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Example 1

Integrate the function sin(x) on the interval [a, b] = [0, &pi]. From calculus, you know that the answer is 2. Continue iterating until ε_step < 1e-5.

Let h = b - a = π. Then

R_{0, 0} = T(h) = ½(sin(0) + sin(π))π = 0

Now, for i = 1, 2, ..., we calculate:

i = 1

R_1,0 = T(π/2) = 1.5707963267948966192

(4 ⋅ 1.5707963267948966192 - 0)/3 = 2.0943951023931954923

i = 2

R_2,0 = T(π/4) = 1.8961188979370399

(4 ⋅ 1.8961188979370399 - 1.5707963267948966)/3 = 2.0045597549844210
(16 ⋅ 2.0045597549844210 - 2.0943951023931955)/15 = 1.9985707318238360

i = 3

R_3,0 = T(π/8) = 1.9742316019455508

(4 ⋅ 1.9742316019455508 - 1.8961188979370399)/3 = 2.0002691699483878
(16 ⋅ 2.0002691699483878 - 2.0045597549844210)/15 = 1.9999831309459856
(64 ⋅ 1.9999831309459856 - 1.9985707318238360)/63 = 2.0000055499796705

i = 4

R_4,0 = T(π/16) = 1.9935703437723393

(4 ⋅ 1.9935703437723393 - 1.9742316019455508)/3 = 2.0000165910479355
(16 ⋅ 2.0000165910479355 - 2.0002691699483878)/15 = 1.9999997524545720
(64 ⋅ 1.9999997524545720 - 1.9999831309459856)/63 = 2.0000000162880417
(256⋅2.0000000162880417 - 2.0000055499796705)/255 = 1.9999999945872902

Finally, |1.9999999945872902 - 2.0000055499796705| ≈ 0.00000556, and thus we may halt and our approximation of the integral is 1.9999999945872902 .

Topic 13.3: Romberg Integration (Examples)

Example 1

i = 1

i = 2

i = 3

i = 4