docs/source/gradient_tutorial.rst - ceres-solver - Git at Google

 .. highlight:: c++

 .. default-domain:: cpp

 .. _chapter-gradient_tutorial:

 ==================================
 General Unconstrained Minimization
 ==================================

 Ceres Solver besides being able to solve non-linear least squares
 problem can also solve general unconstrained problems using just their
 objective function value and gradients. In this chapter we will see
 how to do this.

 Rosenbrock's Function
 =====================

 Consider minimizing the famous `Rosenbrock's function
 <http://en.wikipedia.org/wiki/Rosenbrock_function>`_ [#f1]_.

 The simplest way to minimize is to define a templated functor to
 evaluate the objective value of this function and then use Ceres
 Solver's automatic differentiation to compute its derivatives.

 We begin by defining a templated functor and then using
 ``AutoDiffFirstOrderFunction`` to construct an instance of the
 ``FirstOrderFunction`` interface. This is the object that is
 responsible for computing the objective function value and the
 gradient (if required). This is the analog of the
 :class:`CostFunction` when defining non-linear least squares problems
 in Ceres.

 .. code::

   // f(x,y) = (1-x)^2 + 100(y - x^2)^2;
   struct Rosenbrock {
     template <typename T>
     bool operator()(const T* parameters, T* cost) const {
       const T x = parameters[0];
       const T y = parameters[1];
       cost[0] = (1.0 - x) * (1.0 - x) + 100.0 * (y - x * x) * (y - x * x);
       return true;
     }

     static ceres::FirstOrderFunction* Create() {
       constexpr int kNumParameters = 2;
       return new ceres::AutoDiffFirstOrderFunction<Rosenbrock, kNumParameters>(
           new Rosenbrock);
     }
   };


 Minimizing it then is a straightforward matter of constructing a
 :class:`GradientProblem` object and calling :func:`Solve` on it.

 .. code::

     double parameters[2] = {-1.2, 1.0};

     ceres::GradientProblem problem(Rosenbrock::Create());

     ceres::GradientProblemSolver::Options options;
     options.minimizer_progress_to_stdout = true;
     ceres::GradientProblemSolver::Summary summary;
     ceres::Solve(options, problem, parameters, &summary);

     std::cout << summary.FullReport() << "\n";

 Executing this code results, solve the problem using limited memory
 `BFGS
 <http://en.wikipedia.org/wiki/Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm>`_
 algorithm.

