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// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2015 Google Inc. All rights reserved.
// http://ceres-solver.org/
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are met:
//
// * Redistributions of source code must retain the above copyright notice,
// this list of conditions and the following disclaimer.
// * Redistributions in binary form must reproduce the above copyright notice,
// this list of conditions and the following disclaimer in the documentation
// and/or other materials provided with the distribution.
// * Neither the name of Google Inc. nor the names of its contributors may be
// used to endorse or promote products derived from this software without
// specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
// POSSIBILITY OF SUCH DAMAGE.
//
// Author: keir@google.com (Keir Mierle)
//
// A simple implementation of N-dimensional dual numbers, for automatically
// computing exact derivatives of functions.
//
// While a complete treatment of the mechanics of automatic differentation is
// beyond the scope of this header (see
// http://en.wikipedia.org/wiki/Automatic_differentiation for details), the
// basic idea is to extend normal arithmetic with an extra element, "e," often
// denoted with the greek symbol epsilon, such that e != 0 but e^2 = 0. Dual
// numbers are extensions of the real numbers analogous to complex numbers:
// whereas complex numbers augment the reals by introducing an imaginary unit i
// such that i^2 = -1, dual numbers introduce an "infinitesimal" unit e such
// that e^2 = 0. Dual numbers have two components: the "real" component and the
// "infinitesimal" component, generally written as x + y*e. Surprisingly, this
// leads to a convenient method for computing exact derivatives without needing
// to manipulate complicated symbolic expressions.
//
// For example, consider the function
//
// f(x) = x^2 ,
//
// evaluated at 10. Using normal arithmetic, f(10) = 100, and df/dx(10) = 20.
// Next, augument 10 with an infinitesimal to get:
//
// f(10 + e) = (10 + e)^2
// = 100 + 2 * 10 * e + e^2
// = 100 + 20 * e -+-
// -- |
// | +--- This is zero, since e^2 = 0
// |
// +----------------- This is df/dx!
//
// Note that the derivative of f with respect to x is simply the infinitesimal
// component of the value of f(x + e). So, in order to take the derivative of
// any function, it is only necessary to replace the numeric "object" used in
// the function with one extended with infinitesimals. The class Jet, defined in
// this header, is one such example of this, where substitution is done with
// templates.
//
// To handle derivatives of functions taking multiple arguments, different
// infinitesimals are used, one for each variable to take the derivative of. For
// example, consider a scalar function of two scalar parameters x and y:
//
// f(x, y) = x^2 + x * y
//
// Following the technique above, to compute the derivatives df/dx and df/dy for
// f(1, 3) involves doing two evaluations of f, the first time replacing x with
// x + e, the second time replacing y with y + e.
//
// For df/dx:
//
// f(1 + e, y) = (1 + e)^2 + (1 + e) * 3
// = 1 + 2 * e + 3 + 3 * e
// = 4 + 5 * e
//
// --> df/dx = 5
//
// For df/dy:
//
// f(1, 3 + e) = 1^2 + 1 * (3 + e)
// = 1 + 3 + e
// = 4 + e
//
// --> df/dy = 1
//
// To take the gradient of f with the implementation of dual numbers ("jets") in
// this file, it is necessary to create a single jet type which has components
// for the derivative in x and y, and passing them to a templated version of f:
//
// template<typename T>
// T f(const T &x, const T &y) {
// return x * x + x * y;
// }
//
// // The "2" means there should be 2 dual number components.
// Jet<double, 2> x(0); // Pick the 0th dual number for x.
// Jet<double, 2> y(1); // Pick the 1st dual number for y.
// Jet<double, 2> z = f(x, y);
//
// LOG(INFO) << "df/dx = " << z.v[0]
// << "df/dy = " << z.v[1];
//
// Most users should not use Jet objects directly; a wrapper around Jet objects,
// which makes computing the derivative, gradient, or jacobian of templated
// functors simple, is in autodiff.h. Even autodiff.h should not be used
// directly; instead autodiff_cost_function.h is typically the file of interest.
//
// For the more mathematically inclined, this file implements first-order
// "jets". A 1st order jet is an element of the ring
//
// T[N] = T[t_1, ..., t_N] / (t_1, ..., t_N)^2
//
// which essentially means that each jet consists of a "scalar" value 'a' from T
// and a 1st order perturbation vector 'v' of length N:
//
// x = a + \sum_i v[i] t_i
//
// A shorthand is to write an element as x = a + u, where u is the pertubation.
