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  "body": "\n## Summary\n\nA Gauss–Kronrod quadrature formula extends an $n$-point [[Gauss–Legendre rule|RGGXDRCJVE]] by $n + 1$ new nodes into a rule with $2n + 1$ nodes.\nThe old nodes are kept, and the new nodes and all weights are chosen to give the highest possible [[degree of exactness|ZYT1JNT1Y0]].\nThe difference between the two results is a cheap estimate of the error of the integral.\nFor this reason Gauss–Kronrod pairs are the core of many adaptive integration programs.\n\n## Explanations\n\n### L1 human\n\nTo know how good an approximate integral is, one usually computes it twice with two rules and compares.\nTwo unrelated Gauss rules share almost no sample points, so the second computation costs almost as much as starting over.\nThe Kronrod idea reuses every sample of the first rule and only adds a few new ones in between.\nFor example, the popular pair \"G7–K15\" uses a $7$-point Gauss rule and a $15$-point Kronrod rule that needs only $8$ extra function values.\n",
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