 .. code-block:: bash

      0: f: 2.420000e+01 d: 0.00e+00 g: 2.16e+02 h: 0.00e+00 s: 0.00e+00 e:  0 it: 2.29e-05 tt: 2.29e-05
      1: f: 4.280493e+00 d: 1.99e+01 g: 1.52e+01 h: 2.01e-01 s: 8.62e-04 e:  2 it: 8.39e-05 tt: 1.62e-04
      2: f: 3.571154e+00 d: 7.09e-01 g: 1.35e+01 h: 3.78e-01 s: 1.34e-01 e:  3 it: 2.22e-05 tt: 1.91e-04
      3: f: 3.440869e+00 d: 1.30e-01 g: 1.73e+01 h: 1.36e-01 s: 1.00e+00 e:  1 it: 5.01e-06 tt: 2.01e-04
      4: f: 3.213597e+00 d: 2.27e-01 g: 1.55e+01 h: 1.06e-01 s: 4.59e-01 e:  1 it: 3.81e-06 tt: 2.10e-04
      5: f: 2.839723e+00 d: 3.74e-01 g: 1.05e+01 h: 1.34e-01 s: 5.24e-01 e:  1 it: 4.05e-06 tt: 2.19e-04
      6: f: 2.448490e+00 d: 3.91e-01 g: 1.29e+01 h: 3.04e-01 s: 1.00e+00 e:  1 it: 5.01e-06 tt: 2.28e-04
      7: f: 1.943019e+00 d: 5.05e-01 g: 4.00e+00 h: 8.81e-02 s: 7.43e-01 e:  1 it: 4.05e-06 tt: 2.36e-04
      8: f: 1.731469e+00 d: 2.12e-01 g: 7.36e+00 h: 1.71e-01 s: 4.60e-01 e:  2 it: 1.22e-05 tt: 2.52e-04
      9: f: 1.503267e+00 d: 2.28e-01 g: 6.47e+00 h: 8.66e-02 s: 1.00e+00 e:  1 it: 5.96e-06 tt: 2.66e-04
     10: f: 1.228331e+00 d: 2.75e-01 g: 2.00e+00 h: 7.70e-02 s: 7.90e-01 e:  1 it: 4.05e-06 tt: 2.75e-04
     11: f: 1.016523e+00 d: 2.12e-01 g: 5.15e+00 h: 1.39e-01 s: 3.76e-01 e:  2 it: 9.06e-06 tt: 2.88e-04
     12: f: 9.145773e-01 d: 1.02e-01 g: 6.74e+00 h: 7.98e-02 s: 1.00e+00 e:  1 it: 5.01e-06 tt: 2.97e-04
     13: f: 7.508302e-01 d: 1.64e-01 g: 3.88e+00 h: 5.76e-02 s: 4.93e-01 e:  1 it: 5.01e-06 tt: 3.05e-04
     14: f: 5.832378e-01 d: 1.68e-01 g: 5.56e+00 h: 1.42e-01 s: 1.00e+00 e:  1 it: 4.77e-06 tt: 3.13e-04
     15: f: 3.969581e-01 d: 1.86e-01 g: 1.64e+00 h: 1.17e-01 s: 1.00e+00 e:  1 it: 4.05e-06 tt: 3.20e-04
     16: f: 3.171557e-01 d: 7.98e-02 g: 3.84e+00 h: 1.18e-01 s: 3.97e-01 e:  2 it: 8.82e-06 tt: 3.33e-04
     17: f: 2.641257e-01 d: 5.30e-02 g: 3.27e+00 h: 6.14e-02 s: 1.00e+00 e:  1 it: 4.05e-06 tt: 3.42e-04
     18: f: 1.909730e-01 d: 7.32e-02 g: 5.29e-01 h: 8.55e-02 s: 6.82e-01 e:  1 it: 1.00e-05 tt: 4.64e-04
     19: f: 1.472012e-01 d: 4.38e-02 g: 3.11e+00 h: 1.20e-01 s: 3.47e-01 e:  2 it: 1.29e-05 tt: 4.87e-04
     20: f: 1.093558e-01 d: 3.78e-02 g: 2.97e+00 h: 8.43e-02 s: 1.00e+00 e:  1 it: 5.01e-06 tt: 4.97e-04
     21: f: 6.710346e-02 d: 4.23e-02 g: 1.42e+00 h: 9.64e-02 s: 8.85e-01 e:  1 it: 4.05e-06 tt: 5.06e-04
     22: f: 3.993377e-02 d: 2.72e-02 g: 2.30e+00 h: 1.29e-01 s: 4.63e-01 e:  2 it: 1.00e-05 tt: 5.25e-04
     23: f: 2.911794e-02 d: 1.08e-02 g: 2.55e+00 h: 6.55e-02 s: 1.00e+00 e:  1 it: 5.01e-06 tt: 5.34e-04
     24: f: 1.457683e-02 d: 1.45e-02 g: 2.77e-01 h: 6.37e-02 s: 6.14e-01 e:  1 it: 4.05e-06 tt: 5.42e-04
     25: f: 8.577515e-03 d: 6.00e-03 g: 2.86e+00 h: 1.40e-01 s: 1.00e+00 e:  1 it: 3.81e-06 tt: 5.49e-04
     26: f: 3.486574e-03 d: 5.09e-03 g: 1.76e-01 h: 1.23e-02 s: 1.00e+00 e:  1 it: 4.05e-06 tt: 5.57e-04
     27: f: 1.257570e-03 d: 2.23e-03 g: 1.39e-01 h: 5.08e-02 s: 1.00e+00 e:  1 it: 3.81e-06 tt: 5.65e-04
     28: f: 2.783568e-04 d: 9.79e-04 g: 6.20e-01 h: 6.47e-02 s: 1.00e+00 e:  1 it: 4.05e-06 tt: 5.73e-04
     29: f: 2.533399e-05 d: 2.53e-04 g: 1.68e-02 h: 1.98e-03 s: 1.00e+00 e:  1 it: 4.05e-06 tt: 5.81e-04
     30: f: 7.591572e-07 d: 2.46e-05 g: 5.40e-03 h: 9.27e-03 s: 1.00e+00 e:  1 it: 5.96e-06 tt: 6.30e-04
     31: f: 1.902460e-09 d: 7.57e-07 g: 1.62e-03 h: 1.89e-03 s: 1.00e+00 e:  1 it: 4.05e-06 tt: 6.39e-04
     32: f: 1.003030e-12 d: 1.90e-09 g: 3.50e-05 h: 3.52e-05 s: 1.00e+00 e:  1 it: 3.81e-06 tt: 6.47e-04
     33: f: 4.835994e-17 d: 1.00e-12 g: 1.05e-07 h: 1.13e-06 s: 1.00e+00 e:  1 it: 4.05e-06 tt: 6.59e-04
     34: f: 1.885250e-22 d: 4.84e-17 g: 2.69e-10 h: 1.45e-08 s: 1.00e+00 e:  1 it: 4.05e-06 tt: 6.67e-04

