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(Solved): PLEASE ONLY SOLVE PART 2.(a)(iv). I do NOT want others answered,
you will be reported and I will als ...
PLEASE ONLY SOLVE PART 2.(a)(iv). I do NOT want others answered,
you will be reported and I will also downvote if other answers are
copied as they are for part (a)(i).
2. The \( C_{n} \) quadrature rule for the interval \( [-1,1] \) uses the points at which \( T_{n-1}(t)=\pm 1 \) as its nodes (here \( T_{n-1} \) is the Chebyshev polynomial of degree \( n-1 \) ). The \( C_{3} \) rule is just Simpson's rule because \( T_{2}(t)=2 t^{2}-1 \) (a) (i) Find the nodes and weights for the \( C_{5} \) quadrature rule. (ii) Determine the first nonzero coefficient \( S_{j} \) for the \( C_{5} \) rule. (iii) If the \( C_{5} \) rule and the five-point Newton-Cotes rule are applied on the same number of subintervals, what approximate relationship do you expect the two errors to satisfy? (iv) Suppose that the \( C_{5} \) rule has been applied on \( N \) subintervals, and that all of the function evaluations have been stored. How many new function evaluations are required to apply the \( C_{9} \) rule on the same set of subintervals? Justify your answer.