minpack.cpp

//
// Created by Администратор on 21.01.2018.
//

#include "minpack.h"

# include <cmath>
# include <cstdlib>
# include <ctime>
# include <iomanip>
# include <iostream>

using namespace std;


//****************************************************************************80

void chkder ( int m, int n, double x[], double fvec[], double fjac[],
              int ldfjac, double xp[], double fvecp[], int mode, double err[] )

//****************************************************************************80
//
//  Purpose:
//
//    CHKDER checks the gradients of M functions in N variables.
//
//  Discussion:
//
//    This function checks the gradients of M nonlinear functions
//    in N variables, evaluated at a point x, for consistency with
//    the functions themselves.  The user must call chkder twice,
//    first with mode = 1 and then with mode = 2.
//
//    mode = 1: on input, x must contain the point of evaluation.
//    on output, xp is set to a neighboring point.
//
//    mode = 2. on input, fvec must contain the functions and the
//    rows of fjac must contain the gradients
//    of the respective functions each evaluated
//    at x, and fvecp must contain the functions
//    evaluated at xp.
//    on output, err contains measures of correctness of
//    the respective gradients.
//
//    The function does not perform reliably if cancellation or
//    rounding errors cause a severe loss of significance in the
//    evaluation of a function. therefore, none of the components
//    of x should be unusually small (in particular, zero) or any
//    other value which may cause loss of significance.
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    05 April 2010
//
//  Author:
//
//    Original FORTRAN77 version by Jorge More, Burt Garbow, Ken Hillstrom.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Jorge More, Burton Garbow, Kenneth Hillstrom,
//    User Guide for MINPACK-1,
//    Technical Report ANL-80-74,
//    Argonne National Laboratory, 1980.
//
//  Parameters:
//
//    Input, int M, the number of functions.
//
//    Input, int N, the number of variables.
//
//    Input, double X[N], the point at which the jacobian is to be checked.
//
//       fvec is an array of length m. on input when mode = 2,
//         fvec must contain the functions evaluated at x.
//
//       fjac is an m by n array. on input when mode = 2,
//         the rows of fjac must contain the gradients of
//         the respective functions evaluated at x.
//
//       ldfjac is a positive integer input parameter not less than m
//         which specifies the leading dimension of the array fjac.
//
//       xp is an array of length n. on output when mode = 1,
//         xp is set to a neighboring point of x.
//
//       fvecp is an array of length m. on input when mode = 2,
//         fvecp must contain the functions evaluated at xp.
//
//       mode is an integer input variable set to 1 on the first call
//         and 2 on the second. other values of mode are equivalent
//         to mode = 1.
//
//       err is an array of length m. on output when mode = 2,
//         err contains measures of correctness of the respective
//         gradients. if there is no severe loss of significance,
//         then if err(i) is 1.0 the i-th gradient is correct,
//         while if err(i) is 0.0 the i-th gradient is incorrect.
//         for values of err between 0.0 and 1.0, the categorization
//         is less certain. in general, a value of err(i) greater
//         than 0.5 indicates that the i-th gradient is probably
//         correct, while a value of err(i) less than 0.5 indicates
//         that the i-th gradient is probably incorrect.
//
{
    double eps;
    double epsf;
    double epslog;
    double epsmch;
    const double factor = 100.0;
    int i;
    int j;
    double temp;
//
//  EPSMCH is the machine precision.
//
    epsmch = r8_epsilon ( );
//
    eps = sqrt ( epsmch );
//
//  MODE = 1.
//
    if ( mode == 1 )
    {
        for ( j = 0; j < n; j++ )
        {
            if ( x[j] == 0.0 )
            {
                temp = eps;
            }
            else
            {
                temp = eps * fabs ( x[j] );
            }
            xp[j] = x[j] + temp;
        }
    }
//
//  MODE = 2.
//
    else
    {
        epsf = factor * epsmch;
        epslog = log10 ( eps );
        for ( i = 0; i < m; i++ )
        {
            err[i] = 0.0;
        }

