The use of integral information in the solution of a two-point boundary value problem

Abstract. We study the worst-case \(\varepsilon\)-complexity of a two-point boundary value problem \(u^{\prime\prime}(x)=f(x)u(x)\), \(x \in [0,T]\), \(u(0)=c\), \(u^{\prime}(T)=0\), where \(c,T \in \mathbb{R}\) (\(c \neq 0\), \(T \gt 0\)) and \(f\) is a nonnegative function with \(r\) (\(r\geq 0\)) continuous bounded derivatives. We prove an upper bound on the complexity for linear information showing that a speed-up by two orders of magnitude can be obtained compared to standard information. We define an algorithm based on integral information and analyze its error, which provides an upper bound on the \(\varepsilon\)-complexity.