Ե      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ GHC only experimentalekmett@gmail.com^ is used by  deriveMode but is not exposed  via  # to prevent its abuse by end users  via the AD data type.  is used by  deriveMode but is not exposed  via the  ) class to prevent its abuse by end users  via the AD data type. QIt provides direct access to the result, stripped of its derivative information, K but this is unsafe in general as (lift . primal) would discard derivative N information. The end user is protected from accidentally using this function G by the universal quantification on the various combinators we expose. Embed a constant  Vector sum Scalar-vector multiplication Vector-scalar multiplication Scalar division   'zero' = 'lift' 0 6Higher-order versions of the stock numerical methods.  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNO  'one' = 'lift' 1 PP t provides   instance Lifted $t given supplied instances for  + instance Lifted $t => Primal $t where ... - instance Lifted $t => Jacobian $t where ... The seemingly redundant  $tB constraints are caused by Template Haskell staging restrictions. $Find all the members defined in the  data type QQ f g# provides the following instances: < instance ('Lifted' $f, 'Num' a, 'Enum' a) => 'Enum' ($g a) 8 instance ('Lifted' $f, 'Num' a, 'Eq' a) => 'Eq' ($g a) : instance ('Lifted' $f, 'Num' a, 'Ord' a) => 'Ord' ($g a) A instance ('Lifted' $f, 'Num a, 'Bounded' a) => 'Bounded' ($g a) 3 instance ('Lifted' $f, 'Show' a) => 'Show' ($g a) 1 instance ('Lifted' $f, 'Num' a) => 'Num' ($g a) ? instance ('Lifted' $f, 'Fractional' a) => 'Fractional' ($g a) ; instance ('Lifted' $f, 'Floating' a) => 'Floating' ($g a) = instance ('Lifted' $f, 'RealFloat' a) => 'RealFloat' ($g a) ; instance ('Lifted' $f, 'RealFrac' a) => 'RealFrac' ($g a) 3 instance ('Lifted' $f, 'Real' a) => 'Real' ($g a) R  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQR O PQ !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNR   = !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMN !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQGHC only experimentalekmett@gmail.comRSR* serves as a common wrapper for different  # instances, exposing a traditional X numerical tower. Universal quantification is used to limit the actions in user code to Z machinery that will return the same answers under all AD modes, allowing us to use modes ( interchangeably as both the type level "brand") and dictionary, providing a common API. STUVWXYZ[\]^ RSTUVWXYZ[\]^ WXRSTYZ[\]^UV RSTSTUVVWXYZZ[\]^GHC only experimentalekmett@gmail.com_`abcdef_`abcdefbcd_`aef_`a`abcdcdefGHC only experimentalekmett@gmail.com ghijklmnopq ghijklmnopq ghikjlmnopq ghhijklmnopqGHC only experimentalekmett@gmail.comrstuvThe vY function calculates the first derivative of a scalar-to-scalar function by forward-mode SR  diff sin == cos wThe d'UUV function calculates the result and first derivative of scalar-to-scalar function by Forward SR  d' sin == sin &&& cos  d' f = f &&& d f xThe xN function calculates the first derivative of scalar-to-nonscalar function by Forward SR yThe y\ function calculates the result and first derivative of a scalar-to-non-scalar function by Forward SR zThe dUMS function calculates the first derivative of scalar-to-scalar monadic function by Forward SR {The d'UM` function calculates the result and first derivative of a scalar-to-scalar monadic function by Forward SR |BA fast, simple transposed Jacobian computed with forward-mode AD. }BA fast, simple transposed Jacobian computed with forward-mode AD. ~TCompute the product of a vector with the Hessian using forward-on-forward-mode AD. LCompute the gradient and hessian product using forward-on-forward-mode AD. RSTrstuvwxyz{|}~ ~|}vwxyrstuz{RST rstuvwxyz{|}~GHC only experimentalekmett@gmail.comReverse is a  A using reverse-mode automatic differentiation that provides fast diffFU, diff2FU, grad, grad2 and a fast jacobianF when you have a significantly smaller number of outputs than inputs. A TapeT records the information needed back propagate from the output to each input during    AD. >Used to mark variables for inspection during the reverse pass   +back propagate sensitivities along a tape. 6This returns a list of contributions to the partials. 2 The variable ids returned in the list are likely not unique!  Return an   of $ given bounds for the variable IDs.  