Another way to find the nodes for the La rule is to observe that it should integrate polynomials of degree five exactly,

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Another Way To Find The Nodes For The La Rule Is To Observe That It Should Integrate Polynomials Of Degree Five Exactly 1 (42.42 KiB) Viewed 42 times

Another way to find the nodes for the La rule is to observe that it should integrate polynomials of degree five exactly, because it has six free parameters. Any such polynomial Qs(t) can be expressed in the form Qs(t) = (t? – 1)J2(t)A (t) + B3(t), where Ju(t) = ť? + at + B and Aj(t) and B3(t) are polynomials of degree one and three, respectively. Here, A1 and B3 depend on Q5, but J2 is the same in all cases. Provided Jy is such that (82 - 1))?(t)e" dt = 0, for r=0,1, the argument used to locate the nodes for Gaussian quadrature in the lecture notes will now work (you are not asked to write out the details of this argument). Prove that there are unique values for a and B such that (*) is satisfied, and verify that the roots of J2 are then the same as the nodes found in part (a).