Question

Compute I = ??¹ 1/(x³ + 10) dx using Trapezoidal rule with the number of points 3, 5 and 9. Improve the results using Romberg integration. Hint: For 3, 5 and 9 points, we have h = 1/2, 1/4 and 1/8 respectively. Trapezoidal Rule Algorithm. Suppose the function f, the interval [a, b], the length n of the intitial partition, a positive tolerance ? < 1, and the maximum number of iterations M are given. The following algorithm will compute a sequence of approximations to ??¹ f(x)dx by the Trapezoidal rule, until the estimated relative error is smaller than ?, or the maximum number of computed approximations reach M. The final approximation is stored in I. n := 1; I := 0; it := 0; do { it := it + 1; n := n2; I_old := I; h := (b-a)/n; I := 0; for i := 1, 2, K, n-1 ; I := I + f(a + ih); I := h/2[f(a) + 2I + f(b)] } while (it < M and |I - I_old| > ? * |I|); Output: it, I.

          Compute I = ??¹ 1/(x³ + 10) dx using Trapezoidal rule with the number of points 3, 5 and 9. Improve the
results using Romberg integration.
Hint: For 3, 5 and 9 points, we have h = 1/2, 1/4 and 1/8 respectively.
Trapezoidal Rule Algorithm. Suppose the function f, the interval [a, b], the length n of the intitial
partition, a positive tolerance ? < 1, and the maximum number of iterations M are given. The following
algorithm will compute a sequence of approximations to ??¹ f(x)dx by the Trapezoidal rule, until the
estimated relative error is smaller than ?, or the maximum number of computed approximations reach
M. The final approximation is stored in I.
n := 1; I := 0; it := 0;
do {
it := it + 1; n := n*2; I_old := I;
h := (b-a)/n; I := 0;
for i := 1, 2, K, n-1 ;
I := I + f(a + ih);
I := h/2[f(a) + 2*I + f(b)]
} while (it < M and |I - I_old| > ? * |I|);
Output: it, I.

Compute I = ??¹ 1/(x³ + 10) dx using Trapezoidal rule with the number of points 3, 5 and 9. Improve the
results using Romberg integration.
Hint: For 3, 5 and 9 points, we have h = 1/2, 1/4 and 1/8 respectively.
Trapezoidal Rule Algorithm. Suppose the function f, the interval [a, b], the length n of the intitial
partition, a positive tolerance ? < 1, and the maximum number of iterations M are given. The following
algorithm will compute a sequence of approximations to ??¹ f(x)dx by the Trapezoidal rule, until the
estimated relative error is smaller than ?, or the maximum number of computed approximations reach
M. The final approximation is stored in I.
n := 1; I := 0; it := 0;
do
it := it + 1; n := n*2; Iold := I;
h := (b-a)/n; I := 0;
for i := 1, 2, K, n-1 ;
I := I + f(a + ih);
I := h/2[f(a) + 2*I + f(b)]
while (it < M and |I - Iold| > ? * |I|);
Output: it, I.

Added by Phillip S.

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