Automatic differentiation in Ruby

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Automatic differentiation in Ruby http://codon.com/automatic-differentiation-in-ruby

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A clever trick that makes it easy to do differentiation on a computer

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What the blazes is “differentiation”?

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“function” = relationship between two quantities

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e.g. relationship between time and distance travelled

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cruising 0 sec 1 sec 2 sec time distance

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pulling away 0 sec 1 sec 2 sec time distance

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time distance(time) 0 seconds 0 metres 1 second 1 metre 2 seconds 2 metres 3 seconds 3 metres 4 seconds 4 metres ⋮ ⋮ distance(time) = time

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distance(time) = time × time time distance(time) 0 seconds 0 metres 1 second 1 metre 2 seconds 4 metres 3 seconds 9 metres 4 seconds 16 metres ⋮ ⋮

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“differentiation” = ﬁnding out how fast a function’s result is changing

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distance(time) = … speed(time) = ? differentiate

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time distance(time) speed(time) 0 seconds 0 metres ? 1 second 1 metre ? 2 seconds 4 metres ? 3 seconds 9 metres ? 4 seconds 16 metres ? ⋮ ⋮ ⋮

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cruising 0 sec 1 sec 2 sec 1 m /s time distance

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fast cruising 0 sec 1 sec 2 sec 1 m /s 2 m/s time distance

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slow cruising 0 sec 1 sec 2 sec 0.5 m/s 1 m /s 2 m/s time distance

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pulling away 0 sec 1 sec 2 sec time distance

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pulling away 0 sec 1 sec 2 sec time distance

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pulling away 0 sec 1 sec 2 sec time distance

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distance(time) = … speed(time) = ? differentiate

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Symbolic differentiation

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distance(time) = time × time speed(time) = ?

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distance(time) = time × time speed(time) = ?

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distance(time) = time ² speed(time) = ?

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distance(time) = time ² speed(time) = 2 × time ¹ (by the elementary power rule)

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time distance(time) speed(time) 0 seconds 0 metres 0 m/s 1 second 1 metre 2 m/s 2 seconds 4 metres 4 m/s 3 seconds 9 metres 6 m/s 4 seconds 16 metres 8 m/s ⋮ ⋮ ⋮ distance(time) = time × time speed(time) = 2 × time

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Numerical differentiation

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distance(time) = time × time

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def distance(time:) time * time end

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pulling away 0 sec 1 sec 2 sec time distance

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pulling away 0 sec 1 sec 2 sec time distance

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pulling away 0 sec 1 sec 2 sec time distance

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pulling away 0 sec 1 sec 2 sec time distance

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def speed(time:) time_elapsed = 0.01 distance_before = distance(time: time) distance_after = distance(time: time + time_elapsed) distance_travelled = distance_after - distance_before distance_travelled / time_elapsed end

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>> speed(time: 0) => 0.01 >> speed(time: 1) => 2.0100000000000007 >> speed(time: 2) => 4.009999999999891 >> speed(time: 3) => 6.009999999999849 >> speed(time: 4) => 8.009999999999806

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time distance(time) speed(time) 0 seconds 0 metres 0.01 m/s 1 second 1 metre 2.01 m/s 2 seconds 4 metres 4.01 m/s 3 seconds 9 metres 6.01 m/s 4 seconds 16 metres 8.01 m/s ⋮ ⋮ ⋮ distance(time) = time × time speed(time) = ?

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Automatic differentiation

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Big idea: calculate a function’s rate of change and its value all at once

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distance 3 seconds 9 metres 1 s/s 6 m/s distance(time) = time × time

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i a + b COMPLEX NUMBERS

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DUAL NUMBERS ε a + b

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class DualNumber attr_accessor :real, :dual def initialize(real:, dual:) self.real = real self.dual = dual end def to_s [real, (dual < 0 ? '-' : '+'), dual.abs, 'ε'].join end def inspect "(#{to_s})" end end

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module Kernel def DualNumber(real, dual = 0) case real when DualNumber real else DualNumber.new(real: real, dual: dual) end end end

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def distance(time:) time end >> time_now = DualNumber(3, 1) => (3+1ε) >> distance_now = distance(time: time_now) => (3+1ε) >> distance_now.real => 3 >> distance_now.dual => 1

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distance 3 seconds 3 metres 1 s/s 1 m/s distance(time) = time

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3 metres 1 m/s distance(time) = time × time distance 3 seconds 9 1 s/s ?

