This characterization might be useful: The partial matrix $M$ can be associated with an undirected graph $G$ ($n$ vertices, with vertex $i$ and $j$ connected if $M_{ij}$ is specified). The matrix has a PSD completion if and only if $G$ is a chordal graph (no minimal cycle of length $\geq 4$. There exist efficient tests for chordality, see Testing Chordal Graphs with CUDA.
Carlo Beenakker
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