Minimum asymptotic error of algorithms for solving ODE

✍ Scribed by B.Z Kacewicz

• Publisher: Elsevier Science
• Year: 1988
• Tongue: English
• Weight: 806 KB
• Volume: 4
• Category: Article
• ISSN: 0885-064X
• DOI: 10.1016/0885-064x(88)90016-7

✦ Synopsis

We deal with algorithms for solving systems z'(x) = f(x, z(x)), x E [O, cl, z(O) = 7, where f has r continuous bounded derivatives in [O, c] x UG. We consider algorithms whose sole dependence on f is through the values of n linear continuous functionals at J We show that if these functionals are defined by partial derivatives off then, roughly speaking, the error of an algorithm (for a fixed f) cannot converge to zero faster than n-' as n + +^a. This minimal error is achieved by the Taylor algorithm. If arbitrary linear continuous functionals are allowed, then the error cannot converge to zero faster than n-('+') as n + +=. This minimal error is achieved by the Taylor-integral algorithm which uses integrals off. D 1988 Academic Press, Inc.