Talk:Spring system

Known spring lengths: BWL correct evaluation

The initial values of the matrices,

$\mathbf {L} ={\begin{bmatrix}1\\2\end{bmatrix}}$, $B={\begin{bmatrix}1&0\\-1&1\\0&-1\end{bmatrix}}$ and $W={\begin{bmatrix}1&0\\0&1\end{bmatrix}}$,

as currently set for the simple case of three nodes (vertices) evaluate to $BW\mathbf {L} ={\begin{bmatrix}1\\1\\-2\end{bmatrix}}$ rather than $BW\mathbf {L} ={\begin{bmatrix}-1\\-1\\2\end{bmatrix}}$. This in turn leads to different solution values for $x_{1},x_{2}$ and $x_{3}$.

A further investigation reveals that the problem appears to be not with $BW\mathbf {L} ={\begin{bmatrix}-1\\-1\\2\end{bmatrix}}$, but with incorrectly specified oriented incidence matrix, B. The source of this problem is in page Laplacian matrix, subsection 'Symmetric Laplacian via the incidence matrix'. The definition of the incidence matrix, B in that subsection is incorrect, which has resulted in an error, propagated through the 'Spring system' page.

In particular, redefining correctly the incidence matrix here as

$B={\begin{bmatrix}-1&0\\1&-1\\0&1\end{bmatrix}},$

allows the remaining part of the evaluations to be numerically correct.

If no one responds by tomorrow 10 Dec '24, it is my intention to edit the page, accordingly.

Gozo032 (talk) 11:08, 9 December 2024 (UTC)