        for ( j = 0; j < n; j++ )
        {
            if ( x[j] == 0.0 )
            {
                temp = 1.0;
            }
            else
            {
                temp = fabs ( x[j] );
            }
            for ( i = 0; i < m; i++ )
            {
                err[i] = err[i] + temp * fjac[i+j*ldfjac];
            }
        }

        for ( i = 0; i < m; i++ )
        {
            temp = 1.0;
            if ( fvec[i] != 0.0 &&
                 fvecp[i] != 0.0 &&
                 epsf * fabs ( fvec[i] ) <= fabs ( fvecp[i] - fvec[i] ) )
            {
                temp = eps * fabs ( ( fvecp[i] - fvec[i] ) / eps - err[i] )
                       / ( fabs ( fvec[i] ) + fabs ( fvecp[i] ) );

                if ( temp <= epsmch )
                {
                    err[i] = 1.0;
                }
                else if ( temp < eps )
                {
                    err[i] = ( log10 ( temp ) - epslog ) / epslog;
                }
                else
                {
                    err[i] = 0.0;
                }
            }
        }
    }
    return;
}
//****************************************************************************80

void dogleg ( int n, double r[], int lr, double diag[], double qtb[],
              double delta, double x[], double wa1[], double wa2[] )

//****************************************************************************80
//
//  Purpose:
//
//    DOGLEG combines Gauss-Newton and gradient for a minimizing step.
//
//  Discussion:
//
//    Given an M by N matrix A, an n by n nonsingular diagonal
//    matrix d, an m-vector b, and a positive number delta, the
//    problem is to determine the convex combination x of the
//    gauss-newton and scaled gradient directions that minimizes
//    (a*x - b) in the least squares sense, subject to the
//    restriction that the euclidean norm of d*x be at most delta.
//
//    This function completes the solution of the problem
//    if it is provided with the necessary information from the
//    qr factorization of a.
//
//    That is, if a = q*r, where q has orthogonal columns and r is an upper
//    triangular matrix, then dogleg expects the full upper triangle of r and
//    the first n components of Q'*b.
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    05 April 2010
//
//  Author:
//
//    Original FORTRAN77 version by Jorge More, Burt Garbow, Ken Hillstrom.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Jorge More, Burton Garbow, Kenneth Hillstrom,
//    User Guide for MINPACK-1,
//    Technical Report ANL-80-74,
//    Argonne National Laboratory, 1980.
//
//  Parameters:
//
//    Input, int N, the order of R.
//
//    Input, double R[LR], the upper triangular matrix R stored by rows.
//
//    Input, int LR, the size of the storage for R, which should be at
//    least (n*(n+1))/2.
//
//    Input, double DIAG[N], the diagonal elements of the matrix D.
//
//       qtb is an input array of length n which must contain the first
//         n elements of the vector (q transpose)*b.
//
//       delta is a positive input variable which specifies an upper
//         bound on the euclidean norm of d*x.
//
//       x is an output array of length n which contains the desired
//         convex combination of the gauss-newton direction and the
//         scaled gradient direction.
//
//    Workspace, WA1[N].
//
//    Workspace, WA2[N].
//
{
    double alpha;
    double bnorm;
    double epsmch;
    double gnorm;
    int i;
    int j;
    int jj;
    int jp1;
    int k;
    int l;
    double qnorm;
    double sgnorm;
    double sum;
    double temp;
//
//  EPSMCH is the machine precision.
//
    epsmch = r8_epsilon ( );
//
//  Calculate the Gauss-Newton direction.
//
    jj = ( n * ( n + 1 ) ) / 2 + 1;