Return an   of sparse partials GHC only experimentalekmett@gmail.comThe J function calculates the gradient of a non-scalar-to-scalar function with  AD in a single pass. The U function calculates the result and gradient of a non-scalar-to-scalar function with  AD in a single pass.  g fE function calculates the gradient of a non-scalar-to-scalar function f( with reverse-mode AD in a single pass. L The gradient is combined element-wise with the argument using the function g.  grad == gradWith (\_ dx -> dx)  id == gradWith const  g fG calculates the result and gradient of a non-scalar-to-scalar function f with  AD in a single pass L the gradient is combined element-wise with the argument using the function g. " grad' == gradWith' (\_ dx -> dx) The c function calculates the jacobian of a non-scalar-to-non-scalar function with reverse AD lazily in m passes for m outputs.  An alias for  The b function calculates both the result and the Jacobian of a nonscalar-to-nonscalar function, using m invocations of reverse AD,  where m( is the output dimensionality. Applying fmap snd* to the result will recover the result of   An alias for  ' gradWithF g f'@ calculates the Jacobian of a non-scalar-to-non-scalar function f with reverse AD lazily in m passes for m outputs. kInstead of returning the Jacobian matrix, the elements of the matrix are combined with the input using the g. " gradF == gradWithF (\_ dx -> dx) . gradWithF const == (\f x -> const x <$> f x)  An alias for .  g f'R calculates both the result and the Jacobian of a nonscalar-to-nonscalar function f, using m invocations of reverse AD,  where m( is the output dimensionality. Applying fmap snd* to the result will recover the result of  kInstead of returning the Jacobian matrix, the elements of the matrix are combined with the input using the g. ' jacobian' == gradWithF' (\_ dx -> dx)  An alias for  The d'4 function calculates the value and derivative, as a ' pair, of a scalar-to-scalar function. Compute the hessian via the jacobian of the gradient. gradient is computed in reverse mode and then the jacobian is computed in reverse mode. However, since the 'grad f :: f a -> f a'i is square this is not as fast as using the forward-mode Jacobian of a reverse mode gradient provided by Numeric.AD.hessian in  Numeric.AD. Compute the order 3 Hessian tensor on a non-scalar-to-non-scalar function via the forward-mode Jacobian of the mixed-mode Jacobian of the function. "While this is less efficient than Numeric.AD.hessianTensor from  Numeric.AD or  Numeric.AD.Forward.hessianTensor from Numeric.AD.Forward, the type signature is more permissive with regards to the output non-scalar, and it may be more efficient if only a few coefficients of the result are consumed. zCompute the hessian via the reverse-mode jacobian of the reverse-mode gradient of a non-scalar-to-scalar monadic action. "While this is less efficient than Numeric.AD.hessianTensor from  Numeric.AD or  Numeric.AD.Forward.hessianTensor from Numeric.AD.Forward, the type signature is more permissive with regards to the output non-scalar, and it may be more efficient if only a few coefficients of the result are consumed. # RST#RST GHC only experimentalekmett@gmail.comTower is an AD  B that calculates a tangent tower by forward AD, and provides fast diffsUU, diffsUF  GHC only experimentalekmett@gmail.com RST RST GHC only experimentalekmett@gmail.com The 2 function finds a zero of a scalar function using  Newton':s method; its output is a stream of increasingly accurate ' results. (Modulo the usual caveats.)  Examples:  7 take 10 $ findZero (\\x->x^2-4) 1 -- converge to 2.0  module Data.Complex C take 10 $ findZero ((+1).(^2)) (1 :+ 1) -- converge to (0 :+ 1)@ The  inverseNewton* function inverts a scalar function using  Newton':s method; its output is a stream of increasingly accurate ' results. (Modulo the usual caveats.)  Example: > take 10 $ inverseNewton sqrt 1 (sqrt 10) -- converges to 10 The ( function find a fixedpoint of a scalar  function using Newton'$s method; its output is a stream of = increasingly accurate results. (Modulo the usual caveats.) ? take 10 $ fixedPoint cos 1 -- converges to 0.7390851332151607 The ( function finds an extremum of a scalar  function using Newton',s method; produces a stream of increasingly 0 accurate results. (Modulo the usual caveats.) + take 10 $ extremum cos 1 -- convert to 0 The " function performs a multivariate ? optimization, based on the naive-gradient-descent in the file   stalingrad/examples/ flow-tests/pre-saddle-1a.vlad from the > VLAD compiler Stalingrad sources. Its output is a stream of = increasingly accurate results. (Modulo the usual caveats.) HIt uses reverse mode automatic differentiation to compute the gradient.  