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This is an implementation of so-called downhill simplex method
(https://en.wikipedia.org/wiki/Nelder%E2%80%93Mead_method)

Please, let me know if you have any comments, whoever you'd be.
This commit is contained in:
Alex Leontiev
2013-08-01 20:42:59 +08:00
parent fc37df05ec
commit 554e002747
7 changed files with 575 additions and 7 deletions
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164
modules/optim/doc/downhill_simplex_method.rst Normal file
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Downhill Simplex Method
=======================
.. highlight:: cpp
optim::DownhillSolver
---------------------------------
.. ocv:class:: optim::DownhillSolver
This class is used to perform the non-linear non-constrained *minimization* of a function, given on an *n*-dimensional Euclidean space,
using the **Nelder-Mead method**, also known as **downhill simplex method**. The basic idea about the method can be obtained from
(`http://en.wikipedia.org/wiki/Nelder-Mead\_method <http://en.wikipedia.org/wiki/Nelder-Mead_method>`_). It should be noted, that
this method, although deterministic, is rather a heuristic and therefore may converge to a local minima, not necessary a global one.
It is iterative optimization technique, which at each step uses an information about the values of a function evaluated only at
*n+1* points, arranged as a *simplex* in *n*-dimensional space (hence the second name of the method). At each step new point is
chosen to evaluate function at, obtained value is compared with previous ones and based on this information simplex changes it's shape
, slowly moving to the local minimum.
Algorithm stops when the number of function evaluations done exceeds ``termcrit.maxCount``, when the function values at the
vertices of simplex are within ``termcrit.epsilon`` range or simplex becomes so small that it
can enclosed in a box with ``termcrit.epsilon`` sides, whatever comes first, for some defined by user
positive integer ``termcrit.maxCount`` and positive non-integer ``termcrit.epsilon``.
::
class CV_EXPORTS Solver : public Algorithm
{
public:
class CV_EXPORTS Function
{
public:
virtual ~Function() {}
//! ndim - dimensionality
virtual double calc(const double* x) const = 0;
};
virtual Ptr<Function> getFunction() const = 0;
virtual void setFunction(const Ptr<Function>& f) = 0;
virtual TermCriteria getTermCriteria() const = 0;
virtual void setTermCriteria(const TermCriteria& termcrit) = 0;
// x contain the initial point before the call and the minima position (if algorithm converged) after. x is assumed to be (something that
// after getMat() will return) row-vector or column-vector. *It's size and should
// be consisted with previous dimensionality data given, if any (otherwise, it determines dimensionality)*
virtual double minimize(InputOutputArray x) = 0;
};
class CV_EXPORTS DownhillSolver : public Solver
{
public:
//! returns row-vector, even if the column-vector was given
virtual void getInitStep(OutputArray step) const=0;
//!This should be called at least once before the first call to minimize() and step is assumed to be (something that
//! after getMat() will return) row-vector or column-vector. *It's dimensionality determines the dimensionality of a problem.*
virtual void setInitStep(InputArray step)=0;
};
It should be noted, that ``optim::DownhillSolver`` is a derivative of the abstract interface ``optim::Solver``, which in
turn is derived from the ``Algorithm`` interface and is used to encapsulate the functionality, common to all non-linear optimization
algorithms in the ``optim`` module.
optim::DownhillSolver::getFunction
--------------------------------------------
Getter for the optimized function. The optimized function is represented by ``Solver::Function`` interface, which requires
derivatives to implement the sole method ``calc(double*)`` to evaluate the function.
.. ocv:function:: Ptr<Solver::Function> optim::DownhillSolver::getFunction()
:return: Smart-pointer to an object that implements ``Solver::Function`` interface - it represents the function that is being optimized. It can be empty, if no function was given so far.
optim::DownhillSolver::setFunction
-----------------------------------------------
Setter for the optimized function. *It should be called at least once before the call to* ``DownhillSolver::minimize()``, as
default value is not usable.
.. ocv:function:: void optim::DownhillSolver::setFunction(const Ptr<Solver::Function>& f)
:param f: The new function to optimize.
optim::DownhillSolver::getTermCriteria
----------------------------------------------------
Getter for the previously set terminal criteria for this algorithm.
