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