// Then, the main point about the arithmetic of jets is that the product of
// perturbations is zero:
//
// (a + u) * (b + v) = ab + av + bu + uv
// = ab + (av + bu) + 0
//
// which is what operator* implements below. Addition is simpler:
//
// (a + u) + (b + v) = (a + b) + (u + v).
//
// The only remaining question is how to evaluate the function of a jet, for
// which we use the chain rule:
//
// f(a + u) = f(a) + f'(a) u
//
// where f'(a) is the (scalar) derivative of f at a.
//
// By pushing these things through sufficiently and suitably templated
// functions, we can do automatic differentiation. Just be sure to turn on
// function inlining and common-subexpression elimination, or it will be very
// slow!
//
// WARNING: Most Ceres users should not directly include this file or know the
// details of how jets work. Instead the suggested method for automatic
// derivatives is to use autodiff_cost_function.h, which is a wrapper around
// both jets.h and autodiff.h to make taking derivatives of cost functions for
// use in Ceres easier.
#ifndef CERES_PUBLIC_JET_H_
#define CERES_PUBLIC_JET_H_
#include <cmath>
#include <iosfwd>
#include <iostream> // NOLINT
#include <limits>
#include <string>
#include "Eigen/Core"
#include "ceres/internal/port.h"
namespace ceres {
template <typename T, int N>
struct Jet {
enum { DIMENSION = N };
typedef T Scalar;
// Default-construct "a" because otherwise this can lead to false errors about
// uninitialized uses when other classes relying on default constructed T
// (where T is a Jet<T, N>). This usually only happens in opt mode. Note that
// the C++ standard mandates that e.g. default constructed doubles are
// initialized to 0.0; see sections 8.5 of the C++03 standard.
Jet() : a() {
v.setZero();
}
// Constructor from scalar: a + 0.
explicit Jet(const T& value) {
a = value;
v.setZero();
}
// Constructor from scalar plus variable: a + t_i.
Jet(const T& value, int k) {
a = value;
v.setZero();
v[k] = T(1.0);
}
// Constructor from scalar and vector part
// The use of Eigen::DenseBase allows Eigen expressions
// to be passed in without being fully evaluated until
// they are assigned to v
template<typename Derived>
EIGEN_STRONG_INLINE Jet(const T& a, const Eigen::DenseBase<Derived> &v)
: a(a), v(v) {
}
// Compound operators
Jet<T, N>& operator+=(const Jet<T, N> &y) {
*this = *this + y;
return *this;
}
Jet<T, N>& operator-=(const Jet<T, N> &y) {
*this = *this - y;
return *this;
}
Jet<T, N>& operator*=(const Jet<T, N> &y) {
*this = *this * y;
return *this;
}
Jet<T, N>& operator/=(const Jet<T, N> &y) {
*this = *this / y;
return *this;
}
// Compound with scalar operators.
Jet<T, N>& operator+=(const T& s) {
*this = *this + s;
return *this;
}
Jet<T, N>& operator-=(const T& s) {
*this = *this - s;
return *this;
}
Jet<T, N>& operator*=(const T& s) {
*this = *this * s;
return *this;
}
Jet<T, N>& operator/=(const T& s) {
*this = *this / s;
return *this;
}
// The scalar part.
T a;
// The infinitesimal part.
//
// We allocate Jets on the stack and other places they might not be aligned
// to X(=16 [SSE], 32 [AVX] etc)-byte boundaries, which would prevent the safe
// use of vectorisation. If we have C++11, we can specify the alignment.
// However, the standard gives wide lattitude as to what alignments are valid,
// and it might be that the maximum supported alignment *guaranteed* to be
// supported is < 16, in which case we do not specify an alignment, as this
// implies the host is not a modern x86 machine. If using < C++11, we cannot
// specify alignment.
#if defined(EIGEN_DONT_VECTORIZE)
Eigen::Matrix<T, N, 1, Eigen::DontAlign> v;
#else
// Enable vectorisation iff the maximum supported scalar alignment is >=
// 16 bytes, as this is the minimum required by Eigen for any vectorisation.
//
// NOTE: It might be the case that we could get >= 16-byte alignment even if
// max_align_t < 16. However we can't guarantee that this
// would happen (and it should not for any modern x86 machine) and if it
// didn't, we could get misaligned Jets.
static constexpr int kAlignOrNot =
// Work around a GCC 4.8 bug
// (https://gcc.gnu.org/bugzilla/show_bug.cgi?id=56019) where
// std::max_align_t is misplaced.
