缩放单项式空间

次幂	---	---	---	---	---	---	---	---	---	---	---
$k=0$:						0: $1$
$k=1$:					1:$\bar x$		2:$\bar y$
$k=2$:				3:${\bar x}^2$		4:$\bar x\bar y$		5:${\bar y}^2$
$k=3$:			6:${\bar x}^3$		7:${\bar x}^2\bar y$		8:$\bar x{\bar y}^2$		9:${\bar y}^3$
$k=4$:		10:${\bar x}^4$		11:${\bar x}^3\bar y$		12:${\bar x}^2{\bar y}^2$		13:${\bar x}{\bar y}^3$		14:${\bar y}^4$
$k=5$:	15:${\bar x}^5$		16:${\bar x}^4\bar y$		17:${\bar x}^3{\bar y}^2$		18:${\bar x}^2{\bar y}^3$		19:${\bar x}{\bar y}^4$		20:${\bar y}^5$

令 $K$ 是一个 $R^2$ 上的多边形, 面积为 $|K|$, 尺寸为 $h_K = \sqrt{|K|}$, 重心为 $\boldsymbol x_K = (x_K, y_K)$. 定义: $$ \bar x = \frac{x - x_K}{h_K}, \quad \bar y = \frac{y - y_K}{h_K} $$ 记 $\boldsymbol \alpha = (\alpha_0, \alpha_1)$ 为任一二重非负整数指标, 则 $K$ 上的 **缩放单项式** 可表示为: $$ \boldsymbol m_{\boldsymbol \alpha} = \bar{x}^{\alpha_0} \bar{y}^{\alpha_1} $$ 如上表. 记 $m_k$ 为 $k$ 次多项式组成的向量函数, 按上表中方式排序. $$ \boldsymbol m_k = [m_0, m_1, ..., m_{n_k-1}] $$ 则显然有: $$ \boldsymbol m_k[:-2] = \bar x * \boldsymbol m_{k-1}, \quad \boldsymbol m_k[-1] = \bar y * \boldsymbol m_{k-1}[-1] \tag{1} $$ $K$ 上的 $p$ 次多项式空间的基函数即为: $m_0 \cup m_1 \cup ...\cup m_p$, 对于使用 $(1)$ 式可计算每个基函数. 基函数导数计算公式: $$ \frac{\partial \boldsymbol m_k}{\partial x}[:-2] = [k, k-1, ..., 1]*\boldsymbol m_{k-1}, \quad \frac{\partial \boldsymbol m_k}{\partial x}[-1] = 0 $$ $$ \frac{\partial \boldsymbol m_k}{\partial y}[1:] = [1, 2, ..., k]*\boldsymbol m_{k-1}, \quad \frac{\partial \boldsymbol m_k}{\partial x}[0] = 0 $$ 二阶导计算公式: $$ \frac{\partial^2 \boldsymbol m_k}{\partial x^2}[:-3] = [k*(k-1), (k-1)(k-2), ..., 2*1]*\boldsymbol m_{k-1}, $$ $$ \frac{\partial^2 \boldsymbol m_k}{\partial x^2}[-2:] = 0 $$ $$ \frac{\partial^2 \boldsymbol m_k}{\partial y^2}[2:] = [1*2, 2*3, ..., (k-1)*k]*\boldsymbol m_{k-1}, $$ $$ \frac{\partial^2 \boldsymbol m_k}{\partial y^2}[:2] = 0 $$ $$ \frac{\partial^2 \boldsymbol m_k}{\partial x \partial y}[1:-2] = [1*(k-1), 2*(k-2), ..., (k-1)*1]*\boldsymbol m_{k-1}, $$ $$ \frac{\partial^2 \boldsymbol m_k}{\partial x \partial y}[[0, -1]] = 0 $$