include/ceres/internal/autodiff.h - ceres-solver - Git at Google

 // Ceres Solver - A fast non-linear least squares minimizer
 // Copyright 2010, 2011, 2012 Google Inc. All rights reserved.
 // http://code.google.com/p/ceres-solver/
 //
 // 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)
 //
 // Computation of the Jacobian matrix for vector-valued functions of multiple
 // variables, using automatic differentiation based on the implementation of
 // dual numbers in jet.h. Before reading the rest of this file, it is adivsable
 // to read jet.h's header comment in detail.
 //
 // The helper wrapper AutoDiff::Differentiate() computes the jacobian of
 // functors with templated operator() taking this form:
 //
 //   struct F {
 //     template<typename T>
 //     bool operator(const T *x, const T *y, ..., T *z) {
 //       // Compute z[] based on x[], y[], ...
 //       // return true if computation succeeded, false otherwise.
 //     }
 //   };
 //
 // All inputs and outputs may be vector-valued.
 //
 // To understand how jets are used to compute the jacobian, a
 // picture may help. Consider a vector-valued function, F, returning 3
 // dimensions and taking a vector-valued parameter of 4 dimensions:
 //
 //     y            x
 //   [ * ]    F   [ * ]
 //   [ * ]  <---  [ * ]
 //   [ * ]        [ * ]
 //                [ * ]
 //
 // Similar to the 2-parameter example for f described in jet.h, computing the
 // jacobian dy/dx is done by substutiting a suitable jet object for x and all
 // intermediate steps of the computation of F. Since x is has 4 dimensions, use
 // a Jet<double, 4>.
 //
 // Before substituting a jet object for x, the dual components are set
 // appropriately for each dimension of x:
 //
 //          y                       x
 //   [ * | * * * * ]    f   [ * | 1 0 0 0 ]   x0
 //   [ * | * * * * ]  <---  [ * | 0 1 0 0 ]   x1
 //   [ * | * * * * ]        [ * | 0 0 1 0 ]   x2
 //         ---+---          [ * | 0 0 0 1 ]   x3
 //            |                   ^ ^ ^ ^
 //          dy/dx                 | | | +----- infinitesimal for x3
 //                                | | +------- infinitesimal for x2
 //                                | +--------- infinitesimal for x1
 //                                +----------- infinitesimal for x0
 //
 // The reason to set the internal 4x4 submatrix to the identity is that we wish
 // to take the derivative of y separately with respect to each dimension of x.
 // Each column of the 4x4 identity is therefore for a single component of the
 // independent variable x.
 //
 // Then the jacobian of the mapping, dy/dx, is the 3x4 sub-matrix of the
 // extended y vector, indicated in the above diagram.
 //
 // Functors with multiple parameters
 // ---------------------------------
 // In practice, it is often convenient to use a function f of two or more
 // vector-valued parameters, for example, x[3] and z[6]. Unfortunately, the jet
 // framework is designed for a single-parameter vector-valued input. The wrapper
 // in this file addresses this issue adding support for functions with one or
 // more parameter vectors.
 //
 // To support multiple parameters, all the parameter vectors are concatenated
 // into one and treated as a single parameter vector, except that since the
 // functor expects different inputs, we need to construct the jets as if they
 // were part of a single parameter vector. The extended jets are passed
 // separately for each parameter.
 //
 // For example, consider a functor F taking two vector parameters, p[2] and
 // q[3], and producing an output y[4]:
 //
 //   struct F {
 //     template<typename T>
 //     bool operator(const T *p, const T *q, T *z) {
 //       // ...
 //     }
 //   };
 //
 // In this case, the necessary jet type is Jet<double, 5>. Here is a
 // visualization of the jet objects in this case:
 //
 //          Dual components for p ----+
 //                                    |
 //                                   -+-
 //           y                 [ * | 1 0 | 0 0 0 ]    --- p[0]
 //                             [ * | 0 1 | 0 0 0 ]    --- p[1]
 //   [ * | . . | + + + ]         |
 //   [ * | . . | + + + ]         v
 //   [ * | . . | + + + ]  <--- F(p, q)
 //   [ * | . . | + + + ]            ^
 //         ^^^   ^^^^^              |
 //        dy/dp  dy/dq            [ * | 0 0 | 1 0 0 ] --- q[0]
 //                                [ * | 0 0 | 0 1 0 ] --- q[1]
 //                                [ * | 0 0 | 0 0 1 ] --- q[2]
 //                                            --+--
 //                                              |
 //          Dual components for q --------------+
 //
 // where the 4x2 submatrix (marked with ".") and 4x3 submatrix (marked with "+"
 // of y in the above diagram are the derivatives of y with respect to p and q
 // respectively. This is how autodiff works for functors taking multiple vector
 // valued arguments (up to 6).
 //
 // Jacobian NULL pointers
 // ----------------------
 // In general, the functions below will accept NULL pointers for all or some of
 // the Jacobian parameters, meaning that those Jacobians will not be computed.

