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Forward and central finite-difference Jacobian-vector and Hessian-vector products

Definition

Let $m, n \geq 1$ , $U \subseteq R^{n}$ be open, $x \in U$ , $F : U \to R^{m}$ , $f : U \to R$ , and let $v \in R^{n}$ . Use the Jacobian $J F (x)$ and gradient $\nabla f (x)$ of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, the Hessian $H_{f} (x)$ of The Hessian matrix and critical points of a scalar field, and the published Euclidean vector norm $∥ \cdot ∥_{2}$ of The $p$ -norms $∥ x ∥_{p}$ for rational $p \geq 1$ , and $∥ x ∥_{\infty}$ . The Jacobian-vector products below require $J F (x)$ to exist, and the Hessian-vector products require $H_{f} (x)$ to exist. Each gradient-difference Hessian-vector estimate also requires $\nabla f$ to exist at every point in its stencil. The exact products are $J F (x) v$ and $H_{f} (x) v$ , respectively.

For $h > 0$ with the required points in $U$ , define the forward and central Jacobian-vector estimates

$J_{h}^{+} (F, x, v) : = \frac{F (x + h v) - F (x)}{h}, J_{h}^{0} (F, x, v) : = \frac{F (x + h v) - F (x - h v)}{2 h},$

and the gradient-difference Hessian-vector estimates

$H_{h}^{+} (f, x, v) : = \frac{\nabla f (x + h v) - \nabla f (x)}{h}, H_{h}^{0} (f, x, v) : = \frac{\nabla f (x + h v) - \nabla f (x - h v)}{2 h} .$

For each requested vector value $z (y)$ , write an inexact evaluation as $\hat{z} (y) = z (y) + e$ . When an error bound is specified, it is per call: $∥ e ∥_{2} \leq δ$ for that call's $δ \geq 0$ . Errors may be dependent or random; a stated bound on a random error is understood to hold almost surely. Define the inexact estimates by substitution:

${\hat{J}}_{h}^{+} (F, x, v) : = \frac{\hat{F} (x + h v) - \hat{F} (x)}{h}, {\hat{J}}_{h}^{0} (F, x, v) : = \frac{\hat{F} (x + h v) - \hat{F} (x - h v)}{2 h},$

${\hat{H}}_{h}^{+} (f, x, v) : = \frac{\hat{\nabla f} (x + h v) - \hat{\nabla f} (x)}{h}, {\hat{H}}_{h}^{0} (f, x, v) : = \frac{\hat{\nabla f} (x + h v) - \hat{\nabla f} (x - h v)}{2 h} .$

Depends on

Used by

Directional Jacobian and Hessian differences retain Taylor and evaluation-noise errors Theorem

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Sources

Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, finite differences (standard reference, not scraped)
Andreas Buttenschoen, Introduction to Scientific Computing, Numerical Differentiation (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)