Second derivative with complex numbers

[Albert Chan]

(02-11-2021 12:36 PM)Werner Wrote:  It may avoid roundoff if you want high accuracy.

Yes, but only with first derivative. (assumed im(f(x+hi)) does not hit with catastrophic cancellation)
For second derivative, you still hit with subtraction cancellation errors.

XCas> f(x) := x*(x^3+5*x^2-21*x)

XCas> f1(x,h) := re((f(x+h)-f(x-h))/(2h))                // central difference 1st derivative
XCas> f'(x) .- [f1(x,h), f1(x,h*i)] | x=4, h=1e-3       → [-2.10000508787e-05, 2.10000000607e-05]
XCas> f'(x) .- [f1(x,h), f1(x,h*i)] | x=4, h=1e-6       → [-1.0320036381e-08, 2.09752215596e-11]

XCas> f2(x,h) := re((f(x+h)-2*f(x)+f(x-h))/h^2)     // central difference 2nd derivative
XCas> f''(x) .- [f2(x,h), f2(x,h*i)] | x=4, h=1e-3      → [-2.13206476474e-06, 1.87539626495e-06]
XCas> f''(x) .- [f2(x,h), f2(x,h*i)] | x=4, h=1e-6      → [0.0221821205923, 0.0506038300227]

---

If |h| is not too small, we avoided catastrophic cancellation, error = O(h^2)
We may take advantage of it, with h = ε*√i, which make h^2 purely imaginary.

XCas> f'(x) - f1(x,h) | x=4, h=1e-3*√i        → -2.44426701101e-12
XCas> f''(x) - f2(x,h) | x=4, h=1e-3*√i       → -4.78621586808e-11