pdf往下翻突然看见一个克里斯蒂娜
感觉就像人群当中突然钻出来一个光头!
$\sum\limits_{l<=i<j<=r} {(x_i*y_j-x_j*y_i)}^2$
带修改,
$\sum\limits_{l<=i<j<=r} {x_i}^2*{y_j}^2+\sum\limits_{l<=j<i<=r}{x_i}^2*{y_j}^2-\sum\limits_{l<=i<j<=r}2*x_i*x_j*y_i*y_j$
$\sum\limits_{l<=i,j<=r,[i!=j]} {x_i}^2*{y_j}^2-\sum\limits_{l<=i<j<=r} 2*x_i*x_j*y_i*x_j$
$\sum\limits_{l<=i,j<=r,[i可以=j]} {x_i}^2*{y_j}^2-\sum\limits_{l<=i<=r}x_i*y_i-\sum\limits_{l<=i<j<=r} 2*x_i*x_j*y_i*y_j$
$(\sum\limits_{l<=i<=r}{x_i}^2) (\sum\limits_{l<=i<=r} {y_i}^2) -\sum\limits_{l<=i<=r} {x_i}^2*{y_i}^2-2*\sum\limits_{l<=i<j<=r} x_i*x_j*y_i*y_j$
这里$-\sum\limits_{l<=i<=r} {x_i}^2*{y_i}^2-2*\sum\limits_{l<=i<j<=r} x_i*x_j*y_i*y_j=-(\sum\limits_{l<=i<=r}x_i*y_i)^2$
推导过程:
$2*\sum\limits_{l<=i<j<=r} x_i*x_j*y_i*y_j$用上面的套路拆开
$\sum\limits_{l<=i,j<=r[i!=j]} x_i*x_j*y_i*y_j$
即$\sum\limits_{l<=i,j<=r[i可以=j]} x_i*x_j*y_i*y_j-\sum\limits_{l<=i<=r} {x_i}^2*{y_i}^2$
后面这一部分和前面消掉了
前面这部分即$\sum\limits_{l<=i<=r} x_i*y_j*\sum\limits_{l<=i<=r}x_i*y_j$即$(\sum\limits_{l<=i<=r}x_i*y_i)^2$
$(\sum\limits_{l<=i<=r}{x_i}^2)(\sum\limits_{l<=i<=r}{y_i}^2)-(\sum\limits_{l<=i<=r}x_i*y_i)^2$