MEMBERS OF SLIPINTINF CLASS. 27 Extra bounds When extra bounds are required, they must be implemented in a class derived from SLipInt or SLipIntLp or SLipIntInf classes. double ExtraUpperBound(int dim, double* x, double * param) double ExtraLowerBound(int dim, double* x, double * param) The derived class calculates these extra bounds and returns the computed values. x is the point at which the bounds are calculated and param contains the Lipschitz constant (its current value, when these methods are called when estimating the best value of the Lipschitz constant). See example in a later section. Note that the member variable U seOtherBounds must be set to 1. Useful member variables Most member variables are declared as public, for easy inheritance, and access to these variables from the derived classes. double MaxLipConst The Lipschitz constant compatible with the data set. double g1,g2 2 After calculating the value of the interpolant, g1,g2 contain the lower and upper bounds respectively. The value of the interpolant is g1+g2 int UseOtherBounds Flag indicating whether the range of f (x) should be restricted by the additional upper and lower bounds. The user should set this variable to 1 and imple- ment the virtual member functions ExtraUpperBound() and ExtraLowerBound() in the derived class. The bounds need not be constants, and the syntaxis is ExtraUpperBound(int dim, double* x, double * param), where x is the point at which the bounds are required, and param is an optional parameter passed to this function (currently the current value of the Lipschitz constant is passed). See section 2.4.6 on additional bounds.
MEMBERS OF SLIPINTINF CLASS. This class is derived from SLipIntBasic, and shares most methods with SLipInt. The difference between the two classes is that SLipIntInf uses l∞- norm and not Euclidean distance in the Lipschitz condition. For low dimen- sion (d < 5) this may bring a computational advantage when smoothing the data, as the number of constraints in the LP problem can be reduced. This class also implements a smoothing method in the simplicial distance, which is useful in conjunction with the evaluation methods of STCInterpolant class. This section only details the methods which are different from those of SLipInt. Refer to the description of SLipInt class for information on the other methods. void SmoothLipschitzSimp(int dim, int N, double* X, double* Y, double* T, double LC); × Computes the vector T of the modified (smoothened) data values, consistent with the specified Lipschitz constant LC. The Lipschitz condition is understood in the simplicial distance (i.e., when d() is simplicial distance in (2.2)). dim is the dimension, N is the size of the data set, the abscissae of the data points are in the array X of size N dim, stored in rows, the data values are supplied in Y of size N . The memory for the array T should be provided in the calling routine. void SmoothLipschitzSimpW(int dim, int N, double* X, double* Y, double* T, double LC, double* W); Variation of the previous procedure, with the vector of weights W reflecting relative accuracy of the data.
