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

Statement

Let $m, n \geq 1$ , $U \subseteq R^{n}$ be open, $x \in U$ , $v \in R^{n}$ , and $h > 0$ with the required line segments contained in $U$ . Use the estimators of Forward and central finite-difference Jacobian-vector and Hessian-vector products. For a vector-valued map, $F \in C^{k} (U)$ means that each component $F_{i}$ is in $C^{k} (U)$ . For $F \in C^{2}$ , suppose each component $g_{i} (t) : = F_{i} (x + t v)$ satisfies $∣ g_{i}^{''} (t) ∣ \leq b_{2, i}$ for $0 \leq t \leq h$ , with $b_{2, i} \geq 0$ ; put $M_{2, F} : = (\sum_{i < m} b_{2, i}^{2})^{1 / 2}$ . For $F \in C^{3}$ , suppose $∣ g_{i}^{'''} (t) ∣ \leq b_{3, i}$ for $- h \leq t \leq h$ , with $b_{3, i} \geq 0$ ; put $M_{3, F} : = (\sum_{i < m} b_{3, i}^{2})^{1 / 2}$ . Then

$∥ J_{h}^{+} (F, x, v) - J F (x) v ∥_{2} \leq M_{2, F} h / 2, ∥ J_{h}^{0} (F, x, v) - J F (x) v ∥_{2} \leq M_{3, F} h^{2} / 6.$

For $f \in C^{3}$ , set $q_{j} (t) : = \partial_{j} f (x + t v)$ , suppose $∣ q_{j}^{''} (t) ∣ \leq a_{2, j}$ on $[0, h]$ , with $a_{2, j} \geq 0$ , and put $M_{2, H} : = (\sum_{j < n} a_{2, j}^{2})^{1 / 2}$ . For $f \in C^{4}$ , suppose $∣ q_{j}^{'''} (t) ∣ \leq a_{3, j}$ on $[- h, h]$ , with $a_{3, j} \geq 0$ , and put $M_{3, H} : = (\sum_{j < n} a_{3, j}^{2})^{1 / 2}$ . Then

$∥ H_{h}^{+} (f, x, v) - H_{f} (x) v ∥_{2} \leq M_{2, H} h / 2, ∥ H_{h}^{0} (f, x, v) - H_{f} (x) v ∥_{2} \leq M_{3, H} h^{2} / 6.$

If each vector evaluation has Euclidean error at most $δ$ , the forward evaluation contribution is at most $2 δ / h$ and the central contribution at most $δ / h$ (hence also at most $2 δ / h$ ). For a scalar output coordinate on a common probability space, suppose its two square-integrable evaluation noises are centered, with variances $σ_{1}^{2}, σ_{0}^{2}$ and covariance $c$ . Their forward-difference noise variance is $(σ_{1}^{2} + σ_{0}^{2} - 2 c) / h^{2}$ ; the central formula has the same numerator divided by $4 h^{2}$ . If these centered noises are independent and both have variance $σ^{2}$ , then the formulas reduce to $2 σ^{2} / h^{2}$ forward and $σ^{2} / (2 h^{2})$ central. No variance sum or root-sum-square total-error claim is made without the stated centering and covariance information.

If every evaluation call used in a given estimate has Euclidean error at most a common $δ \geq 0$ , then the inexact estimates of Forward and central finite-difference Jacobian-vector and Hessian-vector products satisfy

$∥ {\hat{J}}_{h}^{+} (F, x, v) - J F (x) v ∥_{2} \leq M_{2, F} h / 2 + 2 δ / h, ∥ {\hat{J}}_{h}^{0} (F, x, v) - J F (x) v ∥_{2} \leq M_{3, F} h^{2} / 6 + δ / h,$

$∥ {\hat{H}}_{h}^{+} (f, x, v) - H_{f} (x) v ∥_{2} \leq M_{2, H} h / 2 + 2 δ / h, ∥ {\hat{H}}_{h}^{0} (f, x, v) - H_{f} (x) v ∥_{2} \leq M_{3, H} h^{2} / 6 + δ / h .$

The central evaluation contribution is at most $δ / h$ , so the conservative $2 δ / h$ contribution is also valid for the central estimates.

Facts & Assumptions

Given: The domains, smoothness, component derivative bounds and evaluation-error hypotheses in the statement.

[F1]

Forward and central finite-difference Jacobian-vector and Hessian-vector products: the exact products and four directional estimators are as stated there; the inexact estimates replace each function or gradient evaluation by its inexact value, with error bounds applying per call.

[F2]

Repeated derivatives along a line expand by the multinomial formula: for a $C^{k}$ scalar field on an open set, derivatives of its restriction to an affine line are the corresponding directional partial-derivative sums through order $k$ .

[F3]

Continuous mixed partials of order $k$ are invariant under permutations: for a $C^{k}$ scalar field and $k \geq 2$ , permuting the order of the $k$ coordinate differentiations does not change the result.

[F4]

The Lagrange and Cauchy forms of Taylor's remainder and Taylor polynomials and their remainders: for a real $C^{2}$ or $C^{3}$ function on an interval, the remainder after degree $1$ or degree $2$ is $g^{''} (ξ) h^{2} / 2$ or $g^{'''} (ξ) h^{3} / 6$ .