#if defined (__GNUC__) && __GNUC__ == 4 && __GNUC_MINOR__ == 8
alignof(::max_align_t) >= 16
#else
alignof(std::max_align_t) >= 16
#endif
? Eigen::AutoAlign : Eigen::DontAlign;
#if defined(EIGEN_MAX_ALIGN_BYTES)
// Eigen >= 3.3 supports AVX & FMA instructions that require 32-byte alignment
// (greater for AVX512). Rather than duplicating the detection logic, use
// Eigen's macro for the alignment size.
//
// NOTE: EIGEN_MAX_ALIGN_BYTES can be > 16 (e.g. 32 for AVX), even though
// kMaxAlignBytes will max out at 16. We are therefore relying on
// Eigen's detection logic to ensure that this does not result in
// misaligned Jets.
#define CERES_JET_ALIGN_BYTES EIGEN_MAX_ALIGN_BYTES
#else
// Eigen < 3.3 only supported 16-byte alignment.
#define CERES_JET_ALIGN_BYTES 16
#endif
// Default to the native alignment if 16-byte alignment is not guaranteed to
// be supported. We cannot use alignof(T) as if we do, GCC 4.8 complains that
// the alignment 'is not an integer constant', although Clang accepts it.
static constexpr size_t kAlignment = kAlignOrNot == Eigen::AutoAlign
? CERES_JET_ALIGN_BYTES : alignof(double);
#undef CERES_JET_ALIGN_BYTES
alignas(kAlignment) Eigen::Matrix<T, N, 1, kAlignOrNot> v;
#endif
};
// Unary +
template<typename T, int N> inline
Jet<T, N> const& operator+(const Jet<T, N>& f) {
return f;
}
// TODO(keir): Try adding __attribute__((always_inline)) to these functions to
// see if it causes a performance increase.
// Unary -
template<typename T, int N> inline
Jet<T, N> operator-(const Jet<T, N>&f) {
return Jet<T, N>(-f.a, -f.v);
}
// Binary +
template<typename T, int N> inline
Jet<T, N> operator+(const Jet<T, N>& f,
const Jet<T, N>& g) {
return Jet<T, N>(f.a + g.a, f.v + g.v);
}
// Binary + with a scalar: x + s
template<typename T, int N> inline
Jet<T, N> operator+(const Jet<T, N>& f, T s) {
return Jet<T, N>(f.a + s, f.v);
}
// Binary + with a scalar: s + x
template<typename T, int N> inline
Jet<T, N> operator+(T s, const Jet<T, N>& f) {
return Jet<T, N>(f.a + s, f.v);
}
// Binary -
template<typename T, int N> inline
Jet<T, N> operator-(const Jet<T, N>& f,
const Jet<T, N>& g) {
return Jet<T, N>(f.a - g.a, f.v - g.v);
}
// Binary - with a scalar: x - s
template<typename T, int N> inline
Jet<T, N> operator-(const Jet<T, N>& f, T s) {
return Jet<T, N>(f.a - s, f.v);
}
// Binary - with a scalar: s - x
template<typename T, int N> inline
Jet<T, N> operator-(T s, const Jet<T, N>& f) {
return Jet<T, N>(s - f.a, -f.v);
}
// Binary *
template<typename T, int N> inline
Jet<T, N> operator*(const Jet<T, N>& f,
const Jet<T, N>& g) {
return Jet<T, N>(f.a * g.a, f.a * g.v + f.v * g.a);
}
// Binary * with a scalar: x * s
template<typename T, int N> inline
Jet<T, N> operator*(const Jet<T, N>& f, T s) {
return Jet<T, N>(f.a * s, f.v * s);
}
// Binary * with a scalar: s * x
template<typename T, int N> inline
Jet<T, N> operator*(T s, const Jet<T, N>& f) {
return Jet<T, N>(f.a * s, f.v * s);
}
// Binary /
template<typename T, int N> inline
Jet<T, N> operator/(const Jet<T, N>& f,
const Jet<T, N>& g) {
// This uses:
//
// a + u (a + u)(b - v) (a + u)(b - v)
// ----- = -------------- = --------------
// b + v (b + v)(b - v) b^2
//
// which holds because v*v = 0.
const T g_a_inverse = T(1.0) / g.a;
const T f_a_by_g_a = f.a * g_a_inverse;
return Jet<T, N>(f.a * g_a_inverse, (f.v - f_a_by_g_a * g.v) * g_a_inverse);
}
// Binary / with a scalar: s / x
template<typename T, int N> inline
Jet<T, N> operator/(T s, const Jet<T, N>& g) {
const T minus_s_g_a_inverse2 = -s / (g.a * g.a);
return Jet<T, N>(s / g.a, g.v * minus_s_g_a_inverse2);
}
// Binary / with a scalar: x / s
template<typename T, int N> inline
Jet<T, N> operator/(const Jet<T, N>& f, T s) {
const T s_inverse = T(1.0) / s;
return Jet<T, N>(f.a * s_inverse, f.v * s_inverse);
}
// Binary comparison operators for both scalars and jets.