    for ( k = 1; k <= n; k++ )
    {
        j = n - k + 1;
        jp1 = j + 1;
        jj = jj - k;
        l = jj + 1;
        sum = 0.0;
        for ( i = jp1; i <= n; i++ )
        {
            sum = sum + r[l-1] * x[i-1];
            l = l + 1;
        }
        temp = r[jj-1];
        if ( temp == 0.0 )
        {
            l = j;
            for ( i = 1; i <= j; i++ )
            {
                temp = r8_max ( temp, fabs ( r[l-1] ) );
                l = l + n - i;
            }
            temp = epsmch * temp;
            if ( temp == 0.0 )
            {
                temp = epsmch;
            }
        }
        x[j-1] = ( qtb[j-1] - sum ) / temp;
    }
//
//  Test whether the Gauss-Newton direction is acceptable.
//
    for ( j = 0; j < n; j++ )
    {
        wa1[j] = 0.0;
        wa2[j] = diag[j] * x[j];
    }
    qnorm = enorm ( n, wa2 );

    if ( qnorm <= delta )
    {
        return;
    }
//
//  The Gauss-Newton direction is not acceptable.
//  Calculate the scaled gradient direction.
//
    l = 0;
    for ( j = 0; j < n; j++ )
    {
        temp = qtb[j];
        for ( i = j; i < n; i++ )
        {
            wa1[i-1] = wa1[i-1] + r[l-1] * temp;
            l = l + 1;
        }
        wa1[j] = wa1[j] / diag[j];
    }
//
//  Calculate the norm of the scaled gradient and test for
//  the special case in which the scaled gradient is zero.
//
    gnorm = enorm ( n, wa1 );
    sgnorm = 0.0;
    alpha = delta / qnorm;
//
//  Calculate the point along the scaled gradient
//  at which the quadratic is minimized.
//
    if ( gnorm != 0.0 )
    {
        for ( j = 0; j < n; j++ )
        {
            wa1[j] = ( wa1[j] / gnorm ) / diag[j];
        }
        l = 0;
        for ( j = 0; j < n; j++ )
        {
            sum = 0.0;
            for ( i = j; i < n; i++ )
            {
                sum = sum + r[l] * wa1[i];
                l = l + 1;
            }
            wa2[j] = sum;
        }
        temp = enorm ( n, wa2 );
        sgnorm = ( gnorm / temp ) / temp;
        alpha = 0.0;
//
//  If the scaled gradient direction is not acceptable,
//  calculate the point along the dogleg at which the quadratic is minimized.
//
        if ( sgnorm < delta)
        {
            bnorm = enorm ( n, qtb );
            temp = ( bnorm / gnorm ) * ( bnorm / qnorm ) * ( sgnorm / delta );
            temp = temp - ( delta / qnorm ) * ( sgnorm / delta ) * ( sgnorm / delta )
                   + sqrt ( pow ( temp - ( delta / qnorm ), 2 )
                            + ( 1.0 - ( delta / qnorm ) * ( delta / qnorm ) )
                              * ( 1.0 - ( sgnorm / delta ) * ( sgnorm / delta ) ) );
            alpha = ( ( delta / qnorm )
                      * ( 1.0 - ( sgnorm / delta ) * ( sgnorm / delta ) ) ) / temp;
        }
    }
//
//  Form appropriate convex combination of the Gauss-Newton
//  direction and the scaled gradient direction.
//
    temp = ( 1.0 - alpha ) * r8_min ( sgnorm, delta );
    for ( j = 0; j < n; j++ )
    {
        x[j] = temp * wa1[j] + alpha * x[j];
    }
    return;
}
//****************************************************************************80

double enorm ( int n, double x[] )

//****************************************************************************80
//
//  Purpose:
//
//    ENORM returns the Euclidean norm of a vector.
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    05 April 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in A.
//
//    Input, double X[N], the vector whose norm is desired.
//
//    Output, double ENORM, the norm of X.
//
{
    int i;
    double value;

    value = 0.0;
    for ( i = 0; i < n; i++ )
    {
        value = value + x[i] * x[i];
    }
    value = sqrt ( value );
    return value;
}
//****************************************************************************80

void fdjac1 ( void fcn ( int n, double x[], double f[], int &iflag ),
              int n, double x[], double fvec[], double fjac[], int ldfjac, int &iflag,
              int ml, int mu, double epsfcn, double wa1[], double wa2[] )

