520 lines
19 KiB
C++
520 lines
19 KiB
C++
/* ----------------------------------------------------------------------------
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* GTSAM Copyright 2010, Georgia Tech Research Corporation,
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* Atlanta, Georgia 30332-0415
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* All Rights Reserved
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* Authors: Frank Dellaert, et al. (see THANKS for the full author list)
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* See LICENSE for the license information
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* -------------------------------------------------------------------------- */
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/**
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* @file numericalDerivative.h
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* @brief Some functions to compute numerical derivatives
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* @author Frank Dellaert
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*/
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// \callgraph
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#pragma once
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#include <boost/function.hpp>
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#ifdef __GNUC__
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#pragma GCC diagnostic push
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#pragma GCC diagnostic ignored "-Wunused-variable"
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#endif
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#include <boost/bind.hpp>
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#ifdef __GNUC__
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#pragma GCC diagnostic pop
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#endif
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#include <gtsam/base/Matrix.h>
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#include <gtsam/base/Manifold.h>
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namespace gtsam {
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/*
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* Note that all of these functions have two versions, a boost.function version and a
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* standard C++ function pointer version. This allows reformulating the arguments of
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* a function to fit the correct structure, which is useful for situations like
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* member functions and functions with arguments not involved in the derivative:
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*
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* Usage of the boost bind version to rearrange arguments:
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* for a function with one relevant param and an optional derivative:
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* Foo bar(const Obj& a, boost::optional<Matrix&> H1)
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* Use boost.bind to restructure:
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* boost::bind(bar, _1, boost::none)
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* This syntax will fix the optional argument to boost::none, while using the first argument provided
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*
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* For member functions, such as below, with an instantiated copy instanceOfSomeClass
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* Foo SomeClass::bar(const Obj& a)
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* Use boost bind as follows to create a function pointer that uses the member function:
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* boost::bind(&SomeClass::bar, ref(instanceOfSomeClass), _1)
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*
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* For additional details, see the documentation:
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* http://www.boost.org/doc/libs/release/libs/bind/bind.html
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*/
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/**
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* Numerically compute gradient of scalar function
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* Class X is the input argument
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* The class X needs to have dim, expmap, logmap
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*/
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template<class X>
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Vector numericalGradient(boost::function<double(const X&)> h, const X& x,
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double delta = 1e-5) {
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double factor = 1.0 / (2.0 * delta);
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BOOST_STATIC_ASSERT_MSG(traits::is_manifold<X>::value,
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"Template argument X must be a manifold type.");
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static const int N = traits::dimension<X>::value;
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BOOST_STATIC_ASSERT_MSG(N>0, "Template argument X must be fixed-size type.");
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typedef DefaultChart<X> ChartX;
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typedef typename ChartX::vector TangentX;
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// get chart at x
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ChartX chartX;
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// Prepare a tangent vector to perturb x with, only works for fixed size
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TangentX d;
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d.setZero();
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Vector g = zero(N); // Can be fixed size
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for (int j = 0; j < N; j++) {
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d(j) = delta;
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double hxplus = h(chartX.retract(x, d));
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d(j) = -delta;
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double hxmin = h(chartX.retract(x, d));
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d(j) = 0;
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g(j) = (hxplus - hxmin) * factor;
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}
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return g;
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}
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/**
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* @brief New-style numerical derivatives using manifold_traits
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* @brief Computes numerical derivative in argument 1 of unary function
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* @param h unary function yielding m-vector
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* @param x n-dimensional value at which to evaluate h
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* @param delta increment for numerical derivative
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* Class Y is the output argument
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* Class X is the input argument
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* @return m*n Jacobian computed via central differencing
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*/
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template<class Y, class X>
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// TODO Should compute fixed-size matrix