#define CERES_DEFINE_JET_COMPARISON_OPERATOR(op) \
template<typename T, int N> inline \
bool operator op(const Jet<T, N>& f, const Jet<T, N>& g) { \
return f.a op g.a; \
} \
template<typename T, int N> inline \
bool operator op(const T& s, const Jet<T, N>& g) { \
return s op g.a; \
} \
template<typename T, int N> inline \
bool operator op(const Jet<T, N>& f, const T& s) { \
return f.a op s; \
}
CERES_DEFINE_JET_COMPARISON_OPERATOR( < ) // NOLINT
CERES_DEFINE_JET_COMPARISON_OPERATOR( <= ) // NOLINT
CERES_DEFINE_JET_COMPARISON_OPERATOR( > ) // NOLINT
CERES_DEFINE_JET_COMPARISON_OPERATOR( >= ) // NOLINT
CERES_DEFINE_JET_COMPARISON_OPERATOR( == ) // NOLINT
CERES_DEFINE_JET_COMPARISON_OPERATOR( != ) // NOLINT
#undef CERES_DEFINE_JET_COMPARISON_OPERATOR
// Pull some functions from namespace std.
//
// This is necessary because we want to use the same name (e.g. 'sqrt') for
// double-valued and Jet-valued functions, but we are not allowed to put
// Jet-valued functions inside namespace std.
using std::abs;
using std::acos;
using std::asin;
using std::atan;
using std::atan2;
using std::cbrt;
using std::ceil;
using std::cos;
using std::cosh;
using std::exp;
using std::exp2;
using std::floor;
using std::fmax;
using std::fmin;
using std::hypot;
using std::isfinite;
using std::isinf;
using std::isnan;
using std::isnormal;
using std::log;
using std::log2;
using std::pow;
using std::sin;
using std::sinh;
using std::sqrt;
using std::tan;
using std::tanh;
// Legacy names from pre-C++11 days.
inline bool IsFinite (double x) { return std::isfinite(x); }
inline bool IsInfinite(double x) { return std::isinf(x); }
inline bool IsNaN (double x) { return std::isnan(x); }
inline bool IsNormal (double x) { return std::isnormal(x); }
// In general, f(a + h) ~= f(a) + f'(a) h, via the chain rule.
// abs(x + h) ~= x + h or -(x + h)
template <typename T, int N> inline
Jet<T, N> abs(const Jet<T, N>& f) {
return f.a < T(0.0) ? -f : f;
}
// log(a + h) ~= log(a) + h / a
template <typename T, int N> inline
Jet<T, N> log(const Jet<T, N>& f) {
const T a_inverse = T(1.0) / f.a;
return Jet<T, N>(log(f.a), f.v * a_inverse);
}
// exp(a + h) ~= exp(a) + exp(a) h
template <typename T, int N> inline
Jet<T, N> exp(const Jet<T, N>& f) {
const T tmp = exp(f.a);
return Jet<T, N>(tmp, tmp * f.v);
}
// sqrt(a + h) ~= sqrt(a) + h / (2 sqrt(a))
template <typename T, int N> inline
Jet<T, N> sqrt(const Jet<T, N>& f) {
const T tmp = sqrt(f.a);
const T two_a_inverse = T(1.0) / (T(2.0) * tmp);
return Jet<T, N>(tmp, f.v * two_a_inverse);
}
// cos(a + h) ~= cos(a) - sin(a) h
template <typename T, int N> inline
Jet<T, N> cos(const Jet<T, N>& f) {
return Jet<T, N>(cos(f.a), - sin(f.a) * f.v);
}
// acos(a + h) ~= acos(a) - 1 / sqrt(1 - a^2) h
template <typename T, int N> inline
Jet<T, N> acos(const Jet<T, N>& f) {
const T tmp = - T(1.0) / sqrt(T(1.0) - f.a * f.a);
return Jet<T, N>(acos(f.a), tmp * f.v);
}
// sin(a + h) ~= sin(a) + cos(a) h
template <typename T, int N> inline
Jet<T, N> sin(const Jet<T, N>& f) {
return Jet<T, N>(sin(f.a), cos(f.a) * f.v);
}
// asin(a + h) ~= asin(a) + 1 / sqrt(1 - a^2) h
template <typename T, int N> inline
Jet<T, N> asin(const Jet<T, N>& f) {
const T tmp = T(1.0) / sqrt(T(1.0) - f.a * f.a);