//****************************************************************************80
//
//  Purpose:
//
//    FDJAC1 estimates an N by N Jacobian matrix using forward differences.
//
//  Discussion:
//
//    This function computes a forward-difference approximation
//    to the N by N jacobian matrix associated with a specified
//    problem of N functions in N variables.
//
//    If the jacobian has a banded form, then function evaluations are saved
//    by only approximating the nonzero terms.
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    05 April 2010
//
//  Author:
//
//    Original FORTRAN77 version by Jorge More, Burt Garbow, Ken Hillstrom.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Jorge More, Burton Garbow, Kenneth Hillstrom,
//    User Guide for MINPACK-1,
//    Technical Report ANL-80-74,
//    Argonne National Laboratory, 1980.
//
//  Parameters:
//
//       fcn is the name of the user-supplied subroutine which
//         calculates the functions. fcn must be declared
//         in an external statement in the user calling
//         program, and should be written as follows.
//
//         subroutine fcn(n,x,fvec,iflag)
//         integer n,iflag
//         double precision x(n),fvec(n)
//         ----------
//         calculate the functions at x and
//         return this vector in fvec.
//         ----------
//         return
//         end
//
//         the value of iflag should not be changed by fcn unless
//         the user wants to terminate execution of fdjac1.
//         in this case set iflag to a negative integer.
//
//    Input, int N, the number of functions and variables.
//
//    Input, double X[N], the evaluation point.
//
//    Input, double FVEC[N], the functions evaluated at X.
//
//    Output, double FJAC[N*N], the approximate jacobian matrix at X.
//
//       ldfjac is a positive integer input variable not less than n
//         which specifies the leading dimension of the array fjac.
//
//       iflag is an integer variable which can be used to terminate
//         the execution of fdjac1. see description of fcn.
//
//       ml is a nonnegative integer input variable which specifies
//         the number of subdiagonals within the band of the
//         jacobian matrix. if the jacobian is not banded, set
//         ml to at least n - 1.
//
//       epsfcn is an input variable used in determining a suitable
//         step length for the forward-difference approximation. this
//         approximation assumes that the relative errors in the
//         functions are of the order of epsfcn. if epsfcn is less
//         than the machine precision, it is assumed that the relative
//         errors in the functions are of the order of the machine
//         precision.
//
//       mu is a nonnegative integer input variable which specifies
//         the number of superdiagonals within the band of the
//         jacobian matrix. if the jacobian is not banded, set
//         mu to at least n - 1.
//
//       wa1 and wa2 are work arrays of length n. if ml + mu + 1 is at
//         least n, then the jacobian is considered dense, and wa2 is
//         not referenced.
{
    double eps;
    double epsmch;
    double h;
    int i;
    int j;
    int k;
    int msum;
    double temp;
//
//  EPSMCH is the machine precision.
//
    epsmch = r8_epsilon ( );

    eps = sqrt ( r8_max ( epsfcn, epsmch ) );
    msum = ml + mu + 1;
//
//  Computation of dense approximate jacobian.
//
    if ( n <= msum )
    {
        for ( j = 0; j < n; j++ )
        {
            temp = x[j];
            h = eps * fabs ( temp );
            if ( h == 0.0 )
            {
                h = eps;
            }
            x[j] = temp + h;
            fcn ( n, x, wa1, iflag );
            if ( iflag < 0 )
            {
                break;
            }
            x[j] = temp;
            for ( i = 0; i < n; i++ )
            {
                fjac[i+j*ldfjac] = ( wa1[i] - fvec[i] ) / h;
            }
        }
    }
//
//  Computation of a banded approximate jacobian.
//
    else
    {
        for ( k = 0; k < msum; k++ )
        {
            for ( j = k; j < n; j = j + msum )
            {
                wa2[j] = x[j];
                h = eps * fabs ( wa2[j] );
                if ( h == 0.0 )
                {
                    h = eps;
                }
                x[j] = wa2[j] + h;
            }
            fcn ( n, x, wa1, iflag );
            if ( iflag < 0 )
            {
                break;
            }
            for ( j = k; j < n; j = j + msum )
            {
                x[j] = wa2[j];
                h = eps * fabs ( wa2[j] );
                if ( h == 0.0 )
                {
                    h = eps;
                }
                for ( i = 0; i < n; i++ )
                {
                    if ( j - mu <= i && i <= j + ml )
                    {
                        fjac[i+j*ldfjac] = ( wa1[i] - fvec[i] ) / h;
                    }
                    else
                    {
                        fjac[i+j*ldfjac] = 0.0;
                    }
                }
            }
        }
    }
    return;
}
//****************************************************************************80

