= = 2. One disadvantage of Gaussian quadrature rules is that they cannot be refined as easily as Newton- Cotes rules, be

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2 One Disadvantage Of Gaussian Quadrature Rules Is That They Cannot Be Refined As Easily As Newton Cotes Rules Be 1 (251.32 KiB) Viewed 23 times

Answer for a)
2=6w_1(t_1)^2+2w_2,
2=10w_1(t_1)^4+(10/9)w_2
Answer for b)
K_3(t) = (-7/6)t^3+t
t=0 , (sqrt42)/7, -(sqrt42)/7
= = 2. One disadvantage of Gaussian quadrature rules is that they cannot be refined as easily as Newton- Cotes rules, because the nodes move if the number of subintervals is increased. However, there is a way to perform a refinement which can be used with any Gaussian rule. This problem is concerned with an extension to the two-point Gaussian rule, which we call the GK5 rule. This has five nodes: t2 = -1/V3, t4 = 1/73 and the others chosen optimally, to minimise error. With function values at all five points we can calculate two estimates: one from the two-point Gaussian rule (which has nodes at t = +1/73) and one from the GK5 rule. (a) Write down a system of equations for the weights and the node locations tı, tz and ts. Use symmetry to reduce the number of unknowns. (b) The node locations ti, tz and t5 are roots of K3(t), which is a cubic polynomial such that P2 (t)K3(t)t" dt = 0 for r= 0, 1, 2. Here, P2(t) is the Legendre polynomial of degree two. Find K3(t) and its roots. This is a lot easier than it first appears — think about the symmetry in the nodes and what this means for the form of K3. Note: this works because we can write an arbitrary polynomial of degree seven in the form Qr(t) = Pz(t)K3(t)Az(t) + Ba(t). The method used to locate the Gauss nodes in the lecture notes will now work for the GK5 rule (you are not asked to write out this argument). (c) (i) Use the result of part (b) and the system of equations from (a) to find the exact values of the weights for the GK5 rule. You should find that W3 = 28/45. (ii) Calculate the exact value of the first nonzero coefficient Sp in the error formula, and the leading-order error in the GK5 rule for a single subinterval of width Ax. (iii) Given that the leading-order error for the two-point Gaussian rule on one subinterval is f(4)(x) E; ~ (Ax), 4320 is it possible to accurately predict the relationship between the errors in the two rules? Justify your answer.