return Jet<T, N>(asin(f.a), tmp * f.v);
}
// tan(a + h) ~= tan(a) + (1 + tan(a)^2) h
template <typename T, int N> inline
Jet<T, N> tan(const Jet<T, N>& f) {
const T tan_a = tan(f.a);
const T tmp = T(1.0) + tan_a * tan_a;
return Jet<T, N>(tan_a, tmp * f.v);
}
// atan(a + h) ~= atan(a) + 1 / (1 + a^2) h
template <typename T, int N> inline
Jet<T, N> atan(const Jet<T, N>& f) {
const T tmp = T(1.0) / (T(1.0) + f.a * f.a);
return Jet<T, N>(atan(f.a), tmp * f.v);
}
// sinh(a + h) ~= sinh(a) + cosh(a) h
template <typename T, int N> inline
Jet<T, N> sinh(const Jet<T, N>& f) {
return Jet<T, N>(sinh(f.a), cosh(f.a) * f.v);
}
// cosh(a + h) ~= cosh(a) + sinh(a) h
template <typename T, int N> inline
Jet<T, N> cosh(const Jet<T, N>& f) {
return Jet<T, N>(cosh(f.a), sinh(f.a) * f.v);
}
// tanh(a + h) ~= tanh(a) + (1 - tanh(a)^2) h
template <typename T, int N> inline
Jet<T, N> tanh(const Jet<T, N>& f) {
const T tanh_a = tanh(f.a);
const T tmp = T(1.0) - tanh_a * tanh_a;
return Jet<T, N>(tanh_a, tmp * f.v);
}
// The floor function should be used with extreme care as this operation will
// result in a zero derivative which provides no information to the solver.
//
// floor(a + h) ~= floor(a) + 0
template <typename T, int N> inline
Jet<T, N> floor(const Jet<T, N>& f) {
return Jet<T, N>(floor(f.a));
}
// The ceil function should be used with extreme care as this operation will
// result in a zero derivative which provides no information to the solver.
//
// ceil(a + h) ~= ceil(a) + 0
template <typename T, int N> inline
Jet<T, N> ceil(const Jet<T, N>& f) {
return Jet<T, N>(ceil(f.a));
}
// Some new additions to C++11:
// cbrt(a + h) ~= cbrt(a) + h / (3 a ^ (2/3))
template <typename T, int N> inline
Jet<T, N> cbrt(const Jet<T, N>& f) {
const T derivative = T(1.0) / (T(3.0) * cbrt(f.a * f.a));
return Jet<T, N>(cbrt(f.a), f.v * derivative);
}
// exp2(x + h) = 2^(x+h) ~= 2^x + h*2^x*log(2)
template <typename T, int N> inline
Jet<T, N> exp2(const Jet<T, N>& f) {
const T tmp = exp2(f.a);
const T derivative = tmp * log(T(2));
return Jet<T, N>(tmp, f.v * derivative);
}
// log2(x + h) ~= log2(x) + h / (x * log(2))
template <typename T, int N> inline
Jet<T, N> log2(const Jet<T, N>& f) {
const T derivative = T(1.0) / (f.a * log(T(2)));
return Jet<T, N>(log2(f.a), f.v * derivative);
}
// Like sqrt(x^2 + y^2),
// but acts to prevent underflow/overflow for small/large x/y.
// Note that the function is non-smooth at x=y=0,
// so the derivative is undefined there.
template <typename T, int N> inline
Jet<T, N> hypot(const Jet<T, N>& x, const Jet<T, N>& y) {
// d/da sqrt(a) = 0.5 / sqrt(a)
// d/dx x^2 + y^2 = 2x
// So by the chain rule:
// d/dx sqrt(x^2 + y^2) = 0.5 / sqrt(x^2 + y^2) * 2x = x / sqrt(x^2 + y^2)
// d/dy sqrt(x^2 + y^2) = y / sqrt(x^2 + y^2)
const T tmp = hypot(x.a, y.a);
return Jet<T, N>(tmp, x.a / tmp * x.v + y.a / tmp * y.v);
}
template <typename T, int N> inline
const Jet<T, N>& fmax(const Jet<T, N>& x, const Jet<T, N>& y) {
return x < y ? y : x;
}
template <typename T, int N> inline
const Jet<T, N>& fmin(const Jet<T, N>& x, const Jet<T, N>& y) {
return y < x ? y : x;
}
// Bessel functions of the first kind with integer order equal to 0, 1, n.
