(35S) 2nd order Derivative (at a point)
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11-11-2020, 12:35 AM
Post: #4
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RE: (35S) 2nd order Derivative (at a point)
(11-10-2020 11:23 PM)trojdor Wrote: What would you suggest as a good example? We wanted a test function with errors dependent on size of h. For polynomials, use quartic or higher degree. XCas> f(x) := x*(x^3 + 5*x^2 - 21*x) XCas> expand(f(x+4)) → x^4+21*x^3+135*x^2+328*x+240 f(4) = 240 f'(4) = 328*1! = 328 f''(4) = 135*2! = 270 // Estimate f'(4), with different h's XCas> (f(x+h) - f(x)) / h | x=4, h=1e-3 → 328.135021001 XCas> (f(x+h) - f(x)) / h | x=4, h=1e-6 → 328.000134999 XCas> (f(x+h) - f(x-h)) / (2h) | x=4, h=1e-3 → 328.000021 XCas> (f(x+h) - f(x-h)) / (2h) | x=4, h=1e-6 → 328.00000001 // Estimate f''(4), with different h's XCas> t1 := (f(x+h) - 2*f(x) + f(x-h))/(h*h) | x=4, h=1e-1 → 270.02 XCas> t2 := (f(x+h) - 2*f(x) + f(x-h))/(h*h) | x=4, h=1e-2 → 270.0002 XCas> t3 := (f(x+h) - 2*f(x) + f(x-h))/(h*h) | x=4, h=1e-3 → 270.000002132 h=1e-3 is slightly too small. Without rounding errors, t3 = 270.000002 (exactly) Richardson Extrapolation from t1, t2 gives excellent estimated for f''(4): XCas> t2 + (t2-t1) / (100-1) → 270.0 |
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Messages In This Thread |
(35S) 2nd order Derivative (at a point) - trojdor - 11-09-2020, 09:51 PM
RE: (35S) 2nd order Derivative (at a point) - Albert Chan - 11-10-2020, 02:20 PM
RE: (35S) 2nd order Derivative (at a point) - trojdor - 11-10-2020, 11:23 PM
RE: (35S) 2nd order Derivative (at a point) - Albert Chan - 11-11-2020 12:35 AM
RE: (35S) 2nd order Derivative (at a point) - trojdor - 11-11-2020, 04:11 AM
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