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Matrix numericalDerivative11(boost::function<Y(const X&)> h, const X& x,
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double delta = 1e-5) {
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using namespace traits;
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BOOST_STATIC_ASSERT_MSG(traits::is_manifold<Y>::value,
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"Template argument Y must be a manifold type.");
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typedef DefaultChart<Y> ChartY;
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typedef typename ChartY::vector TangentY;
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BOOST_STATIC_ASSERT_MSG(traits::is_manifold<X>::value,
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"Template argument X must be a manifold type.");
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static const int N = traits::dimension<X>::value;
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BOOST_STATIC_ASSERT_MSG(N>0, "Template argument X must be fixed-size type.");
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typedef DefaultChart<X> ChartX;
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typedef typename ChartX::vector TangentX;
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// get value at x, and corresponding chart
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Y hx = h(x);
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ChartY chartY;
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// Bit of a hack for now to find number of rows
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TangentY zeroY = chartY.local(hx,hx);
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size_t m = zeroY.size();
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// get chart at x
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ChartX chartX;
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// Prepare a tangent vector to perturb x with, only works for fixed size
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TangentX dx;
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dx.setZero();
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// Fill in Jacobian H
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Matrix H = zeros(m, N);
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double factor = 1.0 / (2.0 * delta);
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for (int j = 0; j < N; j++) {
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dx(j) = delta;
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TangentY dy1 = chartY.local(hx, h(chartX.retract(x, dx)));
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dx(j) = -delta;
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TangentY dy2 = chartY.local(hx, h(chartX.retract(x, dx)));
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dx(j) = 0;
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H.col(j) << (dy1 - dy2) * factor;
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}
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return H;
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}
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/** use a raw C++ function pointer */
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template<class Y, class X>
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Matrix numericalDerivative11(Y (*h)(const X&), const X& x,
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double delta = 1e-5) {
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return numericalDerivative11<Y, X>(boost::bind(h, _1), x, delta);
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}
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/**
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* Compute numerical derivative in argument 1 of binary function
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* @param h binary function yielding m-vector
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* @param x1 n-dimensional first argument value
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* @param x2 second argument value
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* @param delta increment for numerical derivative
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* @return m*n Jacobian computed via central differencing
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*/
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template<class Y, class X1, class X2>
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Matrix numericalDerivative21(const boost::function<Y(const X1&, const X2&)>& h,
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const X1& x1, const X2& x2, double delta = 1e-5) {
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BOOST_STATIC_ASSERT_MSG(traits::is_manifold<Y>::value,
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"Template argument Y must be a manifold type.");
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BOOST_STATIC_ASSERT_MSG(traits::is_manifold<X1>::value,
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"Template argument X1 must be a manifold type.");
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return numericalDerivative11<Y, X1>(boost::bind(h, _1, x2), x1, delta);
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}
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/** use a raw C++ function pointer */
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template<class Y, class X1, class X2>
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inline Matrix numericalDerivative21(Y (*h)(const X1&, const X2&), const X1& x1,
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const X2& x2, double delta = 1e-5) {
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return numericalDerivative21<Y, X1, X2>(boost::bind(h, _1, _2), x1, x2, delta);
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}
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/**
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* Compute numerical derivative in argument 2 of binary function
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* @param h binary function yielding m-vector
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* @param x1 first argument value
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* @param x2 n-dimensional second argument value
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* @param delta increment for numerical derivative
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* @return m*n Jacobian computed via central differencing
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*/
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template<class Y, class X1, class X2>
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Matrix numericalDerivative22(boost::function<Y(const X1&, const X2&)> h,
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const X1& x1, const X2& x2, double delta = 1e-5) {
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BOOST_STATIC_ASSERT_MSG(traits::is_manifold<Y>::value,
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"Template argument Y must be a manifold type.");
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BOOST_STATIC_ASSERT_MSG(traits::is_manifold<X2>::value,
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"Template argument X2 must be a manifold type.");
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return numericalDerivative11<Y, X2>(boost::bind(h, x1, _1), x2, delta);
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}
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/** use a raw C++ function pointer */
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template<class Y, class X1, class X2>
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inline Matrix numericalDerivative22(Y (*h)(const X1&, const X2&), const X1& x1,
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const X2& x2, double delta = 1e-5) {
