∂ — Derivative / VJP
Monadic operator. 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 differentiates real evaluators C⌻, including factored and exponent-table forms, and functions with a derivative that f∂←g attaches. 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]ₓMultivariate 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 ⍝ 12ₓRepeated ∂ requires unit input and output and one variable.
f∂←g attaches g as the derivative of the function that f names, as assignment describes. f still applies itself. f∂X then calls g X, and U f∂ X calls U g X, so g gives the gradient with one argument and the vector–Jacobian product with two. A higher order differentiates g, and ⁻¹ inverts f. ⇄ follows the derivative of a function that has one.
r←{⍵×⍵} ⋄ r∂←{⍺←1 ⋄ ⍺×2×⍵}
r 3 ⍝ 9
r∂3 ⍝ 6
10 r∂ 3 ⍝ 60Errors
RANK: monadic array outputLENGTH: wrong cotangent shapeDOMAIN: unsupported function or nonfinite/non-real data