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Complex-step differentiation is second-order for holomorphic functions real on the real axis

Statement

Let $U \subseteq C$ be open, let $x \in U \cap R$ , and suppose $f : U \to C$ is holomorphic, real-valued on a real interval $I \subseteq U \cap R$ around $x$ , and ${x + i t : 0 \leq t \leq h} \subset U$ for $h > 0$ . Suppose $∣ Re f^{'''} (x + i t) ∣ \leq M_{3}$ for $0 \leq t \leq h$ , with $M_{3} \geq 0$ . Then

$∣ C_{h} f (x) - f^{'} (x) ∣ \leq \frac{M_{3} h^{2}}{6} .$

If the evaluated imaginary component has error at most $δ_{i m}$ , then

$∣ {\hat{C}}_{h} f (x) - f^{'} (x) ∣ \leq \frac{M_{3} h^{2}}{6} + \frac{δ_{i m}}{h} .$

Holomorphy and real-axis reality are essential to the formula: real differentiability of the restriction alone is insufficient.

Facts & Assumptions

Given: The domain, holomorphy, real-axis reality, third-derivative bound, step and evaluation-error hypotheses in the statement.

[F1]

The complex-step derivative estimator and its component-error model: $C_{h} f (x) = Im f (x + i h) / h$ , and an imaginary-component evaluation error $δ_{i m}$ contributes at most $δ_{i m} / h$ .

[F2]

Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions: holomorphy means complex differentiability at every point, with derivative given by the limit of complex difference quotients.

[F3]

Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\overset{ˉ}{z}} f = 0$ , or with the Cauchy–Riemann equations: if $f = u + i v$ is holomorphic, then $f^{'} = u_{x} + i v_{x} = v_{y} - i u_{y}$ and $u_{x} = v_{y}$ , $u_{y} = - v_{x}$ .

[F4]

Holomorphic functions are real analytic and smooth in their two real coordinates: the real and imaginary coordinate functions of a holomorphic function are $C^{k}$ for every natural $k$ .

[F5]

All higher complex derivatives exist and satisfy Cauchy's integral formula on an interior circle and If a holomorphic function has $C^{2}$ components, then its derivative is holomorphic: all iterated complex derivatives exist locally; if a holomorphic function has $C^{2}$ coordinate functions, its derivative is holomorphic.

[F6]

The Lagrange and Cauchy forms of Taylor's remainder and Taylor polynomials and their remainders: a real $C^{3}$ function on the interval equals its degree- $2$ Taylor polynomial plus $g^{'''} (ξ) h^{3} / 3!$ for an intermediate $ξ$ .

[F7]

Real and imaginary parts, complex conjugation, and modulus: $Re$ and $Im$ are the real and imaginary coordinates of a complex number.

[F8]

Absolute value in an ordered field and Basic properties of the absolute value: absolute value is nonnegative and $∣ a b ∣ = ∣ a ∣ ∣ b ∣$ .

[F9]

The triangle inequality: $∣ a + b ∣ \leq ∣ a ∣ + ∣ b ∣$ for real $a, b$ .

Proof

Given: The hypotheses listed above.

1.1givenF3F4F5construct

Put $g (t) : = Im f (x + i t)$ for $0 \leq t \leq h$ . By [F4], its coordinate functions are smooth. By [F3], $g^{'} (t) = v_{y} (x, t) = u_{x} (x, t) = Re f^{'} (x + i t)$ . If $q = p + i r$ is holomorphic, [F3] gives $\partial_{y} p = - \partial_{x} r = - Im q^{'}$ and $\partial_{y} r = \partial_{x} p = Re q^{'}$ . Applying [F5] to $f$ and then to $f^{'}$ makes $f^{'}$ and $f^{''}$ holomorphic, so these identities give $g^{''} (t) = - Im f^{''} (x + i t)$ and $g^{'''} (t) = - Re f^{'''} (x + i t)$ .

2.1givenF2step 1.1algebra

At real arguments near $x$ , $f$ is real by hypothesis, so its real-direction difference quotients show that $f^{'}$ is real there; applying the same argument to the holomorphic function $f^{'}$ shows that $f^{''}$ is real there. Thus $g (0) = 0$ , $g^{'} (0) = f^{'} (x)$ , and $g^{''} (0) = - Im f^{''} (x) = 0$ . By step 1.1, $∣ g^{'''} (t) ∣ \leq M_{3}$ on $[0, h]$ .

3.1givenF1F6F7F8step 2.1algebra

Apply [F6] to $g$ from $0$ to $h$ . For some $ξ \in (0, h)$ , $g (h) = h f^{'} (x) + g^{'''} (ξ) h^{3} / 6$ . Since $h > 0$ , [F8] gives $∣ g (h) / h - f^{'} (x) ∣ = ∣ g^{'''} (ξ) ∣ h^{2} / 6 \leq M_{3} h^{2} / 6$ . By [F1] and [F7], $g (h) / h = C_{h} f (x)$ , proving the exact complex-step error bound.

4.1givenF1F8F9step 3.1algebra∎

By [F1] and [F8], $∣ {\hat{C}}_{h} f (x) - C_{h} f (x) ∣ \leq δ_{i m} / h$ . [F9] and step 3.1 give the inexact bound.

Depends on

Used by

The absolute-value extension gives a zero complex-step quotient despite its real derivative Counterexample

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51 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Nicholas J. Higham, What Is the Complex Step Approximation? (standard reference, not scraped)
Mike Giles, Monte Carlo methods, Lecture 6: Complex Variable Trick (standard reference, not scraped)
R. Boudjemaa, M. G. Cox, A. B. Forbes, and P. M. Harris, Automatic Differentiation Techniques and their Application in Metrology, NPL Report CMSC 26/03 (standard reference, not scraped)