   Solver Summary (v 2.0.0-eigen-(3.3.9)-lapack-suitesparse-(5.8.1)-cxsparse-(3.2.0)-acceleratesparse-eigensparse-no_openmp-no_custom_blas)

   Parameters                                  2
   Line search direction              LBFGS (20)
   Line search type                  CUBIC WOLFE


   Cost:
   Initial                          2.420000e+01
   Final                            1.955192e-27
   Change                           2.420000e+01

   Minimizer iterations                       36

   Time (in seconds):

     Cost evaluation                    0.000000 (0)
     Gradient & cost evaluation         0.000008 (44)
     Polynomial minimization            0.000067
   Total                                0.000721

   Termination:                      CONVERGENCE (Parameter tolerance reached. Relative step_norm: 1.890726e-11 <= 1.000000e-08.)

   Initial x: -1.2 y: 1
   Final   x: 1 y: 1


 If you are unable to use automatic differentiation for some reason
 (say beacause you need to call an external library), then you can
 use numeric differentiation. In that case the functor is defined as
 follows [#f2]_.

 .. code::

   // f(x,y) = (1-x)^2 + 100(y - x^2)^2;
   struct Rosenbrock {
     bool operator()(const double* parameters, double* cost) const {
       const double x = parameters[0];
       const double y = parameters[1];
       cost[0] = (1.0 - x) * (1.0 - x) + 100.0 * (y - x * x) * (y - x * x);
       return true;
     }

     static ceres::FirstOrderFunction* Create() {
       constexpr int kNumParameters = 2;
       return new ceres::NumericDiffFirstOrderFunction<Rosenbrock,
                                                       ceres::CENTRAL,
                                                       kNumParameters>(
           new Rosenbrock);
     }
   };

 And finally, if you would rather compute the derivatives by hand (say
 because the size of the parameter vector is too large to be
 automatically differentiated). Then you should define an instance of
 `FirstOrderFunction`, which is the analog of :class:`CostFunction` for
 non-linear least squares problems [#f3]_.

 .. code::

   // f(x,y) = (1-x)^2 + 100(y - x^2)^2;
   class Rosenbrock final  : public ceres::FirstOrderFunction {
     public:
       ~Rosenbrock() override {}

       bool Evaluate(const double* parameters,
                              double* cost,
                              double* gradient) const override {
          const double x = parameters[0];
          const double y = parameters[1];

          cost[0] = (1.0 - x) * (1.0 - x) + 100.0 * (y - x * x) * (y - x * x);
          if (gradient) {
            gradient[0] = -2.0 * (1.0 - x) - 200.0 * (y - x * x) * 2.0 * x;
            gradient[1] = 200.0 * (y - x * x);
          }
         return true;
      }

      int NumParameters() const override { return 2; }
   };

 .. rubric:: Footnotes

 .. [#f1] `examples/rosenbrock.cc
    <https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/rosenbrock.cc>`_

 .. [#f2] `examples/rosenbrock_numeric_diff.cc
    <https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/rosenbrock_numeric_diff.cc>`_

 .. [#f3] `examples/rosenbrock_analytic_diff.cc
    <https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/rosenbrock_analytic_diff.cc>`_
	.. highlight:: c++

	.. default-domain:: cpp

	.. _chapter-gradient_tutorial:

	==================================
	General Unconstrained Minimization
	==================================

	Ceres Solver besides being able to solve non-linear least squares
	problem can also solve general unconstrained problems using just their
	objective function value and gradients. In this chapter we will see
	how to do this.

