5. Two Improvements

1. Smoothing and Fast Convolution

Let us see now two techniques giving a better approximation of beyond . First an approximate value of may be obtained from by a fast exponential technique based on the following formula:

$$\sum _{k=1}^{k=2n}\rho ^{k-1}\, c\left[ k\right] =\sum _{k=1... ...ight] \otimes \sum _{k=1}^{k=n}\rho ^{k-1}\, c\left[ k\right]$$ (9)

Let us note that a direct application of this formula does not improve the computation complexity. On the contrary it increases it. Of course, there are fewer convolutions to compute, but each of them takes into account functions which have a larger number of pieces, made of polynomials with higher degree: the number of integrations is higher and so is computing time. For a computation based on formula (9) to be possible, smoothing of and functions is necessary so as to decrease both the degree and the number of pieces.

An experimental study shows that applying (9) to the functions obtained by smoothing an exactly known yields an important gain. But this method cannot be applied again, since the second smoothing, again needed to lower both degree and number of pieces, ruins the quality of approximation.

2. Again, the Central Limit Theorem

Another technique for approximating may be obtained as a consequence of central limit theorem: functions, when centered and reduced to a unit variance converge to a normal distribution, since a convolution of pdf's leads to the pdf of the sum of r.v.'s. Therefore, an approximation of even indexed 's is obtained by writing that .

It is in fact better to use which does not introduce new linking points and does not increase the complexity of the next computations. Starting from , may be approximated from and so on until from , which allows the computation of . Thereafter, a convolution with allows the computation of the other terms (whose indices are not ).