void fdjac2 ( void fcn ( int m, int n, double x[], double fvec[], int &iflag ),
              int m, int n, double x[], double fvec[], double fjac[], int ldfjac,
              int &iflag, double epsfcn, double wa[] )

//****************************************************************************80
//
//  Purpose:
//
//    FDJAC2 estimates an M by N Jacobian matrix using forward differences.
//
//  Discussion:
//
//    This function computes a forward-difference approximation
//    to the M by N jacobian matrix associated with a specified
//    problem of M functions in N variables.
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    05 April 2010
//
//  Author:
//
//    Original FORTRAN77 version by Jorge More, Burt Garbow, Ken Hillstrom.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Jorge More, Burton Garbow, Kenneth Hillstrom,
//    User Guide for MINPACK-1,
//    Technical Report ANL-80-74,
//    Argonne National Laboratory, 1980.
//
//  Parameters:
//
//       fcn is the name of the user-supplied subroutine which
//         calculates the functions. fcn must be declared
//         in an external statement in the user calling
//         program, and should be written as follows.
//
//         subroutine fcn(m,n,x,fvec,iflag)
//         integer m,n,iflag
//         double precision x(n),fvec(m)
//         ----------
//         calculate the functions at x and
//         return this vector in fvec.
//         ----------
//         return
//         end
//
//         the value of iflag should not be changed by fcn unless
//         the user wants to terminate execution of fdjac2.
//         in this case set iflag to a negative integer.
//
//    Input, int M, the number of functions.
//
//    Input, int N, the number of variables.  N must not exceed M.
//
//    Input, double X[N], the point at which the jacobian is to be estimated.
//
//       fvec is an input array of length m which must contain the
//         functions evaluated at x.
//
//       fjac is an output m by n array which contains the
//         approximation to the jacobian matrix evaluated at x.
//
//       ldfjac is a positive integer input variable not less than m
//         which specifies the leading dimension of the array fjac.
//
//       iflag is an integer variable which can be used to terminate
//         the execution of fdjac2. see description of fcn.
//
//       epsfcn is an input variable used in determining a suitable
//         step length for the forward-difference approximation. this
//         approximation assumes that the relative errors in the
//         functions are of the order of epsfcn. if epsfcn is less
//         than the machine precision, it is assumed that the relative
//         errors in the functions are of the order of the machine
//         precision.
//
//       wa is a work array of length m.
//
{
    double eps;
    double epsmch;
    double h;
    int i;
    int j;
    double temp;
//
//  EPSMCH is the machine precision.
//
    epsmch = r8_epsilon ( );
    eps = sqrt ( r8_max ( epsfcn, epsmch ) );

    for ( j = 0; j < n; j++ )
    {
        temp = x[j];
        if ( temp == 0.0 )
        {
            h = eps;
        }
        else
        {
            h = eps * fabs ( temp );
        }
        x[j] = temp + h;
        fcn ( m, n, x, wa, iflag );
        if ( iflag < 0 )
        {
            break;
        }
        x[j] = temp;
        for ( i = 0; i < m; i++ )
        {
            fjac[i+j*ldfjac] = ( wa[i] - fvec[i] ) / h;
        }
    }
    return;
}
//****************************************************************************80

int hybrd ( void fcn ( int n, double x[], double fvec[], int &iflag ),
            int n, double x[],
            double fvec[], double xtol, int maxfev, int ml, int mu, double epsfcn,
            double diag[], int mode, double factor, int nprint, int nfev,
            double fjac[], int ldfjac, double r[], int lr, double qtf[], double wa1[],
            double wa2[], double wa3[], double wa4[] )

