HP71B Integral Questions

[Albert Chan]

(02-09-2020 08:33 PM)Wes Loewer Wrote:  

Quote:For this example, extrapolated trapezoid numbers seems to converge with half the error.

This makes perfect sense.
The error bounds for midpoint and trapezoid methods are: |Em | < k(b-a)^3/(24n^2) and |Et | < k(b-a)^3/(12n^2)
So for a given number of intervals (n) the midpoint method has about half the error of the trapezoid method. But for a given number of evaluation points, the trapezoid will have twice as many intervals, so now trapezoid will have about half the error.

The constant k of |Em| and |Et| are not the same k, but depends on actual curve.
Even if the k's are the same, the formulas only give the upper bound, not equality.

Also, INTEGRAL never return raw trapezoid numbers, but a minimum of 3 extrapolations.
(even more extrapolations on top with more than 7 sample points).

When I try compare errors of 2 methods, error ratios goes all over the place.
The two schemes likely bracketed the true area (but, I also found exceptions)

Example, from Kahan's article, Handheld Calculator Evaluates Integrals, page 30

f(x) = sqrt(x)/(x-1) - 1/log(x), x=0 to 1        → Et/Em ≈ -10
u-transformed f(x), u=-1 to 1                    → Et/Em ≈ -0.4
u-transformed f(x), 2 times                      → Et/Em ≈ -0.06

Equivalent formula, with x=w^2 substitution:

g(w) = 2*w^2/((w+1)*(w-1)) - w/log(w), w=0 to 1  → Et/Em ≈ -0.4
u-transformed g(w), u=-1 to 1                    → Et/Em ≈ -0.05
u-transformed g(w), 2 times                      → Et/Em ≈ -0.02 to +0.02

Note: error ratios only consider final few iterations, initial numbers can go wild.

u-transformation make the curve "bumpy", creating area strips errors of opposite signs, cancelling each other.
If u-transformed more than once, area converge faster (at the cost of more time to evaluate functions).