.. ocv:function:: TermCriteria optim::DownhillSolver::getTermCriteria()
:return: Deep copy of the terminal criteria used at the moment.
optim::DownhillSolver::setTermCriteria
------------------------------------------
Set terminal criteria for downhill simplex method. Two things should be noted. First, this method *is not necessary* to be called
before the first call to ``DownhillSolver::minimize()``, as the default value is sensible. Second, the method will raise an error
if ``termcrit.type!=(TermCriteria::MAX_ITER+TermCriteria::EPS)``, ``termcrit.epsilon<=0`` or ``termcrit.maxCount<=0``. That is,
both ``epsilon`` and ``maxCount`` should be set to positive values (non-integer and integer respectively) and they represent
tolerance and maximal number of function evaluations that is allowed.
Algorithm stops when the number of function evaluations done exceeds ``termcrit.maxCount``, when the function values at the
vertices of simplex are within ``termcrit.epsilon`` range or simplex becomes so small that it
can enclosed in a box with ``termcrit.epsilon`` sides, whatever comes first.
.. ocv:function:: void optim::DownhillSolver::setTermCriteria(const TermCriteria& termcrit)
:param termcrit: Terminal criteria to be used, represented as ``TermCriteria`` structure (defined elsewhere in openCV). Mind you, that it should meet ``(termcrit.type==(TermCriteria::MAX_ITER+TermCriteria::EPS) && termcrit.epsilon>0 && termcrit.maxCount>0)``, otherwise the error will be raised.
optim::DownhillSolver::getInitStep
-----------------------------------
Returns the initial step that will be used in downhill simplex algorithm. See the description
of corresponding setter (follows next) for the meaning of this parameter.
.. ocv:function:: void optim::getInitStep(OutputArray step)
:param step: Initial step that will be used in algorithm. Note, that although corresponding setter accepts column-vectors as well as row-vectors, this method will return a row-vector.
optim::DownhillSolver::setInitStep
----------------------------------
Sets the initial step that will be used in downhill simplex algorithm. Step, together with initial point (givin in ``DownhillSolver::minimize``)
are two *n*-dimensional vectors that are used to determine the shape of initial simplex. Roughly said, initial point determines the position
of a simplex (it will become simplex's centroid), while step determines the spread (size in each dimension) of a simplex. To be more precise,
if :math:`s,x_0\in\mathbb{R}^n` are the initial step and initial point respectively, the vertices of a simplex will be: :math:`v_0:=x_0-\frac{1}{2}
s` and :math:`v_i:=x_0+s_i` for :math:`i=1,2,\dots,n` where :math:`s_i` denotes projections of the initial step of *n*-th coordinate (the result
of projection is treated to be vector given by :math:`s_i:=e_i\cdot\left<e_i\cdot s\right>`, where :math:`e_i` form canonical basis)
.. ocv:function:: void optim::setInitStep(InputArray step)
:param step: Initial step that will be used in algorithm. Roughly said, it determines the spread (size in each dimension) of an initial simplex.
optim::DownhillSolver::minimize
-----------------------------------
The main method of the ``DownhillSolver``. It actually runs the algorithm and performs the minimization. The sole input parameter determines the
centroid of the starting simplex (roughly, it tells where to start), all the others (terminal criteria, initial step, function to be minimized)
are supposed to be set via the setters before the call to this method or the default values (not always sensible) will be used.
.. ocv:function:: double optim::DownhillSolver::minimize(InputOutputArray x)
:param x: The initial point, that will become a centroid of an initial simplex. After the algorithm will terminate, it will be setted to the
point where the algorithm stops, the point of possible minimum.
:return: The value of a function at the point found.
Explain parameters.
optim::createDownhillSolver
------------------------------------
This function returns the reference to the ready-to-use ``DownhillSolver`` object. All the parameters are optional, so this procedure can be called
even without parameters at all. In this case, the default values will be used. As default value for terminal criteria are the only sensible ones,
``DownhillSolver::setFunction()`` and ``DownhillSolver::setInitStep()`` should be called upon the obtained object, if the respective parameters
were not given to ``createDownhillSolver()``. Otherwise, the two ways (give parameters to ``createDownhillSolver()`` or miss the out and call the
``DownhillSolver::setFunction()`` and ``DownhillSolver::setInitStep()``) are absolutely equivalent (and will drop the same errors in the same way,
should invalid input be detected).
