摘要 : We mainly study numerical integration of real valued functions defined on the d-dimensional unit cube with all partial derivatives up to some finite order r >= 1 bounded by one. It is well known that optimal algorithms that use n ... 展开 We mainly study numerical integration of real valued functions defined on the d-dimensional unit cube with all partial derivatives up to some finite order r >= 1 bounded by one. It is well known that optimal algorithms that use n function values achieve the error rate n(-r/d), where the hidden constant depends on r and d. Here we prove explicit error bounds without hidden constants and, in particular, show that the optimal order of the error is min{1, d n(-r/d)}, where now the hidden constant only depends on r, not on d. For n = m(d), this optimal order can be achieved by (tensor) product rules. 收起