∂ — Derivative

Keys: Alt-f

f∂X: gradient of f at X. f must return a unit. U f∂ X: vector–Jacobian product, where U has the shape of f’s result. The result has the shape of X.

Currently supports real evaluators C⌻, including factored and exponent-table forms. C⌻ is a train that binds C, the same as C↣⌻.

f←[1 2 3]ₓ⌻ ⋄ f∂2ₓ                ⍝ 14ₓ
f←[1 2 3]ₓ⌻ ⋄ f∂∂2ₓ               ⍝ 6ₓ
f←[1 2 3]ₓ⌻ ⋄ [10 20]ₓ f∂ [1 2]ₓ  ⍝ [80 280]ₓ

Multivariate gradients retain the coordinate enclosure. A shared unit coordinate sums partials.

f←⊂[1 2 0⋄1 0 2]ₓ ⌻ ⋄ f∂⊂[3 4]ₓ      ⍝ ⊂[6 8]ₓ
f←⊂[1 2 0⋄1 0 2]ₓ ⌻ ⋄ f∂3ₓ           ⍝ 12ₓ

Repeated ∂ requires unit input and output and one variable.

Errors

• RANK: monadic array output
• LENGTH: wrong cotangent shape
• DOMAIN: unsupported function or nonfinite/non-real data