By Vandenbussche D., Nemhauser G. L.

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1992) on using simple differencing to estimate derivatives even when functions are available analytically. Although a common practice, derivative estimation by differencing is, in fact, seldom a good idea! 2) where a noise function is added to the smooth signal x and then evaluated. We might assume that has the intuitive characteristics of white noise: mean zero, constant variance, and covariance zero for distinct argument values. 2). For example, the growth data 42 3. Representing functional data as smooth functions clearly have discrete or observational noise.

2. Solve the linear system Φ Φc = Φ y. ˜ c, where the matrix Φ ˜ contains 3. Compute the m inner products Φ the basis functions evaluated at the evaluation arguments. Efﬁcient and stable least squares algorithms can perform these calculations or their equivalents in O[(n + m)K 2 ] operations, and this 46 3. Representing functional data as smooth functions is acceptable provided K is small and ﬁxed relative to n and m. But for large K it is extremely helpful, for both computational economy and numerical stability, if the cross-product matrix Φ Φ has a band structure such that nonzero values appear only in a ﬁxed and limited number of positions on either side of the diagonal.

There are three essential tasks in computing the estimates for general evaluation points: 1. Compute inner products, of which there are K in Φ y and K(K + 1)/2 in Φ Φ. 2. Solve the linear system Φ Φc = Φ y. ˜ c, where the matrix Φ ˜ contains 3. Compute the m inner products Φ the basis functions evaluated at the evaluation arguments. Efﬁcient and stable least squares algorithms can perform these calculations or their equivalents in O[(n + m)K 2 ] operations, and this 46 3. Representing functional data as smooth functions is acceptable provided K is small and ﬁxed relative to n and m.

### A branch-and-cut algorithm for nonconvex quadratic programs with box constraints by Vandenbussche D., Nemhauser G. L.

by Daniel

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