 #ifndef CERES_PUBLIC_INTERNAL_AUTODIFF_H_
 #define CERES_PUBLIC_INTERNAL_AUTODIFF_H_

 #include <stddef.h>

 #include <glog/logging.h>
 #include "ceres/jet.h"
 #include "ceres/internal/fixed_array.h"

 namespace ceres {
 namespace internal {

 // Extends src by a 1st order pertubation for every dimension and puts it in
 // dst. The size of src is N. Since this is also used for perturbations in
 // blocked arrays, offset is used to shift which part of the jet the
 // perturbation occurs. This is used to set up the extended x augmented by an
 // identity matrix. The JetT type should be a Jet type, and T should be a
 // numeric type (e.g. double). For example,
 //
 //             0   1 2   3 4 5   6 7 8
 //   dst[0]  [ * | . . | 1 0 0 | . . . ]
 //   dst[1]  [ * | . . | 0 1 0 | . . . ]
 //   dst[2]  [ * | . . | 0 0 1 | . . . ]
 //
 // is what would get put in dst if N was 3, offset was 3, and the jet type JetT
 // was 8-dimensional.
 template <typename JetT, typename T>
 inline void Make1stOrderPerturbation(int offset, int N, const T *src,
                                      JetT *dst) {
   DCHECK(src);
   DCHECK(dst);
   for (int j = 0; j < N; ++j) {
     dst[j] = JetT(src[j], offset + j);
   }
 }

 // Takes the 0th order part of src, assumed to be a Jet type, and puts it in
 // dst. This is used to pick out the "vector" part of the extended y.
 template <typename JetT, typename T>
 inline void Take0thOrderPart(int M, const JetT *src, T dst) {
   DCHECK(src);
   for (int i = 0; i < M; ++i) {
     dst[i] = src[i].a;
   }
 }

 // Takes N 1st order parts, starting at index N0, and puts them in the M x N
 // matrix 'dst'. This is used to pick out the "matrix" parts of the extended y.
 template <typename JetT, typename T, int M, int N0, int N>
 inline void Take1stOrderPart(const JetT *src, T *dst) {
   DCHECK(src);
   DCHECK(dst);
   // TODO(keir): Change Jet to use a single array, where v[0] is the
   // non-infinitesimal part rather than "a". That way it's possible to use a
   // single memcpy or eigen operation, rather than the explicit loop. The loop
   // doesn't exploit any SSE or other intrinsics.
   for (int i = 0; i < M; ++i) {
     for (int j = 0; j < N; ++j) {
       dst[N * i + j] = src[i].v[N0 + j];
     }
   }
 }

 // This block of quasi-repeated code calls the user-supplied functor, which may
 // take a variable number of arguments. This is accomplished by specializing the
 // struct based on the size of the trailing parameters; parameters with 0 size
 // are assumed missing.
 //
 // Supporting variadic functions is the primary source of complexity in the
 // autodiff implementation.