RSTRST  GHC only experimentalekmett@gmail.comCalculate the Jacobian of a non-scalar-to-non-scalar function, automatically choosing between forward and reverse mode AD based on the number of inputs and outputs. <If you need to support functions where the output is only a   or  , consider Numeric.AD.Reverse.jacobian or  from Numeric.AD.Reverse. Calculate both the answer and Jacobian of a non-scalar-to-non-scalar function, automatically choosing between forward- and reverse- mode AD based on the relative, number of inputs and outputs. <If you need to support functions where the output is only a   or  , consider Numeric.AD.Reverse.jacobian' or  from Numeric.AD.Reverse.  g f calculates the Jacobian of a non-scalar-to-non-scalar function, automatically choosing between forward and reverse mode AD based on the number of inputs and outputs. SThe resulting Jacobian matrix is then recombined element-wise with the input using g. <If you need to support functions where the output is only a   or  , consider Numeric.AD.Reverse.jacobianWith or  from Numeric.AD.Reverse.  g f calculates the answer and Jacobian of a non-scalar-to-non-scalar function, automatically choosing between forward and reverse mode AD based on the number of inputs and outputs. SThe resulting Jacobian matrix is then recombined element-wise with the input using g. <If you need to support functions where the output is only a   or  , consider  Numeric.AD.Reverse.jacobianWith' or  from Numeric.AD.Reverse.  f wv% computes the product of the hessian H$ of a non-scalar-to-scalar function f at w =   $ wv with a vector v = snd  $ wv using " Pearlmutter's method" from  ?http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.6143, which states:  ' H v = (d/dr) grad_w (w + r v) | r = 0 GOr in other words, we take the directional derivative of the gradient.  f wv6 computes both the gradient of a non-scalar-to-scalar f at w =   $ wv and the product of the hessian H at w with a vector v = snd  $ wv using " Pearlmutter's method"8. The outputs are returned wrapped in the same functor.  ' H v = (d/dr) grad_w (w + r v) | r = 0 GOr in other words, we take the directional derivative of the gradient. Compute the hessian via the jacobian of the gradient. gradient is computed in reverse mode and then the jacobian is computed in forward mode. Compute the order 3 Hessian tensor on a non-scalar-to-non-scalar function via the forward-mode Jacobian of the mixed-mode Jacobian of the function. 6 RSTrstuvwxyz{|}6|}vxwyrstuz{RST  GHC only experimentalekmett@gmail.com  RST RST   !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^__`aabcddefghiijkklmnoopqrstuvwxyz{|}~t}~rs } ~ y z { |     o } ~ ad-0.23Numeric.AD.ClassesNumeric.AD.InternalNumeric.AD.Internal.CompositionNumeric.AD.Internal.ForwardNumeric.AD.ForwardNumeric.AD.Internal.ReverseNumeric.AD.ReverseNumeric.AD.Internal.TowerNumeric.AD.TowerNumeric.AD.Newton Numeric.ADNumeric.AD.DirectedJacobianDunarylift1lift1_binarylift2lift2_PrimalprimalModelift<+>*^^*^/zeroLifted showsPrec1==!compare1 fromInteger1+!*!-!negate1signum1abs1/!recip1 fromRational1 toRational1pi1exp1sqrt1log1**!logBase1sin1atan1acos1asin1tan1cos1sinh1atanh1acosh1asinh1tanh1cosh1properFraction1 truncate1floor1ceiling1round1 floatRadix1 floatDigits1 floatRange1 decodeFloat1 encodeFloat1 exponent1 significand1 scaleFloat1isNaN1isIEEE1isNegativeZero1isDenormalized1 isInfinite1atan21succ1pred1toEnum1 fromEnum1 enumFrom1 enumFromThen1 enumFromTo1enumFromThenTo1 minBound1 maxBound1one deriveLifted deriveNumericADrunADPairzipWithTzipWithDefaultTIdprobeunprobeprobedunprobed ComposeModerunComposeModeComposeFunctordecomposeFunctor composeMode decomposeModeForwardtangentunbundlebundleapplybindbind'bindWith bindWith' transposeWithdudu'duFduF'diffdiff'diffFdiffF'diffMdiffM' jacobianT jacobianWithTjacobian jacobianWith jacobian' jacobianWith'gradgrad'gradWith gradWith'hessianProducthessianProduct'ReverseTapeUnaryBinaryVarLiftvarvarId derivative derivative'partials partialArray partialMapunbind unbindWith unbindMapunbindMapWithDefaultgradFgradF' gradWithF gradWithF'gradMgradM' gradWithM gradWithM'hessian hessianTensorhessianMTowergetTowerzeroPadzeroPadF transposePadFdd'tangentswithD getADTowertowerdiffsdiffs0diffsFdiffs0FdiffsMdiffs0Mtaylortaylor0 maclaurin maclaurin0dusdus0dusFdus0FfindZero findZeroMinverseinverseM fixedPoint fixedPointMextremum extremumMgradientDescentgradientAscentgradientDescentMgradientAscentM DirectionMixednegOne withPrimalfromBy fromIntegral1square1on discrete1 discrete2 discrete3varA liftedMembers lowerInstanceIsoisoosipidunpidSrunS backPropagatebaseGHC.ArrArraycontainers-0.3.0.0 Data.IntMapIntMapGHC.BaseFunctorMonad Data.Tuplefst