	Rosenbrock's Function
	=====================

	Consider minimizing the famous `Rosenbrock's function
	<http://en.wikipedia.org/wiki/Rosenbrock_function>`_ [#f1]_.

	The simplest way to minimize is to define a templated functor to
	evaluate the objective value of this function and then use Ceres
	Solver's automatic differentiation to compute its derivatives.

	We begin by defining a templated functor and then using
	``AutoDiffFirstOrderFunction`` to construct an instance of the
	``FirstOrderFunction`` interface. This is the object that is
	responsible for computing the objective function value and the
	gradient (if required). This is the analog of the
	:class:`CostFunction` when defining non-linear least squares problems
	in Ceres.

	.. code::

	// f(x,y) = (1-x)^2 + 100(y - x^2)^2;
	struct Rosenbrock {
	template <typename T>
	bool operator()(const T* parameters, T* cost) const {
	const T x = parameters[0];
	const T y = parameters[1];
	cost[0] = (1.0 - x) * (1.0 - x) + 100.0 * (y - x * x) * (y - x * x);
	return true;
	}

	static ceres::FirstOrderFunction* Create() {
	constexpr int kNumParameters = 2;
	return new ceres::AutoDiffFirstOrderFunction<Rosenbrock, kNumParameters>(
	new Rosenbrock);
	}
	};


	Minimizing it then is a straightforward matter of constructing a
	:class:`GradientProblem` object and calling :func:`Solve` on it.

	.. code::

	double parameters[2] = {-1.2, 1.0};

	ceres::GradientProblem problem(Rosenbrock::Create());

	ceres::GradientProblemSolver::Options options;
	options.minimizer_progress_to_stdout = true;
	ceres::GradientProblemSolver::Summary summary;
	ceres::Solve(options, problem, parameters, &summary);

	std::cout << summary.FullReport() << "\n";

	Executing this code results, solve the problem using limited memory
	`BFGS
	<http://en.wikipedia.org/wiki/Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm>`_
	algorithm.