 template<typename Functor, typename T, int kNumOutputs,
          int N0, int N1, int N2, int N3, int N4, int N5>
 struct VariadicEvaluate {
   static bool Call(const Functor& functor, T const *const *input, T* output) {
     return functor(input[0],
                    input[1],
                    input[2],
                    input[3],
                    input[4],
                    input[5],
                    output);
   }
 };

 template<typename Functor, typename T, int kNumOutputs,
          int N0, int N1, int N2, int N3, int N4>
 struct VariadicEvaluate<Functor, T, kNumOutputs, N0, N1, N2, N3, N4, 0> {
   static bool Call(const Functor& functor, T const *const *input, T* output) {
     return functor(input[0],
                    input[1],
                    input[2],
                    input[3],
                    input[4],
                    output);
   }
 };

 template<typename Functor, typename T, int kNumOutputs,
          int N0, int N1, int N2, int N3>
 struct VariadicEvaluate<Functor, T, kNumOutputs, N0, N1, N2, N3, 0, 0> {
   static bool Call(const Functor& functor, T const *const *input, T* output) {
     return functor(input[0],
                    input[1],
                    input[2],
                    input[3],
                    output);
   }
 };

 template<typename Functor, typename T, int kNumOutputs,
          int N0, int N1, int N2>
 struct VariadicEvaluate<Functor, T, kNumOutputs, N0, N1, N2, 0, 0, 0> {
   static bool Call(const Functor& functor, T const *const *input, T* output) {
     return functor(input[0],
                    input[1],
                    input[2],
                    output);
   }
 };

 template<typename Functor, typename T, int kNumOutputs,
          int N0, int N1>
 struct VariadicEvaluate<Functor, T, kNumOutputs, N0, N1, 0, 0, 0, 0> {
   static bool Call(const Functor& functor, T const *const *input, T* output) {
     return functor(input[0],
                    input[1],
                    output);
   }
 };

 template<typename Functor, typename T, int kNumOutputs, int N0>
 struct VariadicEvaluate<Functor, T, kNumOutputs, N0, 0, 0, 0, 0, 0> {
   static bool Call(const Functor& functor, T const *const *input, T* output) {
     return functor(input[0],
                    output);
   }
 };

 // This is in a struct because default template parameters on a function are not
 // supported in C++03 (though it is available in C++0x). N0 through N5 are the
 // dimension of the input arguments to the user supplied functor.
 template <typename Functor, typename T, int kNumOutputs,
           int N0 = 0, int N1 = 0, int N2 = 0, int N3 = 0, int N4 = 0, int N5=0>
 struct AutoDiff {
   static bool Differentiate(const Functor& functor,
                             T const *const *parameters,
                             T *function_value,
                             T **jacobians) {
     typedef Jet<T, N0 + N1 + N2 + N3 + N4 + N5> JetT;

     DCHECK_GT(N0, 0)
         << "Cost functions must have at least one parameter block.";
     DCHECK((!N1 && !N2 && !N3 && !N4 && !N5) ||
            ((N1 > 0) && !N2 && !N3 && !N4 && !N5) ||
            ((N1 > 0) && (N2 > 0) && !N3 && !N4 && !N5) ||
            ((N1 > 0) && (N2 > 0) && (N3 > 0) && !N4 && !N5) ||
            ((N1 > 0) && (N2 > 0) && (N3 > 0) && (N4 > 0) && !N5) ||
            ((N1 > 0) && (N2 > 0) && (N3 > 0) && (N4 > 0) && (N5 > 0)))
         << "Zero block cannot precede a non-zero block. Block sizes are "
         << "(ignore trailing 0s): " << N0 << ", " << N1 << ", " << N2 << ", "
         << N3 << ", " << N4 << ", " << N5;

     DCHECK_GT(kNumOutputs, 0);

     FixedArray<JetT, (256 * 7) / sizeof(JetT)> x(
         N0 + N1 + N2 + N3 + N4 + N5 + kNumOutputs);

     // It's ugly, but it works.
     const int jet0 = 0;
     const int jet1 = N0;
     const int jet2 = N0 + N1;
     const int jet3 = N0 + N1 + N2;
     const int jet4 = N0 + N1 + N2 + N3;
     const int jet5 = N0 + N1 + N2 + N3 + N4;
     const int jet6 = N0 + N1 + N2 + N3 + N4 + N5;

     const JetT *unpacked_parameters[6] = {
         x.get() + jet0,
         x.get() + jet1,
         x.get() + jet2,
         x.get() + jet3,
         x.get() + jet4,
         x.get() + jet5,
     };
     JetT *output = x.get() + jet6;

