多项式近似法

Combining the definition of CWT and the derivative property of convolution, we constructed a general method to calculate the approximate derivative of signal through CWT by using the first and second derivative of Gaussian function, Haar, and the first derivative of three-order-Spline function as wavelets. As compared with the other approaches of calculating derivative, which include the numerical differentiation, polynomial filters, Fourier transform, and the recently proposed DWT method, fast calculation and simple mathematical operation were remarkable advantages of CWT method. For the signal corrupted by severe noise (Signal-toNoise Ratio=5), the satisfactory results could also obtained via CWT method through appropriately adiusting the dilations.

在此基础上，(1)结合连续小波变换的特点和卷积的微分性质，提出了使用Gaussian函数的一阶和二阶导数，Haar和三次样条函数的一阶导数作为小波函数的连续小波变换计算信号近似导数的一般性方法，与其他导数计算方法(包括数字微分法，多项式滤波法，Fourier变换法和离散小波变换法)相比，本法简单便捷，计算速度快，对于噪声含量较高的信号(S/N为5)，只要适当调节尺度即可获得比较满意的结果。

Firstly, the function sinx that appears in the simple pendulum equation is replaced with a polynomial of degree 3 using Maclaurin series expansion and Chebyshev polynomial approximation. Subsequently, the resulting equation which is a Duffing type equation is approximately solved with combing Newton's method and the harmonic balance method.

首先，利用Maclaurin展开和Chebyshev多项式加速技术将单摆振动方程中的正弦型恢复力用三次多项式近似代替，得到一个Duffing型方程；然后，将牛顿法与谐波平衡法结合起来建立Duffing方程的解析逼近周期及周期解，从而给出单摆振动的解析逼近解。

The polynomial function bases are used to form the approximately matching function. Moreover, the discrete values of grey image is transformed to polinomial function that are exact fit at the considered pixel locations. As a result, The mathematical operations performed such as derivation, integration, and gradient evaluation also give out the polynomial functions, therefore, the system of undetermined coefficient equations is derived to be a system of linear algebraic equations.

我们把立体匹配过程看作泛函的极小化过程，因此选用变分问题近似解法的Ritz法构造数值算法，以多项式函数为基构作匹配函数的近似解，进而又利用多项式过样值点的拟合将离散的灰度图象变换为多项式函数，因此计算中所需的求导，积分，及梯度运算的结果仍然是多项式函数，最终将待定系数方程组化为线性代数方程组。

Radial basis function coupled with a polynomial basis function was used as a trail function, and a Heaviside function was used as a test function of the weighted residual method.

这种无网格方法采用径向基函数耦合多项式基函数来近似试函数，采用Heaviside函数作为加权残值法中的权函数。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。