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diff --git a/extern/ceres/include/ceres/jet.h b/extern/ceres/include/ceres/jet.h new file mode 100644 index 00000000000..a21fd7adb90 --- /dev/null +++ b/extern/ceres/include/ceres/jet.h @@ -0,0 +1,784 @@ +// 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/fpclassify.h" + +namespace ceres { + +template <typename T, int N> +struct Jet { + enum { DIMENSION = N }; + + // 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; + } + + // The scalar part. + T a; + + // The infinitesimal part. + // + // Note the Eigen::DontAlign bit is needed here because this object + // gets allocated on the stack and as part of other arrays and + // structs. Forcing the right alignment there is the source of much + // pain and suffering. Even if that works, passing Jets around to + // functions by value has problems because the C++ ABI does not + // guarantee alignment for function arguments. + // + // Setting the DontAlign bit prevents Eigen from using SSE for the + // various operations on Jets. This is a small performance penalty + // since the AutoDiff code will still expose much of the code as + // statically sized loops to the compiler. But given the subtle + // issues that arise due to alignment, especially when dealing with + // multiple platforms, it seems to be a trade off worth making. + Eigen::Matrix<T, N, 1, Eigen::DontAlign> v; +}; + +// 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 = 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. +// +// TODO(keir): Switch to "using". +inline double abs (double x) { return std::abs(x); } +inline double log (double x) { return std::log(x); } +inline double exp (double x) { return std::exp(x); } +inline double sqrt (double x) { return std::sqrt(x); } +inline double cos (double x) { return std::cos(x); } +inline double acos (double x) { return std::acos(x); } +inline double sin (double x) { return std::sin(x); } +inline double asin (double x) { return std::asin(x); } +inline double tan (double x) { return std::tan(x); } +inline double atan (double x) { return std::atan(x); } +inline double sinh (double x) { return std::sinh(x); } +inline double cosh (double x) { return std::cosh(x); } +inline double tanh (double x) { return std::tanh(x); } +inline double pow (double x, double y) { return std::pow(x, y); } +inline double atan2(double y, double x) { return std::atan2(y, 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); +} + +// Bessel functions of the first kind with integer order equal to 0, 1, n. +inline double BesselJ0(double x) { return j0(x); } +inline double BesselJ1(double x) { return j1(x); } +inline double BesselJn(int n, double x) { return jn(n, x); } + +// 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 IsFinite and 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 (!IsFinite(f.a)) { + return false; + } + for (int i = 0; i < N; ++i) { + if (!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 IsInfinite(const Jet<T, N>& f) { + if (IsInfinite(f.a)) { + return true; + } + for (int i = 0; i < N; i++) { + if (IsInfinite(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 (IsNaN(f.a)) { + return true; + } + for (int i = 0; i < N; ++i) { + if (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 (!IsNormal(f.a)) { + return false; + } + for (int i = 0; i < N; ++i) { + if (!IsNormal(f.v[i])) { + return false; + } + } + return true; +} + +// 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); +} + +// Define the helper functions Eigen needs to embed Jet types. +// +// NOTE(keir): machine_epsilon() and precision() are missing, because they don't +// work with nested template types (e.g. where the scalar is itself templated). +// Among other things, this means that decompositions of Jet's does not work, +// for example +// +// Matrix<Jet<T, N> ... > A, x, b; +// ... +// A.solve(b, &x) +// +// does not work and will fail with a strange compiler error. +// +// TODO(keir): This is an Eigen 2.0 limitation that is lifted in 3.0. When we +// switch to 3.0, also add the rest of the specialization functionality. +template<typename T, int N> inline const Jet<T, N>& ei_conj(const Jet<T, N>& x) { return x; } // NOLINT +template<typename T, int N> inline const Jet<T, N>& ei_real(const Jet<T, N>& x) { return x; } // NOLINT +template<typename T, int N> inline Jet<T, N> ei_imag(const Jet<T, N>& ) { return Jet<T, N>(0.0); } // NOLINT +template<typename T, int N> inline Jet<T, N> ei_abs (const Jet<T, N>& x) { return fabs(x); } // NOLINT +template<typename T, int N> inline Jet<T, N> ei_abs2(const Jet<T, N>& x) { return x * x; } // NOLINT +template<typename T, int N> inline Jet<T, N> ei_sqrt(const Jet<T, N>& x) { return sqrt(x); } // NOLINT +template<typename T, int N> inline Jet<T, N> ei_exp (const Jet<T, N>& x) { return exp(x); } // NOLINT +template<typename T, int N> inline Jet<T, N> ei_log (const Jet<T, N>& x) { return log(x); } // NOLINT +template<typename T, int N> inline Jet<T, N> ei_sin (const Jet<T, N>& x) { return sin(x); } // NOLINT +template<typename T, int N> inline Jet<T, N> ei_cos (const Jet<T, N>& x) { return cos(x); } // NOLINT +template<typename T, int N> inline Jet<T, N> ei_tan (const Jet<T, N>& x) { return tan(x); } // NOLINT +template<typename T, int N> inline Jet<T, N> ei_atan(const Jet<T, N>& x) { return atan(x); } // NOLINT +template<typename T, int N> inline Jet<T, N> ei_sinh(const Jet<T, N>& x) { return sinh(x); } // NOLINT +template<typename T, int N> inline Jet<T, N> ei_cosh(const Jet<T, N>& x) { return cosh(x); } // NOLINT +template<typename T, int N> inline Jet<T, N> ei_tanh(const Jet<T, N>& x) { return tanh(x); } // NOLINT +template<typename T, int N> inline Jet<T, N> ei_pow (const Jet<T, N>& x, Jet<T, N> y) { return pow(x, y); } // NOLINT + +// 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) { + return s << "[" << z.a << " ; " << z.v.transpose() << "]"; +} + +} // 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; + + 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()); + } + + 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 + }; +}; + +} // namespace Eigen + +#endif // CERES_PUBLIC_JET_H_ |