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( i × i = -1 ) a + bi

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= a + c + bi + di = (a + c) + (b + d)i (a + bi) + (c + di) =

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(a + bi) × (c + di) = = (a × c) + (a × di) + (bi × c) + (bi × di) = ac + (ad + bc)i + bdi ² = ac + (ad + bc)i + bd × -1 = ac + (ad + bc)i - bd = (ac - bd) + (ad + bc)i

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a + bε ( ε × ε = 0 )

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= a + c + bε + dε = (a + c) + (b + d)ε (a + bε) + (c + dε) =

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(a + bε) × (c + dε) = = (a × c) + (a × dε) + (bε × c) + (bε × dε) = ac + (ad + bc)ε + bdε² = ac + (ad + bc)ε + bd × 0 = ac + (ad + bc)ε

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class DualNumber def +(other) DualNumber.new \ real: real + other.real, dual: dual + other.dual end def *(other) DualNumber.new \ real: real * other.real, dual: real * other.dual + dual * other.real end end

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>> x = DualNumber(1, 2) => (1+2ε) >> y = DualNumber(3, 4) => (3+4ε) >> x + y => (4+6ε) >> x * y => (3+10ε)

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def distance(time:) time * time end >> time_now = DualNumber(3, 1) => (3+1ε) >> distance_now = distance(time: time_now) => (9+6ε) >> distance_now.real => 9 >> distance_now.dual => 6

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distance 3 seconds 9 metres 1 s/s 6 m/s distance(time) = time × time

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>> x = DualNumber(1, 2) => (1+2ε) >> (x + 3) * 4 NoMethodError: undefined method `dual' for 3:Fixnum

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class DualNumber def +(other) other = DualNumber(other) DualNumber.new \ real: real + other.real, dual: dual + other.dual end def *(other) other = DualNumber(other) DualNumber.new \ real: real * other.real, dual: real * other.dual + dual * other.real end end

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>> x = DualNumber(1, 2) => (1+2ε) >> (x + 3) * 4 => (16+8ε)

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>> x = DualNumber(1, 2) => (1+2ε) >> 3 + (4 * x) TypeError: DualNumber can't be coerced into Fixnum

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coerce(3) 3 ! [!!!, !] +(!) !!!! !!! +(!) !!!!

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class DualNumber def coerce(other) [DualNumber(other), self] end end

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>> x = DualNumber(1, 2) => (1+2ε) >> 3 + (4 * x) => (7+8ε)

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= sin(a)ε = cos(a)ε = exp(a)ε = log(a)ε = sqrt(a)ε ⋮ = sin(a)ε = cos(a)ε = exp(a)ε = log(a)ε = sqrt(a)ε + (b × cos(a))ε (b × sin(a))ε + (b × exp(a))ε + (b ÷ a)ε + (b ÷ (2 × sqrt(a))ε sin(a + bε) = cos(a + bε) = exp(a + bε) = log(a + bε) = sqrt(a + bε) = - (b × sin(a))ε (b × cos(a))ε (b × sin(a))ε (b × exp(a))ε (b ÷ a)ε (b ÷ (2 × sqrt(a))ε

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Math.singleton_class.prepend Module.new { def sin(x) case x when DualNumber DualNumber.new \ real: sin(x.real), dual: x.dual * cos(x.real) else super end end def cos(x) case x when DualNumber DualNumber.new \ real: cos(x.real), dual: -x.dual * sin(x.real) else super end end }

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>> x = DualNumber(Math::PI / 3, 1) => (1.0471975511965976+1ε) >> Math.sin(x) => (0.8660254037844386+0.5000000000000001ε) >> Math.sin(x) + Math.cos(x) => (1.3660254037844388-0.3660254037844385ε) >> Math.sin(x) * Math.cos(x / 2) => (0.75+0.21650635094610984ε)