[F5]

The $p$ -norms $∥ x ∥_{p}$ for rational $p \geq 1$ , and $∥ x ∥_{\infty}$ and Each $∥ \cdot ∥_{p}$ is a norm on $R^{n}$ , and the induced metrics are exactly $d_{1}$ , $d_{2}$ and $d_{\infty}$ of the published metric-spaces page: for nonempty finite-dimensional real output vectors, $∥ \cdot ∥_{2}$ is the Euclidean norm and obeys the triangle inequality and absolute homogeneity.

[F6]

Moments, variance, and covariance on a probability space and Variance and covariance identities for random variables: variance and covariance are centered second moments, and covariance is symmetric and bilinear on finite linear combinations of square-integrable real random variables.

[F7]

Expectations factor over finite products of independent random variables: expectations of products of integrable functions of independent real random variables factor into products of expectations.

[F8]

Absolute value in an ordered field and Basic properties of the absolute value: $∣ r ∣ \geq 0$ and $∣ r s ∣ = ∣ r ∣ ∣ s ∣$ for real $r, s$ .

[F9]

On a probability space, every square-integrable real random variable $X$ is integrable: $2 ∣ X ∣ \leq 1 + X^{2}$ gives $2 E ∣ X ∣ \leq 1 + E [X^{2}] < \infty$ .

Proof

Given: The hypotheses listed above.

1.1givenF1F2F4F5F8algebra

For each component $i$ , [F2] identifies $g_{i}^{'} (0) = (J F (x) v)_{i}$ . Apply [F4] through degree $1$ on $[0, h]$ to obtain $∣ (J_{h}^{+})_{i} - (J F (x) v)_{i} ∣ \leq b_{2, i} h / 2$ . By [F8], these absolute values and bounds are nonnegative; squaring preserves their order because $b^{2} - a^{2} = (b - a) (b + a) \geq 0$ when $0 \leq a \leq b$ . Sum over $i < m$ and use [F5] and the definition of $M_{2, F}$ to obtain the stated Euclidean vector bound.

1.2givenF1F5algebra

For either the Jacobian-vector or gradient-difference Hessian-vector estimate, the forward evaluation perturbation is $(e_{1} - e_{0}) / h$ . [F5] bounds its norm by $(∥ e_{1} ∥_{2} + ∥ e_{0} ∥_{2}) / h \leq 2 δ / h$ . For the central quotient the denominator is $2 h$ , giving $δ / h$ .

1.3givenF6F7F9algebra

For centered square-integrable coordinate noises $E_{1}, E_{0}$ , [F6] and the definitions of variance and covariance give $Var (E_{1} - E_{0}) = Cov (E_{1} - E_{0}, E_{1} - E_{0})$ . Bilinearity and symmetry then yield $σ_{1}^{2} + σ_{0}^{2} - 2 c$ . Dividing by $h^{2}$ or $(2 h)^{2}$ gives the forward and central formulas. If the noises are independent with common variance $σ^{2}$ , [F9] makes them integrable, so [F7] gives $E [E_{1} E_{0}] = E [E_{1}] E [E_{0}] = 0$ ; their covariance is $0$ by [F6], yielding $2 σ^{2} / h^{2}$ and $σ^{2} / (2 h^{2})$ .

2.1givenF1F2F4F5F8step 1.1algebra

For each component $i$ , apply [F4] through degree $2$ at both $h$ and $- h$ to $g_{i}$ . The constant and quadratic terms cancel in the central quotient, leaving $∣ (J_{h}^{0})_{i} - (J F (x) v)_{i} ∣ \leq b_{3, i} h^{2} / 6$ . The same nonnegative-square comparison from step 1.1, [F8], and the sum defining $M_{3, F}$ give the central Jacobian-vector bound by [F5].

2.2givenF1F2F3F4F5step 1.1algebra

For $q_{j} (t) = \partial_{j} f (x + t v)$ , [F2] gives $q_{j}^{'} (0) = \sum_{k} \partial_{k} \partial_{j} f (x) v_{k}$ . By [F3] and the Hessian convention in [F1], this equals $(H_{f} (x) v)_{j}$ . Applying [F4] through degree $1$ to each $q_{j}$ yields the component forward bound $a_{2, j} h / 2$ ; the nonnegative-square comparison from step 1.1 gives the sum bound, and [F5] then yields $∥ H_{h}^{+} - H_{f} v ∥_{2} \leq M_{2, H} h / 2$ .

2.3givenF1F2F3F4F5step 1.1algebra

Applying [F4] through degree $2$ at $h$ and $- h$ to each $q_{j}$ cancels the even terms and yields component error at most $a_{3, j} h^{2} / 6$ . The same nonnegative-square comparison from step 1.1, together with [F5], gives the central Hessian-vector bound.

3.1step 1.1step 2.1step 2.2step 2.3step 1.2step 1.3F5∎

Combining the two Jacobian-vector truncation bounds in steps 1.1 and 2.1, the two Hessian-vector truncation bounds in steps 2.2 and 2.3, and the corresponding evaluation bounds in step 1.2 by the Euclidean triangle inequality [F5] gives all four hatted-estimate bounds in the Statement. For random errors, the covariance identity in step 1.3 applies; absent centering/covariance hypotheses no root-sum-square conclusion follows.

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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)