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return numericalDerivative22<Y, X1, X2>(boost::bind(h, _1, _2), x1, x2, delta);
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}
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/**
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* Compute numerical derivative in argument 1 of ternary function
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* @param h ternary function yielding m-vector
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* @param x1 n-dimensional first argument value
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* @param x2 second argument value
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* @param x3 third argument value
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* @param delta increment for numerical derivative
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* @return m*n Jacobian computed via central differencing
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* All classes Y,X1,X2,X3 need dim, expmap, logmap
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*/
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template<class Y, class X1, class X2, class X3>
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Matrix numericalDerivative31(
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boost::function<Y(const X1&, const X2&, const X3&)> h, const X1& x1,
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const X2& x2, const X3& x3, double delta = 1e-5) {
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BOOST_STATIC_ASSERT_MSG(traits::is_manifold<Y>::value,
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"Template argument Y must be a manifold type.");
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BOOST_STATIC_ASSERT_MSG(traits::is_manifold<X1>::value,
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"Template argument X1 must be a manifold type.");
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return numericalDerivative11<Y, X1>(boost::bind(h, _1, x2, x3), x1, delta);
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}
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template<class Y, class X1, class X2, class X3>
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inline Matrix numericalDerivative31(Y (*h)(const X1&, const X2&, const X3&),
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const X1& x1, const X2& x2, const X3& x3, double delta = 1e-5) {
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return numericalDerivative31<Y, X1, X2, X3>(boost::bind(h, _1, _2, _3), x1,
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x2, x3, delta);
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}
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/**
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* Compute numerical derivative in argument 2 of ternary function
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* @param h ternary function yielding m-vector
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* @param x1 n-dimensional first argument value
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* @param x2 second argument value
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* @param x3 third argument value
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* @param delta increment for numerical derivative
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* @return m*n Jacobian computed via central differencing
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* All classes Y,X1,X2,X3 need dim, expmap, logmap
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*/
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template<class Y, class X1, class X2, class X3>
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Matrix numericalDerivative32(
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boost::function<Y(const X1&, const X2&, const X3&)> h, const X1& x1,
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const X2& x2, const X3& x3, double delta = 1e-5) {
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BOOST_STATIC_ASSERT_MSG(traits::is_manifold<Y>::value,
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"Template argument Y must be a manifold type.");
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BOOST_STATIC_ASSERT_MSG(traits::is_manifold<X2>::value,
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"Template argument X2 must be a manifold type.");
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return numericalDerivative11<Y, X2>(boost::bind(h, x1, _1, x3), x2, delta);
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}
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template<class Y, class X1, class X2, class X3>
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inline Matrix numericalDerivative32(Y (*h)(const X1&, const X2&, const X3&),
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const X1& x1, const X2& x2, const X3& x3, double delta = 1e-5) {
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return numericalDerivative32<Y, X1, X2, X3>(boost::bind(h, _1, _2, _3), x1,
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x2, x3, delta);
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}
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/**
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* Compute numerical derivative in argument 3 of ternary function
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* @param h ternary function yielding m-vector
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* @param x1 n-dimensional first argument value
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* @param x2 second argument value
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* @param x3 third argument value
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* @param delta increment for numerical derivative
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* @return m*n Jacobian computed via central differencing
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* All classes Y,X1,X2,X3 need dim, expmap, logmap
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*/
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template<class Y, class X1, class X2, class X3>
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Matrix numericalDerivative33(
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boost::function<Y(const X1&, const X2&, const X3&)> h, const X1& x1,
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const X2& x2, const X3& x3, double delta = 1e-5) {
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BOOST_STATIC_ASSERT_MSG(traits::is_manifold<Y>::value,
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"Template argument Y must be a manifold type.");
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BOOST_STATIC_ASSERT_MSG(traits::is_manifold<X3>::value,
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"Template argument X3 must be a manifold type.");
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return numericalDerivative11<Y, X3>(boost::bind(h, x1, x2, _1), x3, delta);
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}
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template<class Y, class X1, class X2, class X3>
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inline Matrix numericalDerivative33(Y (*h)(const X1&, const X2&, const X3&),
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const X1& x1, const X2& x2, const X3& x3, double delta = 1e-5) {
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return numericalDerivative33<Y, X1, X2, X3>(boost::bind(h, _1, _2, _3), x1,
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x2, x3, delta);
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}
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/**
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* Compute numerical Hessian matrix. Requires a single-argument Lie->scalar
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* function. This is implemented simply as the derivative of the gradient.