.. ocv:function:: Ptr<optim::DownhillSolver> optim::createDownhillSolver(const Ptr<Solver::Function>& f,InputArray initStep, TermCriteria termcrit)
:param f: Pointer to the function that will be minimized, similarly to the one you submit via ``DownhillSolver::setFunction``.
:param step: Initial step, that will be used to construct the initial simplex, similarly to the one you submit via ``DownhillSolver::setInitStep``.
:param termcrit: Terminal criteria to the algorithm, similarly to the one you submit via ``DownhillSolver::setTermCriteria``.

1
modules/optim/doc/optim.rst
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@@ -8,3 +8,4 @@ optim. Generic numerical optimization
:maxdepth: 2
linear_programming
downhill_simplex_method

39
modules/optim/include/opencv2/optim.hpp
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@@ -47,6 +47,45 @@
namespace cv{namespace optim
{
class CV_EXPORTS Solver : public Algorithm
{
public:
class CV_EXPORTS Function
{
public:
virtual ~Function() {}
//! ndim - dimensionality
virtual double calc(const double* x) const = 0;
};
virtual Ptr<Function> getFunction() const = 0;
virtual void setFunction(const Ptr<Function>& f) = 0;
virtual TermCriteria getTermCriteria() const = 0;
virtual void setTermCriteria(const TermCriteria& termcrit) = 0;
// x contain the initial point before the call and the minima position (if algorithm converged) after. x is assumed to be (something that
// after getMat() will return) row-vector or column-vector. *It's size and should
// be consisted with previous dimensionality data given, if any (otherwise, it determines dimensionality)*
virtual double minimize(InputOutputArray x) = 0;
};
//! downhill simplex class
class CV_EXPORTS DownhillSolver : public Solver
{
public:
//! returns row-vector, even if the column-vector was given
virtual void getInitStep(OutputArray step) const=0;
//!This should be called at least once before the first call to minimize() and step is assumed to be (something that
//! after getMat() will return) row-vector or column-vector. *It's dimensionality determines the dimensionality of a problem.*
virtual void setInitStep(InputArray step)=0;
};
// both minRange & minError are specified by termcrit.epsilon; In addition, user may specify the number of iterations that the algorithm does.
CV_EXPORTS_W Ptr<DownhillSolver> createDownhillSolver(const Ptr<Solver::Function>& f=Ptr<Solver::Function>(),
InputArray initStep=Mat_<double>(1,1,0.0),
TermCriteria termcrit=TermCriteria(TermCriteria::MAX_ITER+TermCriteria::EPS,5000,0.000001));
//!the return codes for solveLP() function
enum
{

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modules/optim/src/debug.hpp Normal file
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namespace cv{namespace optim{
#ifdef ALEX_DEBUG
#define dprintf(x) printf x
static void print_matrix(const Mat& x){
printf("\ttype:%d vs %d,\tsize: %d-on-%d\n",x.type(),CV_64FC1,x.rows,x.cols);
for(int i=0;i<x.rows;i++){
printf("\t[");
for(int j=0;j<x.cols;j++){
printf("%g, ",x.at<double>(i,j));
}
printf("]\n");
}
}
#else
#define dprintf(x)
#define print_matrix(x)
#endif
}}

8
modules/optim/src/lpsolver.cpp
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@@ -2,16 +2,12 @@
#include <climits>
#include <algorithm>
#include <cstdarg>
#include <debug.hpp>
namespace cv{namespace optim{
using std::vector;
#ifdef ALEX_DEBUG
