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  "body": "\n## Summary\n\nPseudospectral optimal control is a form of [[direct collocation|T58W34D60G]] that uses a single global polynomial for the state on the whole time interval.\nThe nodes are the points of a Gaussian quadrature rule, such as the Legendre–Gauss–Lobatto points, and derivatives are taken with a differentiation matrix.\nFor smooth problems the error decreases faster than any power of the number of nodes, so few nodes give high accuracy.\nThe method is used in aerospace guidance, for example to plan spacecraft manoeuvres.\n\n## Explanations\n\n### L1 human\n\nLow-order collocation draws a trajectory with many short pieces.\nA pseudospectral method draws it with one high-degree polynomial through well-chosen points that cluster near the ends of the interval.\nClustering avoids the wild oscillations that equally spaced points would cause.\nWith three nodes $-1, 0, 1$, the method uses a parabola and integrates the cost with Simpson's rule.\nFor smooth problems, adding nodes improves the accuracy very quickly.\n",
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