//
// Microsoft has deprecated the j[0,1,n]() POSIX Bessel functions in favour of
// _j[0,1,n](). Where available on MSVC, use _j[0,1,n]() to avoid deprecated
// function errors in client code (the specific warning is suppressed when
// Ceres itself is built).
inline double BesselJ0(double x) {
#if defined(CERES_MSVC_USE_UNDERSCORE_PREFIXED_BESSEL_FUNCTIONS)
return _j0(x);
#else
return j0(x);
#endif
}
inline double BesselJ1(double x) {
#if defined(CERES_MSVC_USE_UNDERSCORE_PREFIXED_BESSEL_FUNCTIONS)
return _j1(x);
#else
return j1(x);
#endif
}
inline double BesselJn(int n, double x) {
#if defined(CERES_MSVC_USE_UNDERSCORE_PREFIXED_BESSEL_FUNCTIONS)
return _jn(n, x);
#else
return jn(n, x);
#endif
}
// For the formulae of the derivatives of the Bessel functions see the book:
// Olver, Lozier, Boisvert, Clark, NIST Handbook of Mathematical Functions,
// Cambridge University Press 2010.
//
// Formulae are also available at http://dlmf.nist.gov
// See formula http://dlmf.nist.gov/10.6#E3
// j0(a + h) ~= j0(a) - j1(a) h
template <typename T, int N> inline
Jet<T, N> BesselJ0(const Jet<T, N>& f) {
return Jet<T, N>(BesselJ0(f.a),
-BesselJ1(f.a) * f.v);
}
// See formula http://dlmf.nist.gov/10.6#E1
// j1(a + h) ~= j1(a) + 0.5 ( j0(a) - j2(a) ) h
template <typename T, int N> inline
Jet<T, N> BesselJ1(const Jet<T, N>& f) {
return Jet<T, N>(BesselJ1(f.a),
T(0.5) * (BesselJ0(f.a) - BesselJn(2, f.a)) * f.v);
}
// See formula http://dlmf.nist.gov/10.6#E1
// j_n(a + h) ~= j_n(a) + 0.5 ( j_{n-1}(a) - j_{n+1}(a) ) h
template <typename T, int N> inline
Jet<T, N> BesselJn(int n, const Jet<T, N>& f) {
return Jet<T, N>(BesselJn(n, f.a),
T(0.5) * (BesselJn(n - 1, f.a) - BesselJn(n + 1, f.a)) * f.v);
}
// Jet Classification. It is not clear what the appropriate semantics are for
// these classifications. This picks that std::isfinite and std::isnormal are "all"
// operations, i.e. all elements of the jet must be finite for the jet itself
// to be finite (or normal). For IsNaN and IsInfinite, the answer is less
// clear. This takes a "any" approach for IsNaN and IsInfinite such that if any
// part of a jet is nan or inf, then the entire jet is nan or inf. This leads
// to strange situations like a jet can be both IsInfinite and IsNaN, but in
// practice the "any" semantics are the most useful for e.g. checking that
// derivatives are sane.
// The jet is finite if all parts of the jet are finite.
template <typename T, int N> inline
bool isfinite(const Jet<T, N>& f) {
if (!std::isfinite(f.a)) {
return false;
}
for (int i = 0; i < N; ++i) {
if (!std::isfinite(f.v[i])) {
return false;
}
}
return true;
}
// The jet is infinite if any part of the Jet is infinite.
template <typename T, int N> inline
bool isinf(const Jet<T, N>& f) {
if (std::isinf(f.a)) {
return true;
}
for (int i = 0; i < N; ++i) {
if (std::isinf(f.v[i])) {
return true;
}
}
return false;
}
// The jet is NaN if any part of the jet is NaN.
template <typename T, int N> inline
bool isnan(const Jet<T, N>& f) {
if (std::isnan(f.a)) {
return true;
}
for (int i = 0; i < N; ++i) {
if (std::isnan(f.v[i])) {
return true;
}
}
return false;
}
// The jet is normal if all parts of the jet are normal.