#define dprintf(x) printf x
static void print_matrix(const Mat& x){
print(x);
printf("\n");
}
static void print_simplex_state(const Mat& c,const Mat& b,double v,const std::vector<int> N,const std::vector<int> B){
printf("\tprint simplex state\n");
@@ -32,8 +28,6 @@ static void print_simplex_state(const Mat& c,const Mat& b,double v,const std::ve
printf("\n");
}
#else
#define dprintf(x)
#define print_matrix(x)
#define print_simplex_state(c,b,v,N,B)
#endif

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modules/optim/src/simplex.cpp Normal file
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#include "precomp.hpp"
#include "debug.hpp"
namespace cv{namespace optim{
class DownhillSolverImpl : public DownhillSolver
{
public:
void getInitStep(OutputArray step) const;
void setInitStep(InputArray step);
Ptr<Function> getFunction() const;
void setFunction(const Ptr<Function>& f);
TermCriteria getTermCriteria() const;
DownhillSolverImpl();
void setTermCriteria(const TermCriteria& termcrit);
double minimize(InputOutputArray x);
protected:
Ptr<Solver::Function> _Function;
TermCriteria _termcrit;
Mat _step;
private:
inline void compute_coord_sum(Mat_<double>& points,Mat_<double>& coord_sum);
inline void create_initial_simplex(Mat_<double>& simplex,Mat& step);
inline double inner_downhill_simplex(cv::Mat_<double>& p,double MinRange,double MinError,int& nfunk,
const Ptr<Solver::Function>& f,int nmax);
inline double try_new_point(Mat_<double>& p,Mat_<double>& y,Mat_<double>& coord_sum,const Ptr<Solver::Function>& f,int ihi,
double fac,Mat_<double>& ptry);
};
double DownhillSolverImpl::try_new_point(
Mat_<double>& p,
Mat_<double>& y,
Mat_<double>& coord_sum,
const Ptr<Solver::Function>& f,
int ihi,
double fac,
Mat_<double>& ptry
)
{
int ndim=p.cols;
int j;
double fac1,fac2,ytry;
fac1=(1.0-fac)/ndim;
fac2=fac1-fac;
for (j=0;j<ndim;j++)
{
ptry(j)=coord_sum(j)*fac1-p(ihi,j)*fac2;
}
ytry=f->calc((double*)ptry.data);
if (ytry < y(ihi))
{
y(ihi)=ytry;
for (j=0;j<ndim;j++)
{
coord_sum(j) += ptry(j)-p(ihi,j);
p(ihi,j)=ptry(j);
}
}
return ytry;
}
/*
Performs the actual minimization of Solver::Function f (after the initialization was done)
The matrix p[ndim+1][1..ndim] represents ndim+1 vertices that
form a simplex - each row is an ndim vector.
On output, nfunk gives the number of function evaluations taken.
*/
double DownhillSolverImpl::inner_downhill_simplex(
cv::Mat_<double>& p,
double MinRange,
double MinError,
int& nfunk,
const Ptr<Solver::Function>& f,
int nmax
)
{
int ndim=p.cols;
double res;
int i,ihi,ilo,inhi,j,mpts=ndim+1;
double error, range,ysave,ytry;
Mat_<double> coord_sum(1,ndim,0.0),buf(1,ndim,0.0),y(1,ndim,0.0);
nfunk = 0;
for(i=0;i<ndim+1;++i)
{
y(i) = f->calc(p[i]);
}
nfunk = ndim+1;
compute_coord_sum(p,coord_sum);
for (;;)
{
ilo=0;
/* find highest (worst), next-to-worst, and lowest
(best) points by going through all of them. */
ihi = y(0)>y(1) ? (inhi=1,0) : (inhi=0,1);
for (i=0;i<mpts;i++)
{
if (y(i) <= y(ilo))
ilo=i;
if (y(i) > y(ihi))
{
inhi=ihi;
ihi=i;
}
else if (y(i) > y(inhi) && i != ihi)
inhi=i;
}
/* check stop criterion */
error=fabs(y(ihi)-y(ilo));
range=0;
for(i=0;i<ndim;++i)
{
double min = p(0,i);
double max = p(0,i);
double d;
for(j=1;j<=ndim;++j)
{
if( min > p(j,i) ) min = p(j,i);
if( max < p(j,i) ) max = p(j,i);
}
d = fabs(max-min);
if(range < d) range = d;
}
if(range <= MinRange || error <= MinError)