 #define CERES_MAKE_1ST_ORDER_PERTURBATION(i) \
     if (N ## i) { \
       internal::Make1stOrderPerturbation(jet ## i, \
                                          N ## i, \
                                          parameters[i], \
                                          x.get() + jet ## i); \
     }
     CERES_MAKE_1ST_ORDER_PERTURBATION(0);
     CERES_MAKE_1ST_ORDER_PERTURBATION(1);
     CERES_MAKE_1ST_ORDER_PERTURBATION(2);
     CERES_MAKE_1ST_ORDER_PERTURBATION(3);
     CERES_MAKE_1ST_ORDER_PERTURBATION(4);
     CERES_MAKE_1ST_ORDER_PERTURBATION(5);
 #undef CERES_MAKE_1ST_ORDER_PERTURBATION

     if (!VariadicEvaluate<Functor, JetT, kNumOutputs,
                           N0, N1, N2, N3, N4, N5>::Call(
         functor, unpacked_parameters, output)) {
       return false;
     }

     internal::Take0thOrderPart(kNumOutputs, output, function_value);

 #define CERES_TAKE_1ST_ORDER_PERTURBATION(i) \
     if (N ## i) { \
       if (jacobians[i]) { \
         internal::Take1stOrderPart<JetT, T, \
                                    kNumOutputs, \
                                    jet ## i, \
                                    N ## i>(output, \
                                           jacobians[i]); \
       } \
     }
     CERES_TAKE_1ST_ORDER_PERTURBATION(0);
     CERES_TAKE_1ST_ORDER_PERTURBATION(1);
     CERES_TAKE_1ST_ORDER_PERTURBATION(2);
     CERES_TAKE_1ST_ORDER_PERTURBATION(3);
     CERES_TAKE_1ST_ORDER_PERTURBATION(4);
     CERES_TAKE_1ST_ORDER_PERTURBATION(5);
 #undef CERES_TAKE_1ST_ORDER_PERTURBATION
     return true;
   }
 };

