c) = f(f(b, c)) = √√(bob) c. a) Show that the absolute forward error (total error) can be estimated via |ŷ y ≤ 0(√emach|b| + Emach √b2 - c) (for a sufficiently small machine precision mach). You can assume that the underlying floating-point system satisfies the IEEE-standard 754, i.e., we have u v = fl(u *v) = (1 + c)(u*v) for all machine numbers u, v, every arithmetic operation * E {+,-,,/,√}, and some & with || ≤ Emach. (In particular, we also have fl(√u) = (1 + ε)√u). To simplify the analysis, you can further assume that b and c are machine numbers. Hint: The estimate √u+v≤ √u + √u can be useful. b) Construct an example and choose b, c such that lŷ-y| > Emach/b]. (E.g., you can use single and double precision in MATLAB or Python to compare and calculate y and ŷ). - - In this problem, we want to analyze numerical errors when calculating square roots. Specifically, for given b, c ER, we consider the function and the associated floating-point approximation y = f(b, c) = √b² - c := ŷ= f(b, c) = f(f(b, c)) = √√(bob) c. a) Show that the absolute forward error (total error) can be estimated via |ŷ y ≤ 0(√emach|b| + Emach √b2 - c) (for a sufficiently small machine precision mach). You can assume that the underlying floating-point system satisfies the IEEE-standard 754, i.e., we have u v = fl(u *v) = (1 + c)(u*v) for all machine numbers u, v, every arithmetic operation * E {+,-,,/,√}, and some & with || ≤ Emach. (In particular, we also have fl(√u) = (1 + ε)√u). To simplify the analysis, you can further assume that b and c are machine numbers. Hint: The estimate √u+v≤ √u + √u can be useful. b) Construct an example and choose b, c such that lŷ-y| > Emach/b]. (E.g., you can use single and double precision in MATLAB or Python to compare and calculate y and ŷ). - -

Answer rating: 100% (QA)

we show that the absetute Sawarders we have 2 1 4 1 41 1 0 1 y fb c c mutyfb... View the full answer