	.. code-block:: bash

	0: f: 2.420000e+01 d: 0.00e+00 g: 2.16e+02 h: 0.00e+00 s: 0.00e+00 e: 0 it: 2.29e-05 tt: 2.29e-05
	1: f: 4.280493e+00 d: 1.99e+01 g: 1.52e+01 h: 2.01e-01 s: 8.62e-04 e: 2 it: 8.39e-05 tt: 1.62e-04
	2: f: 3.571154e+00 d: 7.09e-01 g: 1.35e+01 h: 3.78e-01 s: 1.34e-01 e: 3 it: 2.22e-05 tt: 1.91e-04
	3: f: 3.440869e+00 d: 1.30e-01 g: 1.73e+01 h: 1.36e-01 s: 1.00e+00 e: 1 it: 5.01e-06 tt: 2.01e-04
	4: f: 3.213597e+00 d: 2.27e-01 g: 1.55e+01 h: 1.06e-01 s: 4.59e-01 e: 1 it: 3.81e-06 tt: 2.10e-04
	5: f: 2.839723e+00 d: 3.74e-01 g: 1.05e+01 h: 1.34e-01 s: 5.24e-01 e: 1 it: 4.05e-06 tt: 2.19e-04
	6: f: 2.448490e+00 d: 3.91e-01 g: 1.29e+01 h: 3.04e-01 s: 1.00e+00 e: 1 it: 5.01e-06 tt: 2.28e-04
	7: f: 1.943019e+00 d: 5.05e-01 g: 4.00e+00 h: 8.81e-02 s: 7.43e-01 e: 1 it: 4.05e-06 tt: 2.36e-04
	8: f: 1.731469e+00 d: 2.12e-01 g: 7.36e+00 h: 1.71e-01 s: 4.60e-01 e: 2 it: 1.22e-05 tt: 2.52e-04
	9: f: 1.503267e+00 d: 2.28e-01 g: 6.47e+00 h: 8.66e-02 s: 1.00e+00 e: 1 it: 5.96e-06 tt: 2.66e-04
	10: f: 1.228331e+00 d: 2.75e-01 g: 2.00e+00 h: 7.70e-02 s: 7.90e-01 e: 1 it: 4.05e-06 tt: 2.75e-04
	11: f: 1.016523e+00 d: 2.12e-01 g: 5.15e+00 h: 1.39e-01 s: 3.76e-01 e: 2 it: 9.06e-06 tt: 2.88e-04
	12: f: 9.145773e-01 d: 1.02e-01 g: 6.74e+00 h: 7.98e-02 s: 1.00e+00 e: 1 it: 5.01e-06 tt: 2.97e-04
	13: f: 7.508302e-01 d: 1.64e-01 g: 3.88e+00 h: 5.76e-02 s: 4.93e-01 e: 1 it: 5.01e-06 tt: 3.05e-04
	14: f: 5.832378e-01 d: 1.68e-01 g: 5.56e+00 h: 1.42e-01 s: 1.00e+00 e: 1 it: 4.77e-06 tt: 3.13e-04
	15: f: 3.969581e-01 d: 1.86e-01 g: 1.64e+00 h: 1.17e-01 s: 1.00e+00 e: 1 it: 4.05e-06 tt: 3.20e-04
	16: f: 3.171557e-01 d: 7.98e-02 g: 3.84e+00 h: 1.18e-01 s: 3.97e-01 e: 2 it: 8.82e-06 tt: 3.33e-04
	17: f: 2.641257e-01 d: 5.30e-02 g: 3.27e+00 h: 6.14e-02 s: 1.00e+00 e: 1 it: 4.05e-06 tt: 3.42e-04
	18: f: 1.909730e-01 d: 7.32e-02 g: 5.29e-01 h: 8.55e-02 s: 6.82e-01 e: 1 it: 1.00e-05 tt: 4.64e-04
	19: f: 1.472012e-01 d: 4.38e-02 g: 3.11e+00 h: 1.20e-01 s: 3.47e-01 e: 2 it: 1.29e-05 tt: 4.87e-04
	20: f: 1.093558e-01 d: 3.78e-02 g: 2.97e+00 h: 8.43e-02 s: 1.00e+00 e: 1 it: 5.01e-06 tt: 4.97e-04
	21: f: 6.710346e-02 d: 4.23e-02 g: 1.42e+00 h: 9.64e-02 s: 8.85e-01 e: 1 it: 4.05e-06 tt: 5.06e-04
	22: f: 3.993377e-02 d: 2.72e-02 g: 2.30e+00 h: 1.29e-01 s: 4.63e-01 e: 2 it: 1.00e-05 tt: 5.25e-04
	23: f: 2.911794e-02 d: 1.08e-02 g: 2.55e+00 h: 6.55e-02 s: 1.00e+00 e: 1 it: 5.01e-06 tt: 5.34e-04
	24: f: 1.457683e-02 d: 1.45e-02 g: 2.77e-01 h: 6.37e-02 s: 6.14e-01 e: 1 it: 4.05e-06 tt: 5.42e-04
	25: f: 8.577515e-03 d: 6.00e-03 g: 2.86e+00 h: 1.40e-01 s: 1.00e+00 e: 1 it: 3.81e-06 tt: 5.49e-04
	26: f: 3.486574e-03 d: 5.09e-03 g: 1.76e-01 h: 1.23e-02 s: 1.00e+00 e: 1 it: 4.05e-06 tt: 5.57e-04
	27: f: 1.257570e-03 d: 2.23e-03 g: 1.39e-01 h: 5.08e-02 s: 1.00e+00 e: 1 it: 3.81e-06 tt: 5.65e-04
	28: f: 2.783568e-04 d: 9.79e-04 g: 6.20e-01 h: 6.47e-02 s: 1.00e+00 e: 1 it: 4.05e-06 tt: 5.73e-04
	29: f: 2.533399e-05 d: 2.53e-04 g: 1.68e-02 h: 1.98e-03 s: 1.00e+00 e: 1 it: 4.05e-06 tt: 5.81e-04
	30: f: 7.591572e-07 d: 2.46e-05 g: 5.40e-03 h: 9.27e-03 s: 1.00e+00 e: 1 it: 5.96e-06 tt: 6.30e-04
	31: f: 1.902460e-09 d: 7.57e-07 g: 1.62e-03 h: 1.89e-03 s: 1.00e+00 e: 1 it: 4.05e-06 tt: 6.39e-04
	32: f: 1.003030e-12 d: 1.90e-09 g: 3.50e-05 h: 3.52e-05 s: 1.00e+00 e: 1 it: 3.81e-06 tt: 6.47e-04
	33: f: 4.835994e-17 d: 1.00e-12 g: 1.05e-07 h: 1.13e-06 s: 1.00e+00 e: 1 it: 4.05e-06 tt: 6.59e-04
	34: f: 1.885250e-22 d: 4.84e-17 g: 2.69e-10 h: 1.45e-08 s: 1.00e+00 e: 1 it: 4.05e-06 tt: 6.67e-04