 }  // namespace internal
 }  // namespace ceres

 #endif  // CERES_PUBLIC_INTERNAL_AUTODIFF_H_
	// Ceres Solver - A fast non-linear least squares minimizer
	// Copyright 2010, 2011, 2012 Google Inc. All rights reserved.
	// http://code.google.com/p/ceres-solver/
	//
	// 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)
	//
	// Computation of the Jacobian matrix for vector-valued functions of multiple
	// variables, using automatic differentiation based on the implementation of
	// dual numbers in jet.h. Before reading the rest of this file, it is adivsable
	// to read jet.h's header comment in detail.
	//
	// The helper wrapper AutoDiff::Differentiate() computes the jacobian of
	// functors with templated operator() taking this form:
	//
	// struct F {
	// template<typename T>
	// bool operator(const T x, const T y, ..., T *z) {
	// // Compute z[] based on x[], y[], ...
	// // return true if computation succeeded, false otherwise.
	// }
	// };
	//
	// All inputs and outputs may be vector-valued.
	//
	// To understand how jets are used to compute the jacobian, a
	// picture may help. Consider a vector-valued function, F, returning 3
	// dimensions and taking a vector-valued parameter of 4 dimensions:
	//
	// y x
	// [ * ] F [ * ]
	// [ * ] <--- [ * ]
	// [ * ] [ * ]
	// [ * ]
	//
	// Similar to the 2-parameter example for f described in jet.h, computing the
	// jacobian dy/dx is done by substutiting a suitable jet object for x and all
	// intermediate steps of the computation of F. Since x is has 4 dimensions, use
	// a Jet<double, 4>.
	//
	// Before substituting a jet object for x, the dual components are set
	// appropriately for each dimension of x:
	//
	// y x
	// [ * \| * * * * ] f [ * \| 1 0 0 0 ] x0
	// [ * \| * * * * ] <--- [ * \| 0 1 0 0 ] x1
	// [ * \| * * * * ] [ * \| 0 0 1 0 ] x2
	// ---+--- [ * \| 0 0 0 1 ] x3
	// \| ^ ^ ^ ^
	// dy/dx \| \| \| +----- infinitesimal for x3
	// \| \| +------- infinitesimal for x2
	// \| +--------- infinitesimal for x1
	// +----------- infinitesimal for x0
	//
	// The reason to set the internal 4x4 submatrix to the identity is that we wish
	// to take the derivative of y separately with respect to each dimension of x.
	// Each column of the 4x4 identity is therefore for a single component of the
	// independent variable x.
	//
	// Then the jacobian of the mapping, dy/dx, is the 3x4 sub-matrix of the
	// extended y vector, indicated in the above diagram.
	//
	// Functors with multiple parameters
	// ---------------------------------
	// In practice, it is often convenient to use a function f of two or more
	// vector-valued parameters, for example, x[3] and z[6]. Unfortunately, the jet
	// framework is designed for a single-parameter vector-valued input. The wrapper
	// in this file addresses this issue adding support for functions with one or
	// more parameter vectors.
	//
	// To support multiple parameters, all the parameter vectors are concatenated
	// into one and treated as a single parameter vector, except that since the
	// functor expects different inputs, we need to construct the jets as if they
	// were part of a single parameter vector. The extended jets are passed
	// separately for each parameter.
	//
	// For example, consider a functor F taking two vector parameters, p[2] and
	// q[3], and producing an output y[4]:
	//
	// struct F {
	// template<typename T>
	// bool operator(const T p, const T q, T *z) {
	// // ...
	// }
	// };
	//
	// In this case, the necessary jet type is Jet<double, 5>. Here is a
	// visualization of the jet objects in this case:
	//
	// Dual components for p ----+
	// \|
	// -+-
	// y [ * \| 1 0 \| 0 0 0 ] --- p[0]
	// [ * \| 0 1 \| 0 0 0 ] --- p[1]
	// [ * \| . . \| + + + ] \|
	// [ * \| . . \| + + + ] v
	// [ * \| . . \| + + + ] <--- F(p, q)
	// [ * \| . . \| + + + ] ^
	// ^^^ ^^^^^ \|
	// dy/dp dy/dq [ * \| 0 0 \| 1 0 0 ] --- q[0]
	// [ * \| 0 0 \| 0 1 0 ] --- q[1]
	// [ * \| 0 0 \| 0 0 1 ] --- q[2]
	// --+--
	// \|
	// Dual components for q --------------+
	//
	// where the 4x2 submatrix (marked with ".") and 4x3 submatrix (marked with "+"
	// of y in the above diagram are the derivatives of y with respect to p and q
	// respectively. This is how autodiff works for functors taking multiple vector
	// valued arguments (up to 6).
	//
	// Jacobian NULL pointers
	// ----------------------
	// In general, the functions below will accept NULL pointers for all or some of
	// the Jacobian parameters, meaning that those Jacobians will not be computed.

	#ifndef CERES_PUBLIC_INTERNAL_AUTODIFF_H_
	#define CERES_PUBLIC_INTERNAL_AUTODIFF_H_

	#include <stddef.h>

	#include <glog/logging.h>
	#include "ceres/jet.h"
	#include "ceres/internal/fixed_array.h"

	namespace ceres {
	namespace internal {

	// Extends src by a 1st order pertubation for every dimension and puts it in
	// dst. The size of src is N. Since this is also used for perturbations in
	// blocked arrays, offset is used to shift which part of the jet the
	// perturbation occurs. This is used to set up the extended x augmented by an
	// identity matrix. The JetT type should be a Jet type, and T should be a
	// numeric type (e.g. double). For example,
	//
	// 0 1 2 3 4 5 6 7 8
	// dst[0] [ * \| . . \| 1 0 0 \| . . . ]
	// dst[1] [ * \| . . \| 0 1 0 \| . . . ]
	// dst[2] [ * \| . . \| 0 0 1 \| . . . ]
	//
	// is what would get put in dst if N was 3, offset was 3, and the jet type JetT
	// was 8-dimensional.
	template <typename JetT, typename T>
	inline void Make1stOrderPerturbation(int offset, int N, const T *src,
	JetT *dst) {
	DCHECK(src);
	DCHECK(dst);
	for (int j = 0; j < N; ++j) {
	dst[j] = JetT(src[j], offset + j);
	}
	}

