Question

让x是一个间距不均匀且严格增加的数组（例如x = np.array([0, .1, .3, .7, .99])）。我有以下样条函数：

from scipy.interpolate import InterpolatedUnivariateSpline
a_ = InterpolatedUnivariateSpline(x, a)
b_ = InterpolatedUnivariateSpline(x, b)
c_ = InterpolatedUnivariateSpline(x, c)
d_ = InterpolatedUnivariateSpline(x, d)
e_ = InterpolatedUnivariateSpline(x, e)
f_ = InterpolatedUnivariateSpline(x, f)
g_ = InterpolatedUnivariateSpline(x, g)
w_ = InterpolatedUnivariateSpline(x, w)

由长度与a,b,c,d,e,f,g,w相同的numpy数组x构建。

让w是形状为(2,len(x))的数组。我现在正在执行以下循环：

y1 = []
y2 = []
for ii in range(len(x)): 
    xi = x[ii]
    
    A_1 = np.array([[  a_(xi), -a_(xi)], 
                    [       0, -b_(xi)]])
    
    B_1 = np.array([[ -a_(xi),      0], 
                    [  b_(xi),      0]])
    
    C_1 = np.array([[       1,      0], 
                    [  c_(xi),      0]])
    
    D_1 = np.array([[       1,      0], 
                    [ -d_(xi),      1]])
    
    D_2 = np.array([[      -1,      0], 
                    [       0,  e(xi)]])
    
    Amat = A_1 + B_1 @ np.linalg.inv(D_1) @ C_1
    Bmat = B_1 @ np.linalg.inv(D_1) @ D_2 
    Cmat = np.linalg.inv(D_1) @ C_1
    Dmat = np.linalg.inv(D_1) @ D_2

    K1 = np.array([f_(xi), g_(xi)])

    y1 += [-Amat.T@w  - Cmat.T@K1]
    y2 += [ Dmat.T@K1 + Bmat.T@w ]

这有效，但是它非常慢，因为x很大，我需要多次执行此循环。

我的感觉是，我应该能够对所有这些代码进行矢量化处理，并且它将非常快。但是，当我尝试这样做时，我遇到了@矩阵乘法的问题：

MemoryError: Unable to allocate array with shape (4800, 4800, 4800) and data type float64

这是我的尝试，带有产生该错误的代码：

nil = np.zeros(len(x))
A_1 = np.array([[ a_(x), -a_(x)], 
                [   nil, -b_(x)]])

B_1 = np.array([[-a_(x),    nil], 
                [ b_(x),    nil]])

C_1 = np.array([[ 1+nil,    nil], 
                [ c_(x),    nil]])

D_1 = np.array([[ 1+nil,    nil], 
                [-d_(x),  1+nil]])

D_2 = np.array([[ nil-1,    nil], 
                [   nil,  e_(x)]])

Amat = A_1 + B_1 @ np.linalg.pinv(D_1) @ C_1
Bmat = B_1 @ np.linalg.pinv(D_1) @ D_2
Cmat = np.linalg.pinv(D_1) @ C_1
Dmat = np.linalg.pinv(D_1) @ D_2

K1 = np.array([f_(x), g_(x)])

Cmat.T@K1 # produces the error

从根本上讲，我可以将A_1等矩阵的创建移出循环，从而加快了速度。但是，是否可以进一步加快速度？

在NumPy中向量化矩阵乘法

0 个答案: