The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case

Depends on

• Directional derivatives and partial derivatives of a map $U\subseteq\mathbb{R}^m\to\mathbb{R}^n$
• The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$
• The standard list $e : n \to F^{n}$ with $e_i(i) = 1_F$ and $e_i(j) = 0_F$ for $j \ne i$ is an ordered basis of $F^{n}$; hence $\dim_F F^{n} = n$, and $F^{0}$ is the zero space with basis $\varnothing$ and dimension $0$

Used by

• Agile online mirror descent Algorithm
• Damped Newton method Algorithm
• Finite-budget bracketing and zoom for a strong Wolfe step Algorithm
• Finite-budget forward-difference Newton for a square vector residual Algorithm
• Finite-budget inexact Newton–Krylov with a descent safeguard Algorithm
• Finite-budget Levenberg-Marquardt with a trust-ratio safeguard Algorithm
• Finite-budget multidimensional Krawczyk root certification Algorithm
• Finite-budget Newton and chord solves with scheduled factorization reuse Algorithm
• Finite-budget Newton for a square vector residual Algorithm
• Finite-budget residual-merit damped Newton Algorithm
• Fletcher–Reeves and safeguarded Polak–Ribière nonlinear conjugate gradients Algorithm
• Forward-difference Jacobian on an open domain Algorithm
• Gauss-Newton linearized least-squares step with QR/SVD rank safeguards Algorithm
• Gradient descent Algorithm
• Gram-matrix AdaGrad with full or singular diagonal metrics Algorithm
• Heavy-ball momentum Algorithm
• Ideal exact gradient line-search step Algorithm
• Nesterov accelerated gradient Algorithm
• One-step episodic actor–critic with a bootstrapped critic Algorithm
• Projected online gradient descent Algorithm
• Randomized coordinate gradient descent Algorithm
• Randomized proximal coordinate descent with certified inner solves Algorithm
• Two-stage second-order Rosenbrock W-method with shared factorization Algorithm
• Unadjusted Langevin algorithm (ULA) Algorithm
• For one regular constraint, the objective gradient is a scalar multiple of the constraint gradient Corollary
• Green's first identity on a glued elementary solid region Corollary
• Green's second identity on a glued elementary solid region Corollary
• Lagrange multipliers for a regular graph constraint y=ψ(x) Corollary
• The curl of a curl is the gradient of the divergence minus the Laplacian Corollary
• The Jacobian determinant of a holomorphic map is |f'|² and is positive exactly where f'≠0 Corollary
• The subdifferential of a differentiable convex function is its gradient singleton Corollary
• Vector forms: the boundary integrals of fn and of n× F Corollary
• A critical value can have a smooth level set Counterexample
• A curl-free C¹ field on the complement of a line that is not conservative Counterexample
• A strong Wolfe step can leave PR+ ascending; the safeguard restarts Counterexample
• An isolated boundary point obstructs pointwise-zero Green data Counterexample
• The cone x²+y²=z² has a rank drop at its apex Counterexample
• The cusp y²=x³ has a rank drop at the origin Counterexample
• The degenerate constraint x²+y²=0 defeats the multiplier conclusion Counterexample
• The projected OGD bound does not cover infeasible comparators Counterexample

…and 154 more results.

Dependency tree · two levels

31 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

• J. Lebl, Basic Analysis I, §8.3 (standard reference, not scraped)