	// Takes the 0th order part of src, assumed to be a Jet type, and puts it in
	// dst. This is used to pick out the "vector" part of the extended y.
	template <typename JetT, typename T>
	inline void Take0thOrderPart(int M, const JetT *src, T dst) {
	DCHECK(src);
	for (int i = 0; i < M; ++i) {
	dst[i] = src[i].a;
	}
	}

	// Takes N 1st order parts, starting at index N0, and puts them in the M x N
	// matrix 'dst'. This is used to pick out the "matrix" parts of the extended y.
	template <typename JetT, typename T, int M, int N0, int N>
	inline void Take1stOrderPart(const JetT src, T dst) {
	DCHECK(src);
	DCHECK(dst);
	// TODO(keir): Change Jet to use a single array, where v[0] is the
	// non-infinitesimal part rather than "a". That way it's possible to use a
	// single memcpy or eigen operation, rather than the explicit loop. The loop
	// doesn't exploit any SSE or other intrinsics.
	for (int i = 0; i < M; ++i) {
	for (int j = 0; j < N; ++j) {
	dst[N * i + j] = src[i].v[N0 + j];
	}
	}
	}

	// This block of quasi-repeated code calls the user-supplied functor, which may
	// take a variable number of arguments. This is accomplished by specializing the
	// struct based on the size of the trailing parameters; parameters with 0 size
	// are assumed missing.
	//
	// Supporting variadic functions is the primary source of complexity in the
	// autodiff implementation.

	template<typename Functor, typename T, int kNumOutputs,
	int N0, int N1, int N2, int N3, int N4, int N5>
	struct VariadicEvaluate {
	static bool Call(const Functor& functor, T const const input, T* output) {
	return functor(input[0],
	input[1],
	input[2],
	input[3],
	input[4],
	input[5],
	output);
	}
	};

	template<typename Functor, typename T, int kNumOutputs,
	int N0, int N1, int N2, int N3, int N4>
	struct VariadicEvaluate<Functor, T, kNumOutputs, N0, N1, N2, N3, N4, 0> {
	static bool Call(const Functor& functor, T const const input, T* output) {
	return functor(input[0],
	input[1],
	input[2],
	input[3],
	input[4],
	output);
	}
	};

	template<typename Functor, typename T, int kNumOutputs,
	int N0, int N1, int N2, int N3>
	struct VariadicEvaluate<Functor, T, kNumOutputs, N0, N1, N2, N3, 0, 0> {
	static bool Call(const Functor& functor, T const const input, T* output) {
	return functor(input[0],
	input[1],
	input[2],
	input[3],
	output);
	}
	};

	template<typename Functor, typename T, int kNumOutputs,
	int N0, int N1, int N2>
	struct VariadicEvaluate<Functor, T, kNumOutputs, N0, N1, N2, 0, 0, 0> {
	static bool Call(const Functor& functor, T const const input, T* output) {
	return functor(input[0],
	input[1],
	input[2],
	output);
	}
	};

	template<typename Functor, typename T, int kNumOutputs,
	int N0, int N1>
	struct VariadicEvaluate<Functor, T, kNumOutputs, N0, N1, 0, 0, 0, 0> {
	static bool Call(const Functor& functor, T const const input, T* output) {
	return functor(input[0],
	input[1],
	output);
	}
	};

	template<typename Functor, typename T, int kNumOutputs, int N0>
	struct VariadicEvaluate<Functor, T, kNumOutputs, N0, 0, 0, 0, 0, 0> {
	static bool Call(const Functor& functor, T const const input, T* output) {
	return functor(input[0],
	output);
	}
	};

	// This is in a struct because default template parameters on a function are not
	// supported in C++03 (though it is available in C++0x). N0 through N5 are the
	// dimension of the input arguments to the user supplied functor.
	template <typename Functor, typename T, int kNumOutputs,
	int N0 = 0, int N1 = 0, int N2 = 0, int N3 = 0, int N4 = 0, int N5=0>
	struct AutoDiff {
	static bool Differentiate(const Functor& functor,
	T const const parameters,
	T *function_value,
	T **jacobians) {
	typedef Jet<T, N0 + N1 + N2 + N3 + N4 + N5> JetT;