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* @param f A function taking a Lie object as input and returning a scalar
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* @param x The center point for computing the Hessian
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* @param delta The numerical derivative step size
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* @return n*n Hessian matrix computed via central differencing
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*/
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template<class X>
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inline Matrix numericalHessian(boost::function<double(const X&)> f, const X& x,
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double delta = 1e-5) {
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BOOST_STATIC_ASSERT_MSG(traits::is_manifold<X>::value,
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"Template argument X must be a manifold type.");
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typedef boost::function<double(const X&)> F;
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typedef boost::function<Vector(F, const X&, double)> G;
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G ng = static_cast<G>(numericalGradient<X> );
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return numericalDerivative11<Vector, X>(boost::bind(ng, f, _1, delta), x,
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delta);
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}
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template<class X>
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inline Matrix numericalHessian(double (*f)(const X&), const X& x, double delta =
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1e-5) {
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return numericalHessian(boost::function<double(const X&)>(f), x, delta);
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}
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/** Helper class that computes the derivative of f w.r.t. x1, centered about
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* x1_, as a function of x2
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*/
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template<class X1, class X2>
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class G_x1 {
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const boost::function<double(const X1&, const X2&)>& f_;
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X1 x1_;
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double delta_;
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public:
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G_x1(const boost::function<double(const X1&, const X2&)>& f, const X1& x1,
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double delta) :
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f_(f), x1_(x1), delta_(delta) {
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}
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Vector operator()(const X2& x2) {
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return numericalGradient<X1>(boost::bind(f_, _1, x2), x1_, delta_);
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}
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};
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template<class X1, class X2>
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inline Matrix numericalHessian212(
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boost::function<double(const X1&, const X2&)> f, const X1& x1, const X2& x2,
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double delta = 1e-5) {
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G_x1<X1, X2> g_x1(f, x1, delta);
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return numericalDerivative11<Vector, X2>(
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boost::function<Vector(const X2&)>(
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boost::bind<Vector>(boost::ref(g_x1), _1)), x2, delta);
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}
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template<class X1, class X2>
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inline Matrix numericalHessian212(double (*f)(const X1&, const X2&),
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const X1& x1, const X2& x2, double delta = 1e-5) {
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return numericalHessian212(boost::function<double(const X1&, const X2&)>(f),
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x1, x2, delta);
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}
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template<class X1, class X2>
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inline Matrix numericalHessian211(
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boost::function<double(const X1&, const X2&)> f, const X1& x1, const X2& x2,
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double delta = 1e-5) {
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Vector (*numGrad)(boost::function<double(const X1&)>, const X1&,
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double) = &numericalGradient<X1>;
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boost::function<double(const X1&)> f2(boost::bind(f, _1, x2));
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return numericalDerivative11<Vector, X1>(
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boost::function<Vector(const X1&)>(boost::bind(numGrad, f2, _1, delta)),
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x1, delta);
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}
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template<class X1, class X2>
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inline Matrix numericalHessian211(double (*f)(const X1&, const X2&),
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const X1& x1, const X2& x2, double delta = 1e-5) {
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return numericalHessian211(boost::function<double(const X1&, const X2&)>(f),
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x1, x2, delta);
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}
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template<class X1, class X2>
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inline Matrix numericalHessian222(
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boost::function<double(const X1&, const X2&)> f, const X1& x1, const X2& x2,
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double delta = 1e-5) {
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Vector (*numGrad)(boost::function<double(const X2&)>, const X2&,
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double) = &numericalGradient<X2>;
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boost::function<double(const X2&)> f2(boost::bind(f, x1, _1));
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return numericalDerivative11<Vector, X2>(
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boost::function<Vector(const X2&)>(boost::bind(numGrad, f2, _1, delta)),
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x2, delta);
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}
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template<class X1, class X2>
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inline Matrix numericalHessian222(double (*f)(const X1&, const X2&),
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const X1& x1, const X2& x2, double delta = 1e-5) {
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return numericalHessian222(boost::function<double(const X1&, const X2&)>(f),
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x1, x2, delta);
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}
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/**
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* Numerical Hessian for tenary functions
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*/
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/* **************************************************************** */
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template<class X1, class X2, class X3>
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inline Matrix numericalHessian311(
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boost::function<double(const X1&, const X2&, const X3&)> f, const X1& x1,
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const X2& x2, const X3& x3, double delta = 1e-5) {
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Vector (*numGrad)(boost::function<double(const X1&)>, const X1&,
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double) = &numericalGradient<X1>;