template <typename T, int N> inline
bool isnormal(const Jet<T, N>& f) {
if (!std::isnormal(f.a)) {
return false;
}
for (int i = 0; i < N; ++i) {
if (!std::isnormal(f.v[i])) {
return false;
}
}
return true;
}
// Legacy functions from the pre-C++11 days.
template <typename T, int N>
inline bool IsFinite(const Jet<T, N>& f) {
return isfinite(f);
}
template <typename T, int N>
inline bool IsNaN(const Jet<T, N>& f) {
return isnan(f);
}
template <typename T, int N>
inline bool IsNormal(const Jet<T, N>& f) {
return isnormal(f);
}
// The jet is infinite if any part of the jet is infinite.
template <typename T, int N> inline
bool IsInfinite(const Jet<T, N>& f) {
return isinf(f);
}
// atan2(b + db, a + da) ~= atan2(b, a) + (- b da + a db) / (a^2 + b^2)
//
// In words: the rate of change of theta is 1/r times the rate of
// change of (x, y) in the positive angular direction.
template <typename T, int N> inline
Jet<T, N> atan2(const Jet<T, N>& g, const Jet<T, N>& f) {
// Note order of arguments:
//
// f = a + da
// g = b + db
T const tmp = T(1.0) / (f.a * f.a + g.a * g.a);
return Jet<T, N>(atan2(g.a, f.a), tmp * (- g.a * f.v + f.a * g.v));
}
// pow -- base is a differentiable function, exponent is a constant.
// (a+da)^p ~= a^p + p*a^(p-1) da
template <typename T, int N> inline
Jet<T, N> pow(const Jet<T, N>& f, double g) {
T const tmp = g * pow(f.a, g - T(1.0));
return Jet<T, N>(pow(f.a, g), tmp * f.v);
}
// pow -- base is a constant, exponent is a differentiable function.
// We have various special cases, see the comment for pow(Jet, Jet) for
// analysis:
//
// 1. For f > 0 we have: (f)^(g + dg) ~= f^g + f^g log(f) dg
//
// 2. For f == 0 and g > 0 we have: (f)^(g + dg) ~= f^g
//
// 3. For f < 0 and integer g we have: (f)^(g + dg) ~= f^g but if dg
// != 0, the derivatives are not defined and we return NaN.
template <typename T, int N> inline
Jet<T, N> pow(double f, const Jet<T, N>& g) {
if (f == 0 && g.a > 0) {
// Handle case 2.
return Jet<T, N>(T(0.0));
}
if (f < 0 && g.a == floor(g.a)) {
// Handle case 3.
Jet<T, N> ret(pow(f, g.a));
for (int i = 0; i < N; i++) {
if (g.v[i] != T(0.0)) {
// Return a NaN when g.v != 0.
ret.v[i] = std::numeric_limits<T>::quiet_NaN();
}
}
return ret;
}
// Handle case 1.
T const tmp = pow(f, g.a);
return Jet<T, N>(tmp, log(f) * tmp * g.v);
}
// pow -- both base and exponent are differentiable functions. This has a
// variety of special cases that require careful handling.
//
// 1. For f > 0:
// (f + df)^(g + dg) ~= f^g + f^(g - 1) * (g * df + f * log(f) * dg)
// The numerical evaluation of f * log(f) for f > 0 is well behaved, even for
// extremely small values (e.g. 1e-99).
//
// 2. For f == 0 and g > 1: (f + df)^(g + dg) ~= 0
// This cases is needed because log(0) can not be evaluated in the f > 0
// expression. However the function f*log(f) is well behaved around f == 0
// and its limit as f-->0 is zero.
//
// 3. For f == 0 and g == 1: (f + df)^(g + dg) ~= 0 + df
//
// 4. For f == 0 and 0 < g < 1: The value is finite but the derivatives are not.
//
// 5. For f == 0 and g < 0: The value and derivatives of f^g are not finite.
//
// 6. For f == 0 and g == 0: The C standard incorrectly defines 0^0 to be 1
// "because there are applications that can exploit this definition". We
// (arbitrarily) decree that derivatives here will be nonfinite, since that
// is consistent with the behavior for f == 0, g < 0 and 0 < g < 1.
// Practically any definition could have been justified because mathematical
// consistency has been lost at this point.
//
// 7. For f < 0, g integer, dg == 0: (f + df)^(g + dg) ~= f^g + g * f^(g - 1) df
// This is equivalent to the case where f is a differentiable function and g
// is a constant (to first order).