	DCHECK_GT(N0, 0)
	<< "Cost functions must have at least one parameter block.";
	DCHECK((!N1 && !N2 && !N3 && !N4 && !N5) \|\|
	((N1 > 0) && !N2 && !N3 && !N4 && !N5) \|\|
	((N1 > 0) && (N2 > 0) && !N3 && !N4 && !N5) \|\|
	((N1 > 0) && (N2 > 0) && (N3 > 0) && !N4 && !N5) \|\|
	((N1 > 0) && (N2 > 0) && (N3 > 0) && (N4 > 0) && !N5) \|\|
	((N1 > 0) && (N2 > 0) && (N3 > 0) && (N4 > 0) && (N5 > 0)))
	<< "Zero block cannot precede a non-zero block. Block sizes are "
	<< "(ignore trailing 0s): " << N0 << ", " << N1 << ", " << N2 << ", "
	<< N3 << ", " << N4 << ", " << N5;

	DCHECK_GT(kNumOutputs, 0);

	FixedArray<JetT, (256 * 7) / sizeof(JetT)> x(
	N0 + N1 + N2 + N3 + N4 + N5 + kNumOutputs);

	// It's ugly, but it works.
	const int jet0 = 0;
	const int jet1 = N0;
	const int jet2 = N0 + N1;
	const int jet3 = N0 + N1 + N2;
	const int jet4 = N0 + N1 + N2 + N3;
	const int jet5 = N0 + N1 + N2 + N3 + N4;
	const int jet6 = N0 + N1 + N2 + N3 + N4 + N5;

	const JetT *unpacked_parameters[6] = {
	x.get() + jet0,
	x.get() + jet1,
	x.get() + jet2,
	x.get() + jet3,
	x.get() + jet4,
	x.get() + jet5,
	};
	JetT *output = x.get() + jet6;

	#define CERES_MAKE_1ST_ORDER_PERTURBATION(i) \
	if (N ## i) { \
	internal::Make1stOrderPerturbation(jet ## i, \
	N ## i, \
	parameters[i], \
	x.get() + jet ## i); \
	}
	CERES_MAKE_1ST_ORDER_PERTURBATION(0);
	CERES_MAKE_1ST_ORDER_PERTURBATION(1);
	CERES_MAKE_1ST_ORDER_PERTURBATION(2);
	CERES_MAKE_1ST_ORDER_PERTURBATION(3);
	CERES_MAKE_1ST_ORDER_PERTURBATION(4);
	CERES_MAKE_1ST_ORDER_PERTURBATION(5);
	#undef CERES_MAKE_1ST_ORDER_PERTURBATION

	if (!VariadicEvaluate<Functor, JetT, kNumOutputs,
	N0, N1, N2, N3, N4, N5>::Call(
	functor, unpacked_parameters, output)) {
	return false;
	}

	internal::Take0thOrderPart(kNumOutputs, output, function_value);

	#define CERES_TAKE_1ST_ORDER_PERTURBATION(i) \
	if (N ## i) { \
	if (jacobians[i]) { \
	internal::Take1stOrderPart<JetT, T, \
	kNumOutputs, \
	jet ## i, \
	N ## i>(output, \
	jacobians[i]); \
	} \
	}
	CERES_TAKE_1ST_ORDER_PERTURBATION(0);
	CERES_TAKE_1ST_ORDER_PERTURBATION(1);
	CERES_TAKE_1ST_ORDER_PERTURBATION(2);
	CERES_TAKE_1ST_ORDER_PERTURBATION(3);
	CERES_TAKE_1ST_ORDER_PERTURBATION(4);
	CERES_TAKE_1ST_ORDER_PERTURBATION(5);
	#undef CERES_TAKE_1ST_ORDER_PERTURBATION
	return true;
	}
	};

	} // namespace internal
	} // namespace ceres

	#endif // CERES_PUBLIC_INTERNAL_AUTODIFF_H_