{ /* Put best point and value in first slot. */
std::swap(y(0),y(ilo));
for (i=0;i<ndim;i++)
{
std::swap(p(0,i),p(ilo,i));
}
break;
}
if (nfunk >= nmax){
dprintf(("nmax exceeded\n"));
return y(ilo);
}
nfunk += 2;
/*Begin a new iteration. First, reflect the worst point about the centroid of others */
ytry = try_new_point(p,y,coord_sum,f,ihi,-1.0,buf);
if (ytry <= y(ilo))
{ /*If that's better than the best point, go twice as far in that direction*/
ytry = try_new_point(p,y,coord_sum,f,ihi,2.0,buf);
}
else if (ytry >= y(inhi))
{ /* The new point is worse than the second-highest, but better
than the worst so do not go so far in that direction */
ysave = y(ihi);
ytry = try_new_point(p,y,coord_sum,f,ihi,0.5,buf);
if (ytry >= ysave)
{ /* Can't seem to improve things. Contract the simplex to good point
in hope to find a simplex landscape. */
for (i=0;i<mpts;i++)
{
if (i != ilo)
{
for (j=0;j<ndim;j++)
{
p(i,j) = coord_sum(j) = 0.5*(p(i,j)+p(ilo,j));
}
y(i)=f->calc((double*)coord_sum.data);
}
}
nfunk += ndim;
compute_coord_sum(p,coord_sum);
}
} else --(nfunk); /* correct nfunk */
dprintf(("this is simplex on iteration %d\n",nfunk));
print_matrix(p);
} /* go to next iteration. */
res = y(0);
return res;
}
void DownhillSolverImpl::compute_coord_sum(Mat_<double>& points,Mat_<double>& coord_sum){
for (int j=0;j<points.cols;j++) {
double sum=0.0;
for (int i=0;i<points.rows;i++){
sum += points(i,j);
coord_sum(0,j)=sum;
}
}
}
void DownhillSolverImpl::create_initial_simplex(Mat_<double>& simplex,Mat& step){
for(int i=1;i<=step.cols;++i)
{
simplex.row(0).copyTo(simplex.row(i));
simplex(i,i-1)+= 0.5*step.at<double>(0,i-1);
}
simplex.row(0) -= 0.5*step;
dprintf(("this is simplex\n"));
print_matrix(simplex);
}
double DownhillSolverImpl::minimize(InputOutputArray x){
dprintf(("hi from minimize\n"));
CV_Assert(_Function.empty()==false);
dprintf(("termcrit:\n\ttype: %d\n\tmaxCount: %d\n\tEPS: %g\n",_termcrit.type,_termcrit.maxCount,_termcrit.epsilon));
dprintf(("step\n"));
print_matrix(_step);
Mat x_mat=x.getMat();
CV_Assert(MIN(x_mat.rows,x_mat.cols)==1);
CV_Assert(MAX(x_mat.rows,x_mat.cols)==_step.cols);
CV_Assert(x_mat.type()==CV_64FC1);
Mat_<double> proxy_x;
if(x_mat.rows>1){
proxy_x=x_mat.t();
}else{
proxy_x=x_mat;
}
int count=0;
int ndim=_step.cols;
Mat_<double> simplex=Mat_<double>(ndim+1,ndim,0.0);
simplex.row(0).copyTo(proxy_x);
create_initial_simplex(simplex,_step);
double res = inner_downhill_simplex(
simplex,_termcrit.epsilon, _termcrit.epsilon, count,_Function,_termcrit.maxCount);
simplex.row(0).copyTo(proxy_x);
dprintf(("%d iterations done\n",count));
if(x_mat.rows>1){
Mat(x_mat.rows, 1, CV_64F, (double*)proxy_x.data).copyTo(x);
}
return res;
}
DownhillSolverImpl::DownhillSolverImpl(){
_Function=Ptr<Function>();
_step=Mat_<double>();
}
Ptr<Solver::Function> DownhillSolverImpl::getFunction()const{
return _Function;
}
void DownhillSolverImpl::setFunction(const Ptr<Function>& f){
_Function=f;
}
TermCriteria DownhillSolverImpl::getTermCriteria()const{
return _termcrit;
}
void DownhillSolverImpl::setTermCriteria(const TermCriteria& termcrit){
CV_Assert(termcrit.type==(TermCriteria::MAX_ITER+TermCriteria::EPS) && termcrit.epsilon>0 && termcrit.maxCount>0);
_termcrit=termcrit;
}
// both minRange & minError are specified by termcrit.epsilon; In addition, user may specify the number of iterations that the algorithm does.
