[7] gave a 20/3 factor

approximation algorithm for UkM, which was further improved to 3.25 factor by Charikar and Li in [8].

According to the

approximation methods in [5], using backward-forward systems of Swift-Hohenberg equation and u = [P.sub.c]u + [P.sub.s]u = [u.sub.c] + [u.sub.s], we assume the following reduced system:

In the second, we recall most of the preliminary notions and the necessary definitions, and we prove the third

approximation in the general case.

Firstly, consider the Euler-Maruyama

approximation scheme.

We also discuss weighted

approximation properties of these new operators (3) and (69) and compare with the ones (2) by graphics and the absolute error bound of numerical analysis; we will show that the new ones (3) are better than (2) when approximating to functions f.

The aim of this paper is to introduce an abstract framework of certain

approximation processes using a cosine operator functions concept.

The mean-value first-order saddlepoint

approximation (MVFOSA) [4] is an alternative to FOSA.

Different from one-piecewise linear

approximation used in [6], a three-piecewise linear

approximation is presented to compute the initial seed in this paper.

2, we plot contours of the relative error in the first-order

approximation multiplied by [10.sup.6] (i.e., parts per million).

Under these conditions, the

approximation (2) is named as generalized asymptotic expansion.

Note that to apply Ito formula to Black-Scholes function, because the derivatives of this function are not bounded, we have to use an

approximation to the identity and the dominated convergence theorem as it is done, for example, in [3].

The

approximation methods can be classified into two categories: the Nystrom method [11,12] and random Fourier features [13,14].