	Solver Summary (v 2.0.0-eigen-(3.3.9)-lapack-suitesparse-(5.8.1)-cxsparse-(3.2.0)-acceleratesparse-eigensparse-no_openmp-no_custom_blas)

	Parameters 2
	Line search direction LBFGS (20)
	Line search type CUBIC WOLFE


	Cost:
	Initial 2.420000e+01
	Final 1.955192e-27
	Change 2.420000e+01

	Minimizer iterations 36

	Time (in seconds):

	Cost evaluation 0.000000 (0)
	Gradient & cost evaluation 0.000008 (44)
	Polynomial minimization 0.000067
	Total 0.000721

	Termination: CONVERGENCE (Parameter tolerance reached. Relative step_norm: 1.890726e-11 <= 1.000000e-08.)

	Initial x: -1.2 y: 1
	Final x: 1 y: 1


	If you are unable to use automatic differentiation for some reason
	(say beacause you need to call an external library), then you can
	use numeric differentiation. In that case the functor is defined as
	follows [#f2]_.

	.. code::

	// f(x,y) = (1-x)^2 + 100(y - x^2)^2;
	struct Rosenbrock {
	bool operator()(const double* parameters, double* cost) const {
	const double x = parameters[0];
	const double y = parameters[1];
	cost[0] = (1.0 - x) * (1.0 - x) + 100.0 * (y - x * x) * (y - x * x);
	return true;
	}

	static ceres::FirstOrderFunction* Create() {
	constexpr int kNumParameters = 2;
	return new ceres::NumericDiffFirstOrderFunction<Rosenbrock,
	ceres::CENTRAL,
	kNumParameters>(
	new Rosenbrock);
	}
	};

	And finally, if you would rather compute the derivatives by hand (say
	because the size of the parameter vector is too large to be
	automatically differentiated). Then you should define an instance of
	`FirstOrderFunction`, which is the analog of :class:`CostFunction` for
	non-linear least squares problems [#f3]_.

	.. code::

	// f(x,y) = (1-x)^2 + 100(y - x^2)^2;
	class Rosenbrock final : public ceres::FirstOrderFunction {
	public:
	~Rosenbrock() override {}

	bool Evaluate(const double* parameters,
	double* cost,
	double* gradient) const override {
	const double x = parameters[0];
	const double y = parameters[1];

	cost[0] = (1.0 - x) * (1.0 - x) + 100.0 * (y - x * x) * (y - x * x);
	if (gradient) {
	gradient[0] = -2.0 * (1.0 - x) - 200.0 * (y - x * x) * 2.0 * x;
	gradient[1] = 200.0 * (y - x * x);
	}
	return true;
	}

	int NumParameters() const override { return 2; }
	};

	.. rubric:: Footnotes

	.. [#f1] `examples/rosenbrock.cc
	<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/rosenbrock.cc>`_

	.. [#f2] `examples/rosenbrock_numeric_diff.cc
	<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/rosenbrock_numeric_diff.cc>`_

	.. [#f3] `examples/rosenbrock_analytic_diff.cc
	<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/rosenbrock_analytic_diff.cc>`_