| OCR Text |
Show 52 rather than the pseudo-distance, is used to select better knots, and in error computations to give the specified error a more intuitive meaning for the user. .. The author's modified smoothing norm (see (2.5), page 41) is used. The rational interpolation suggested by Dierclo< is used to solve for an appropriate value of the parameter p. The modified smoothing spline uses values of p in the interval (0, 1 ), since this range of values is easy to handle in a computational environment. In part, the modified smoothing spline was developed because of difficulties in finding a minimized-jump smoothing approximation with an error approaching that of the least squares approximation. The value of p required to produce an approximation with such an error approaches infinity, in the original formulation of the minimized-jump smoothing (Equation (2.4)). Floating point overflow becomes difficult to avoid when such a value must be computed. A version of the minimized-jump smoothing was implemented using a factor p that varied instead over (0, 1). In comparisons with the modified smoothing spline, the most common values of p that were used were very near 1, which is to be expected, since neither a parametric polynomial nor a straight line, which result from p near zero for each of the smoothing metrics, are likely to to be adequate approximations for most of the data encountered in design. Even though all computations were performed in double precision, the rational interpolation sometimes would produce factors which were outside the interval that the solution was known to lie in. Theoretically, this should not occur, but finite precision makes it possible. It is important for the reader to note that since the true distance estimates were in use, the error is not always monotonically decreasing, so that each new estimate of p produced by rational interpolation may need to be coerced into the interval that brackets the solution. But the estimated factor p for the minimized-jump smoothing repeatedly fell outside of the bracketing interval, even when the error was monotonically *As discussed in section 2.3. |
