A large literature specifies conditions under which the information complexity for a sequence of numerical problems defined for dimensions 1 , 2 , … grows at a moderate rate, i.e., the sequence of problems is tractable . Here, we focus on the situation where the space of available information consists of all linear functionals, and the problems are defined as linear operator mappings between Hilbert spaces . We unify the proofs of known tractability results and generalize a number of existing results. These generalizations are expressed as five theorems that provide equivalent conditions for (strong) tractability in terms of sums of functions of the singular values of the solution operators.

Random scrambling of deterministic (t,m,s)-nets and (t,s)-sequences eliminates their inherent bias while retaining their low-discrepancy properties. This article describes an implementation of two types of random scrambling, one proposed by Owen and another proposed by Faure and Tezuka. The four different constructions of digital sequences implemented are those proposed by Sobol', Faure, Niederreiter, and Niederreiter and Xing. Because the random scrambling involves manipulating all digits of each point, the code must be written carefully to minimize the execution time. Computed root mean square discrepancies of the scrambled sequences are compared to known theoretical results. Furthermore, the performances of these sequences on various test problems are discussed.