Ptr<DownhillSolver> createDownhillSolver(const Ptr<Solver::Function>& f, InputArray initStep, TermCriteria termcrit){
DownhillSolver *DS=new DownhillSolverImpl();
DS->setFunction(f);
DS->setInitStep(initStep);
DS->setTermCriteria(termcrit);
return Ptr<DownhillSolver>(DS);
}
void DownhillSolverImpl::getInitStep(OutputArray step)const{
step.create(1,_step.cols,CV_64FC1);
_step.copyTo(step);
}
void DownhillSolverImpl::setInitStep(InputArray step){
//set dimensionality and make a deep copy of step
Mat m=step.getMat();
dprintf(("m.cols=%d\nm.rows=%d\n",m.cols,m.rows));
CV_Assert(MIN(m.cols,m.rows)==1 && m.type()==CV_64FC1);
int ndim=MAX(m.cols,m.rows);
if(ndim!=_step.cols){
_step=Mat_<double>(1,ndim);
}
if(m.rows==1){
m.copyTo(_step);
}else{
Mat step_t=Mat_<double>(ndim,1,(double*)_step.data);
m.copyTo(step_t);
}
}
}}

63
modules/optim/test/test_downhill_simplex.cpp Normal file
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#include "test_precomp.hpp"
#include <cstdlib>
#include <cmath>
#include <algorithm>
static void mytest(cv::Ptr<cv::optim::DownhillSolver> solver,cv::Ptr<cv::optim::Solver::Function> ptr_F,cv::Mat& x,cv::Mat& step,
cv::Mat& etalon_x,double etalon_res){
solver->setFunction(ptr_F);
int ndim=MAX(step.cols,step.rows);
solver->setInitStep(step);
cv::Mat settedStep;
solver->getInitStep(settedStep);
ASSERT_TRUE(settedStep.rows==1 && settedStep.cols==ndim);
ASSERT_TRUE(std::equal(step.begin<double>(),step.end<double>(),settedStep.begin<double>()));
std::cout<<"step setted:\n\t"<<step<<std::endl;
double res=solver->minimize(x);
std::cout<<"res:\n\t"<<res<<std::endl;
std::cout<<"x:\n\t"<<x<<std::endl;
std::cout<<"etalon_res:\n\t"<<etalon_res<<std::endl;
std::cout<<"etalon_x:\n\t"<<etalon_x<<std::endl;
double tol=solver->getTermCriteria().epsilon;
ASSERT_TRUE(std::abs(res-etalon_res)<tol);
/*for(cv::Mat_<double>::iterator it1=x.begin<double>(),it2=etalon_x.begin<double>();it1!=x.end<double>();it1++,it2++){
ASSERT_TRUE(std::abs((*it1)-(*it2))<tol);
}*/
std::cout<<"--------------------------\n";
}
class SphereF:public cv::optim::Solver::Function{
public:
double calc(const double* x)const{
return x[0]*x[0]+x[1]*x[1];
}
};
class RosenbrockF:public cv::optim::Solver::Function{
double calc(const double* x)const{
return 100*(x[1]-x[0]*x[0])*(x[1]-x[0]*x[0])+(1-x[0])*(1-x[0]);
}
};
TEST(Optim_Downhill, regression_basic){
cv::Ptr<cv::optim::DownhillSolver> solver=cv::optim::createDownhillSolver();
#if 1
{
cv::Ptr<cv::optim::Solver::Function> ptr_F(new SphereF());
cv::Mat x=(cv::Mat_<double>(1,2)<<1.0,1.0),
step=(cv::Mat_<double>(2,1)<<-0.5,-0.5),
etalon_x=(cv::Mat_<double>(1,2)<<-0.0,0.0);
double etalon_res=0.0;
mytest(solver,ptr_F,x,step,etalon_x,etalon_res);
}
#endif
#if 1
{
cv::Ptr<cv::optim::Solver::Function> ptr_F(new RosenbrockF());
cv::Mat x=(cv::Mat_<double>(2,1)<<0.0,0.0),
step=(cv::Mat_<double>(2,1)<<0.5,+0.5),
etalon_x=(cv::Mat_<double>(2,1)<<1.0,1.0);
double etalon_res=0.0;
mytest(solver,ptr_F,x,step,etalon_x,etalon_res);
}
#endif
}