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Richardson Extrapolation

ME6758, Dr. Ferri. Richardson Extrapolation. Start with some integration rule, N(h), and some general trend of the error as a function of h:. (1). Now, divide h by 2 (double the number of intervals). (2). To eliminate the lowest order error term, take 2*Eq2 – Eq1. Call this N 2 (h).

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Richardson Extrapolation

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  1. ME6758, Dr. Ferri Richardson Extrapolation Start with some integration rule, N(h), and some general trend of the error as a function of h: (1) Now, divide h by 2 (double the number of intervals) (2) To eliminate the lowest order error term, take 2*Eq2 – Eq1 Call this N2(h)

  2. “weighted average” Now, (3) Halve h again (double the number of intervals): (4) Eliminate h2 terms by taking 4*Eq4 – Eq3 Call this N3(h)

  3. This is accurate to order O(h3) Now, The O(h4) approximation is given by: The O(h5) approximation is given by: This is accurate to order O(hj) In general:

  4. Construct a Table O(h) O(h2) O(h3) O(h4) N2(h) N2(h/2) N2(h/4) N3(h) N3(h/2) N4(h) N1(h) N1(h/2) N1(h/4) N1(h/8)

  5. Romberg Integration Start with composite trapezoidal rule with m-intervals, (m+1) points a < m < b h = (b-a)/m xj =a + j*h Perform n trapezoidal integrations using m1 = 1 interval, m2 = 2 intervals, m3 = 4 intervals, … mn = 2n-1 intervals. In each case hk = (b-a)/mk = (b-a)/ 2k-1 Call this Rk,1

  6. h3 h2 h1 R1,1 R2,1 R3,1 h1/2 just R1,1/2

  7. h3 h2 h1 R1,1 R2,1 R3,1 h2/2 just R2,1/2 odd multiples of hk

  8. R1,1 new point R2,1 new point new point R3,1 h4 R4,1 h5 R5,1 R6,1 2 endpoints, 2k-1-1 interior points, 2k-2 additional points Rk,1 odd multiples of hk

  9. Example Converges, but rather slowly…

  10. Subtract top equation from 4 times bottom equation. After algebra, get Re-write as Now, extrapolate again to eliminate O(hk4) terms to obtain an O(hk6) result, etc

  11. Define: Error: O(hk2j) Generalize to: R1,1 R2,1 R3,1 R4,1 Rn,1 R2,2 R3,2 R4,2 Rn,2 R3,3 R4,3 Rn,3 R4,4 Rn,4 Rn,n Obtain from a single composite trapezoidal integration Construct Romberg Table I=Rn,n + O(hn2n) … Obtain by simple “averaging”

  12. Example Error in R6,6 is only 6.61026789e-011 % !!!! Accurate to O(h612) h6 = p/32= 9.817e-002, h612 = 8.017e-013

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