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boost::function<double(const X1&)> f2(boost::bind(f, _1, x2, x3));
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return numericalDerivative11<Vector, X1>(
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boost::function<Vector(const X1&)>(boost::bind(numGrad, f2, _1, delta)),
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x1, delta);
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}
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template<class X1, class X2, class X3>
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inline Matrix numericalHessian311(double (*f)(const X1&, const X2&, const X3&),
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const X1& x1, const X2& x2, const X3& x3, double delta = 1e-5) {
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return numericalHessian311(
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boost::function<double(const X1&, const X2&, const X3&)>(f), x1, x2, x3,
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delta);
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}
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/* **************************************************************** */
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template<class X1, class X2, class X3>
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inline Matrix numericalHessian322(
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boost::function<double(const X1&, const X2&, const X3&)> f, const X1& x1,
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const X2& x2, const X3& x3, double delta = 1e-5) {
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Vector (*numGrad)(boost::function<double(const X2&)>, const X2&,
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double) = &numericalGradient<X2>;
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boost::function<double(const X2&)> f2(boost::bind(f, x1, _1, x3));
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return numericalDerivative11<Vector, X2>(
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boost::function<Vector(const X2&)>(boost::bind(numGrad, f2, _1, delta)),
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x2, delta);
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}
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template<class X1, class X2, class X3>
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inline Matrix numericalHessian322(double (*f)(const X1&, const X2&, const X3&),
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const X1& x1, const X2& x2, const X3& x3, double delta = 1e-5) {
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return numericalHessian322(
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boost::function<double(const X1&, const X2&, const X3&)>(f), x1, x2, x3,
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delta);
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}
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/* **************************************************************** */
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template<class X1, class X2, class X3>
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inline Matrix numericalHessian333(
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boost::function<double(const X1&, const X2&, const X3&)> f, const X1& x1,
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const X2& x2, const X3& x3, double delta = 1e-5) {
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Vector (*numGrad)(boost::function<double(const X3&)>, const X3&,
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double) = &numericalGradient<X3>;
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boost::function<double(const X3&)> f2(boost::bind(f, x1, x2, _1));
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return numericalDerivative11<Vector, X3>(
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boost::function<Vector(const X3&)>(boost::bind(numGrad, f2, _1, delta)),
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x3, delta);
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}
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template<class X1, class X2, class X3>
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inline Matrix numericalHessian333(double (*f)(const X1&, const X2&, const X3&),
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const X1& x1, const X2& x2, const X3& x3, double delta = 1e-5) {
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return numericalHessian333(
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boost::function<double(const X1&, const X2&, const X3&)>(f), x1, x2, x3,
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|
delta);
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}
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|
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/* **************************************************************** */
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template<class X1, class X2, class X3>
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inline Matrix numericalHessian312(
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boost::function<double(const X1&, const X2&, const X3&)> f, const X1& x1,
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const X2& x2, const X3& x3, double delta = 1e-5) {
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return numericalHessian212<X1, X2>(
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boost::function<double(const X1&, const X2&)>(boost::bind(f, _1, _2, x3)),
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|
x1, x2, delta);
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}
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|
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template<class X1, class X2, class X3>
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inline Matrix numericalHessian313(
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|
boost::function<double(const X1&, const X2&, const X3&)> f, const X1& x1,
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const X2& x2, const X3& x3, double delta = 1e-5) {
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|
return numericalHessian212<X1, X3>(
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|
boost::function<double(const X1&, const X3&)>(boost::bind(f, _1, x2, _2)),
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|
x1, x3, delta);
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}
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|
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template<class X1, class X2, class X3>
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inline Matrix numericalHessian323(
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|
boost::function<double(const X1&, const X2&, const X3&)> f, const X1& x1,
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const X2& x2, const X3& x3, double delta = 1e-5) {
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|
return numericalHessian212<X2, X3>(
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|
boost::function<double(const X2&, const X3&)>(boost::bind(f, x1, _1, _2)),
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|
x2, x3, delta);
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|
}
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|
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/* **************************************************************** */
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|
template<class X1, class X2, class X3>
|
|
inline Matrix numericalHessian312(double (*f)(const X1&, const X2&, const X3&),
|
|
const X1& x1, const X2& x2, const X3& x3, double delta = 1e-5) {
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|
return numericalHessian312(
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|
boost::function<double(const X1&, const X2&, const X3&)>(f), x1, x2, x3,
|
|
delta);
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|
}
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|
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template<class X1, class X2, class X3>
|
|
inline Matrix numericalHessian313(double (*f)(const X1&, const X2&, const X3&),
|
|
const X1& x1, const X2& x2, const X3& x3, double delta = 1e-5) {
|
|
return numericalHessian313(
|
|
boost::function<double(const X1&, const X2&, const X3&)>(f), x1, x2, x3,
|
|
delta);
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|
}
|
|
|
|
template<class X1, class X2, class X3>
|
|
inline Matrix numericalHessian323(double (*f)(const X1&, const X2&, const X3&),
|
|
const X1& x1, const X2& x2, const X3& x3, double delta = 1e-5) {
|
|
return numericalHessian323(
|
|
boost::function<double(const X1&, const X2&, const X3&)>(f), x1, x2, x3,
|
|
delta);
|
|
}
|
|
}
|