We compute the syzygies of \(\mathbb{P}^n\) via numerical linear algebra and high throughput/performance computing. (While our actual computations are focused on the case of \(\mathbb{P}^2\), we describe the approach for general \(n\).) Continuing the notation from Background page \(K_{p,q}(n,b;d)\) is the cohomology of the complex: \[\bigwedge^{p+1}S_{d}\otimes S_{b+(q-1)d}\xrightarrow{\quad \partial_{p+1} \quad}\bigwedge^{p}S_{d}\otimes S_{b+qd}\xrightarrow{\quad \partial_{p} \quad}\bigwedge^{p-1}S_{d}\otimes S_{b+(q+1)d},\] where the differentials are given by \[\partial_{p}(m_1\wedge m_2\wedge \cdots \wedge m_p\otimes f)=\sum_{k=1}^p(-1)^km_1\wedge m_2\wedge \cdots \wedge \hat{m}_{k}\wedge \cdots \wedge m_p\otimes(m_kf).\] Since these differentials respect the \(\mathbb{Z}^{n+1}\)-grading on \(S\), the above complex may be decomposed into its multigraded strands. Thus for any multidegree \(\textbf{a}\), we may compute \(K_{p,q}(n,b;d)_{\textbf{a}}\) via the cohomology of: \[\left(\bigwedge^{p+1}S_{d}\otimes S_{b+(q-1)d}\right)_{\textbf{a}}\xrightarrow{\quad \partial_{p+1,\textbf{a}} \quad}\left(\bigwedge^{p}S_{d}\otimes S_{b+qd}\right)_{\textbf{a}}\xrightarrow{\quad \partial_{p,\textbf{a}} \quad}\left(\bigwedge^{p-1}S_{d}\otimes S_{b+(q+1)d}\right)_{\textbf{a}}.\] It thus suffices to compute the ranks of the linear transformations \(\partial_{p+1,\textbf{a}}\) and \(\partial_{p+1,\textbf{a}}\). With this in mind our computation can roughly be broken down into three steps. These are described in detail in Sections 3, 4, and 5 of our paper arXiv:1711.03513.