Automatic Differentiation

Category: Math

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Forward-, reverse- and mixed- mode automatic differentiation combinators with a common API. Type-level \"branding\" is used to both prevent the end user from confusing infinitesimals and to limit unsafe access to the implementation details of each Mode. Each mode has a separate module full of combinators. * @Numeric.AD.Mode.Forward@ provides basic forward-mode AD. It is good for computing simple derivatives. * @Numeric.AD.Mode.Reverse@ uses benign side-effects to compute reverse-mode AD. It is good for computing gradients in one pass. * @Numeric.AD.Mode.Sparse@ computes a sparse forward-mode AD tower. It is good for higher derivatives or large numbers of outputs. * @Numeric.AD.Mode.Tower@ computes a dense forward-mode AD tower useful for higher derivatives of single input functions. * @Numeric.AD.Mode.Mixed@ computes using whichever mode or combination thereof is suitable to each individual combinator. This mode is the default, re-exported by @Numeric.AD@ While not every mode can provide all operations, the following basic operations are supported, modified as appropriate by the suffixes below: * 'grad' computes the gradient (partial derivatives) of a function at a point. * 'jacobian' computes the Jacobian matrix of a function at a point. * 'diff' computes the derivative of a function at a point. * 'du' computes a directional derivative of a function at a point. * 'hessian' computes the Hessian matrix (matrix of second partial derivatives) of a function at a point. The following suffixes alter the meanings of the functions above as follows: * @\'@ -- also return the answer * @With@ lets the user supply a function to blend the input with the output * @F@ is a version of the base function lifted to return a 'Traversable' (or 'Functor') result * @s@ means the function returns all higher derivatives in a list or f-branching 'Stream' * @T@ means the result is transposed with respect to the traditional formulation. * @0@ means that the resulting derivative list is padded with 0s at the end. Changes since 1.3 * Dependency bump to be compatible with ghc 7.4.1 Changes since 1.2 * Compiles with template haskell 2.6, changed interface to comply with the lack of Eq and Show as superclasses of Num in the new GHC. Changes since 1.1.3 * Made primal calculations strict where possible. Changes since 1.1.0 * Introduced a much faster topological sort into the reverse mode AD implementation by Anthony Cowley. This fixes a space leak and a stack overflow problem on very large (>2000 variable) problem sets. * Made bound calculations in reverse mode more strict. Changes since 1.0.0 * Changed the way 'Show' was derived to comply with changes in instance resolution in ghc >= 7.0 && <= 7.1 Changes since 0.45.0 * Converted 'Stream' to use the external 'comonad' package Changes since 0.44.5 * Added Halley's method Changes since 0.40.0 * Fixed bug fix for @'(/)' :: (Mode s, Fractional a) => AD s a@ * Improved documentation * Regularized naming conventions * Exposed 'Id', probe, and lower methods via @Numeric.AD.Types@ * Removed monadic combinators * Retuned the 'Mixed' mode jacobian calculations to only require a 'Functor'-based result. * Added unsafe variadic 'vgrad', 'vgrad'', and 'vgrads' combinators


Edward Kmett





(c) Edward Kmett 2010-2011, (c) Barak Pearlmutter and Jeffrey Mark Siskind 2008-2009




base, data-reify, containers, array, comonad, free, template-haskell


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