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The forward and central error bounds have explicit balancing steps

Statement

Let $H > 0$ . Consider the conservative error bounds from Forward and central differences have first- and second-order error bounds on a permitted step interval $(0, H]$ , where every stencil with $h \leq H$ stays inside the domain. If $M_{2} > 0$ and $δ > 0$ , the forward bound

$E_{+} (h) = M_{2} h / 2 + 2 δ / h$

has the unique unconstrained minimizer $h_{+} = \sqrt{4 δ / M_{2}}$ and minimum $2 \sqrt{M_{2} δ}$ . On $(0, H]$ the minimizer is $\min {h_{+}, H}$ . If $M_{3} > 0$ and $δ > 0$ , the central conservative bound

$E_{0} (h) = M_{3} h^{2} / 6 + 2 δ / h$

has the unique unconstrained minimizer $h_{0} = \sqrt[3]{6 δ / M_{3}}$ and minimum $M_{3} h_{0}^{2} / 2$ ; on $(0, H]$ the minimizer is $\min {h_{0}, H}$ . These optimize the displayed error bounds, not the unknown actual error.

If $δ = 0$ and the corresponding truncation constant is positive, the bound has infimum $0$ as $h ↓ 0$ and no minimizer for $h > 0$ . If the corresponding truncation constant is $0$ and $δ > 0$ , the bound decreases with $h$ , so its capped minimum is at $H$ . If $δ$ and the corresponding truncation constant are both zero, every permitted step is optimal.

Facts & Assumptions

Given: The derivative bounds, evaluation-error bound and permitted cap described above.

[F1]

Forward and central differences have first- and second-order error bounds: the forward and central conservative bounds are $M_{2} h / 2 + 2 δ / h$ and $M_{3} h^{2} / 6 + 2 δ / h$ .

[F2]

Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ and Existence and uniqueness of $n$ -th roots: a unique $a^{1 / n} \geq 0$ with $(a^{1 / n})^{n} = a$ : positive real numbers have unique positive square and cube roots.

Proof

Given: The hypotheses listed above.

1.1givenF1F2algebra

Set $r : = h / h_{+} > 0$ . The identity $h_{+}^{2} = 4 δ / M_{2}$ gives $E_{+} (h) = \frac{M_{2} h_{+}}{2} (r + r^{- 1})$ , while $E_{+} (h_{+}) = M_{2} h_{+}$ . Since $r + r^{- 1} - 2 = (r - 1)^{2} / r \geq 0$ , equality holds exactly at $r = 1$ . Thus $h_{+}$ is the unique unconstrained minimizer and its value is $2 \sqrt{M_{2} δ}$ .

1.2givenF1F2algebra

Set $r : = h / h_{0}$ and $a : = M_{3} h_{0}^{2} / 6 > 0$ . From $h_{0}^{3} = 6 δ / M_{3}$ , $E_{0} (h) = a (r^{2} + 2 / r)$ and $E_{0} (h_{0}) = 3 a$ . The difference is $a (r^{2} + 2 / r - 3) = a (r - 1)^{2} (r + 2) / r \geq 0$ , with equality only at $r = 1$ . Thus $h_{0}$ uniquely minimizes the unconstrained bound, and the minimum is $3 a = M_{3} h_{0}^{2} / 2$ .

2.1givenstep 1.1algebra

If $0 < r_{1} < r_{2} \leq 1$ , then $(r_{1} + r_{1}^{- 1}) - (r_{2} + r_{2}^{- 1}) = (r_{2} - r_{1}) (1 / (r_{1} r_{2}) - 1) > 0$ , since $r_{1} r_{2} < 1$ . Hence $E_{+}$ strictly decreases up to $h_{+}$ ; the global bound from step 1.1 gives the cap minimizer $\min {h_{+}, H}$ .

2.2givenstep 1.2algebra

If $0 < r_{1} < r_{2} \leq 1$ , then $(r_{1}^{2} + 2 / r_{1}) - (r_{2}^{2} + 2 / r_{2}) = (r_{2} - r_{1}) (2 / (r_{1} r_{2}) - (r_{1} + r_{2})) > 0$ , since $r_{1} r_{2} < 1$ and $r_{1} + r_{2} < 2$ . Hence the central bound strictly decreases up to $h_{0}$ , so the cap minimizer is $\min {h_{0}, H}$ .

3.1givenF1algebra∎

If $δ = 0$ and the corresponding truncation constant is positive, the bound in [F1] is a positive multiple of $h$ or $h^{2}$ and has infimum $0$ as $h ↓ 0$ , not attained for $h > 0$ . If a truncation constant is $0$ and $δ > 0$ , the bound is a positive multiple of $h^{- 1}$ and decreases to $0$ as $h \to \infty$ , while on a permitted capped interval it is minimized at $H$ . If both constants vanish the bound is identically zero.

Depends on

Used by

A quadratic attains the forward truncation bound and exhibits the balanced noisy step Example

Dependency tree · two levels

27 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)
Andreas Buttenschoen, Introduction to Scientific Computing, Numerical Differentiation (standard reference, not scraped)