I'm reading through Fast Simulation of Mass-Spring Systems and I'm trying to understand the paper. To a first read it seems easy enough.
However in section 4 formula (11) doesn't convince me
$$ \frac{1}{2} \sum_{i=1}^s k_i \left\lVert p_{i_1} - p_{i_2} - d_i \right\rVert^2 = \frac{1}{2}x^T L x - x^T Jd $$
Essentially on the lhs each $p_{i_1}, p_{i_2} \in \mathbb{R}^3$ and $d_i \in \mathbb{R}^3$ and $k_i \in \mathbb{R}$ on the rhs $x \in \mathbb{R}^{3m}$ $d \in \mathbb{R}^{3s}$, $L \in \mathbb{R}^{3m \times 3m}$ and $J \in \mathbb{R}^{3m \times 3s}$.
Unless I'm missing something isn't the formula wrong? Indeed if for all $i_1, i_2$ we set $p_{i_1} = p_{i_2} = 0$ we end up with
$$ \frac{1}{2} \sum_{i=1}^s k_i \left\lVert d_i\right\rVert^2 = 0 $$
Which holds iff $d_i = 0$ for $i = 1,\ldots s$. The formula to me should be corrected as
$$ \frac{1}{2} \sum_{i=1}^s k_i \left\lVert p_{i_1} - p_{i_2} - d_i \right\rVert^2 = \frac{1}{2}x^T L x - x^T Jd + \frac{1}{2} d^T A d $$
where
$$ \frac{1}{2} d^T A d = \frac{1}{2} \sum_{i=1}^s k_i \left\lVert d_i\right\rVert^2 $$
So again, isn't the formula in the paper wrong? (which I'd find a bit strange since the paper was published at Siggraph 2013).