//
// 8. For f < 0, g integer, dg != 0: The value is finite but the derivatives are
// not, because any change in the value of g moves us away from the point
// with a real-valued answer into the region with complex-valued answers.
//
// 9. For f < 0, g noninteger: The value and derivatives of f^g are not finite.
template <typename T, int N> inline
Jet<T, N> pow(const Jet<T, N>& f, const Jet<T, N>& g) {
if (f.a == 0 && g.a >= 1) {
// Handle cases 2 and 3.
if (g.a > 1) {
return Jet<T, N>(T(0.0));
}
return f;
}
if (f.a < 0 && g.a == floor(g.a)) {
// Handle cases 7 and 8.
T const tmp = g.a * pow(f.a, g.a - T(1.0));
Jet<T, N> ret(pow(f.a, g.a), tmp * f.v);
for (int i = 0; i < N; i++) {
if (g.v[i] != T(0.0)) {
// Return a NaN when g.v != 0.
ret.v[i] = std::numeric_limits<T>::quiet_NaN();
}
}
return ret;
}
// Handle the remaining cases. For cases 4,5,6,9 we allow the log() function
// to generate -HUGE_VAL or NaN, since those cases result in a nonfinite
// derivative.
T const tmp1 = pow(f.a, g.a);
T const tmp2 = g.a * pow(f.a, g.a - T(1.0));
T const tmp3 = tmp1 * log(f.a);
return Jet<T, N>(tmp1, tmp2 * f.v + tmp3 * g.v);
}
// Note: This has to be in the ceres namespace for argument dependent lookup to
// function correctly. Otherwise statements like CHECK_LE(x, 2.0) fail with
// strange compile errors.
template <typename T, int N>
inline std::ostream &operator<<(std::ostream &s, const Jet<T, N>& z) {
s << "[" << z.a << " ; ";
for (int i = 0; i < N; ++i) {
s << z.v[i];
if (i != N - 1) {
s << ", ";
}
}
s << "]";
return s;
}
} // namespace ceres
namespace Eigen {
// Creating a specialization of NumTraits enables placing Jet objects inside
// Eigen arrays, getting all the goodness of Eigen combined with autodiff.
template<typename T, int N>
struct NumTraits<ceres::Jet<T, N>> {
typedef ceres::Jet<T, N> Real;
typedef ceres::Jet<T, N> NonInteger;
typedef ceres::Jet<T, N> Nested;
typedef ceres::Jet<T, N> Literal;
static typename ceres::Jet<T, N> dummy_precision() {
return ceres::Jet<T, N>(1e-12);
}
static inline Real epsilon() {
return Real(std::numeric_limits<T>::epsilon());
}
static inline int digits10() { return NumTraits<T>::digits10(); }
enum {
IsComplex = 0,
IsInteger = 0,
IsSigned,
ReadCost = 1,
AddCost = 1,
// For Jet types, multiplication is more expensive than addition.
MulCost = 3,
HasFloatingPoint = 1,
RequireInitialization = 1
};
template<bool Vectorized>
struct Div {
enum {
#if defined(EIGEN_VECTORIZE_AVX)
AVX = true,
#else
AVX = false,
#endif
// Assuming that for Jets, division is as expensive as
// multiplication.
Cost = 3
};
};
static inline Real highest() { return Real(std::numeric_limits<T>::max()); }
static inline Real lowest() { return Real(-std::numeric_limits<T>::max()); }
};
#if EIGEN_VERSION_AT_LEAST(3, 3, 0)
// Specifying the return type of binary operations between Jets and scalar types
// allows you to perform matrix/array operations with Eigen matrices and arrays
// such as addition, subtraction, multiplication, and division where one Eigen
// matrix/array is of type Jet and the other is a scalar type. This improves
// performance by using the optimized scalar-to-Jet binary operations but
// is only available on Eigen versions >= 3.3
template <typename BinaryOp, typename T, int N>
struct ScalarBinaryOpTraits<ceres::Jet<T, N>, T, BinaryOp> {
typedef ceres::Jet<T, N> ReturnType;
};
template <typename BinaryOp, typename T, int N>
struct ScalarBinaryOpTraits<T, ceres::Jet<T, N>, BinaryOp> {
typedef ceres::Jet<T, N> ReturnType;
};
#endif // EIGEN_VERSION_AT_LEAST(3, 3, 0)
} // namespace Eigen
#endif // CERES_PUBLIC_JET_H_