Rootfinding through global Newton iteration and Chebyshev polynomials for the amplitude of an electronic oscillator

The amplitude z and oscillator gain k of a balanced oscillator are connected by the transcendental equation f(z, k) = 0 where f(z,k)≡(2/z)∫01tanh(zsin(2πt))sin(2πt)dt-(1/k) where the k ∈ [1, ∞] and z ∈ [0, ∞]. It is trivial to find k(z) through numerical quadrature, but the inverse function z(k) is a non-trivial exercise in rootfinding. First, we derive a power series for small k and an asymptotic expansion for large k. By blending these, we obtain an analytic approximation to initialize Newton’s iteration; it is found that the iteration converges to full machine precision within at most four iterations over the entire parameter range. However, the Newton method is rather slow because each iteration requires two numerical quadratures. Because the integrands are periodic, these can be done by the rectangle rule with an accuracy that increases exponentially with the number of quadrature points N, but N must be large when z is large. A faster method is to apply the Newton/quadrature algorithm to generate z(k) at a few selected points and apply Chebyshev interpolation twice. Each Chebyshev expansion is to choose the mimic the corresponding power series or asymptotic series so that singularities are avoided. Just 15 terms of each series plus an asymptotic approximation for very large k are sufficient to approximate z(k) to within about eight decimal places over the entire parameter range.