//****************************************************************************80
//
//  Purpose:
//
//    HYBRD finds a zero of a system of N nonlinear equations.
//
//  Discussion:
//
//    The purpose of HYBRD is to find a zero of a system of
//    N nonlinear functions in N variables by a modification
//    of the Powell hybrid method.
//
//    The user must provide FCN, which calculates the functions.
//
//    The jacobian is calculated by a forward-difference approximation.
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    08 April 2010
//
//  Author:
//
//    Original FORTRAN77 version by Jorge More, Burt Garbow, Ken Hillstrom.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Jorge More, Burton Garbow, Kenneth Hillstrom,
//    User Guide for MINPACK-1,
//    Technical Report ANL-80-74,
//    Argonne National Laboratory, 1980.
//
//  Parameters:
//
//       fcn is the name of the user-supplied subroutine which
//         calculates the functions. fcn must be declared
//         in an external statement in the user calling
//         program, and should be written as follows.
//
//         subroutine fcn(n,x,fvec,iflag)
//         integer n,iflag
//         double precision x(n),fvec(n)
//         ----------
//         calculate the functions at x and
//         return this vector in fvec.
//         ---------
//         return
//         end
//
//         the value of iflag should not be changed by fcn unless
//         the user wants to terminate execution of hybrd.
//         in this case set iflag to a negative integer.
//
//    Input, int N, the number of functions and variables.
//
//    Input/output, double X[N].  On input an initial estimate of the solution.
//    On output, the final estimate of the solution.
//
//    Output, double FVEC[N], the functions evaluated at the output value of X.
//
//    Input, double XTOL, a nonnegative value.  Termination occurs when the
//    relative error between two consecutive iterates is at most XTOL.
//
//    Input, int MAXFEV.  Termination occurs when the number of calls to FCN
//    is at least MAXFEV by the end of an iteration.
//
//       ml is a nonnegative integer input variable which specifies
//         the number of subdiagonals within the band of the
//         jacobian matrix. if the jacobian is not banded, set
//         ml to at least n - 1.
//
//       mu is a nonnegative integer input variable which specifies
//         the number of superdiagonals within the band of the
//         jacobian matrix. if the jacobian is not banded, set
//         mu to at least n - 1.
//
//       epsfcn is an input variable used in determining a suitable
//         step length for the forward-difference approximation. this
//         approximation assumes that the relative errors in the
//         functions are of the order of epsfcn. if epsfcn is less
//         than the machine precision, it is assumed that the relative
//         errors in the functions are of the order of the machine
//         precision.
//
//       diag is an array of length n. if mode = 1 (see
//         below), diag is internally set. if mode = 2, diag
//         must contain positive entries that serve as
//         multiplicative scale factors for the variables.
//
//       mode is an integer input variable. if mode = 1, the
//         variables will be scaled internally. if mode = 2,
//         the scaling is specified by the input diag. other
//         values of mode are equivalent to mode = 1.
//
//       factor is a positive input variable used in determining the
//         initial step bound. this bound is set to the product of
//         factor and the euclidean norm of diag*x if nonzero, or else
//         to factor itself. in most cases factor should lie in the
//         interval (.1,100.). 100. is a generally recommended value.
//
//       nprint is an integer input variable that enables controlled
//         printing of iterates if it is positive. in this case,
//         fcn is called with iflag = 0 at the beginning of the first
//         iteration and every nprint iterations thereafter and
//         immediately prior to return, with x and fvec available
//         for printing. if nprint is not positive, no special calls
//         of fcn with iflag = 0 are made.
//
//       info is an integer output variable. if the user has
//         terminated execution, info is set to the (negative)
//         value of iflag. see description of fcn. otherwise,
//         info is set as follows.
//
//         info = 0   improper input parameters.
//
//         info = 1   relative error between two consecutive iterates
//                    is at most xtol.
//
//         info = 2   number of calls to fcn has reached or exceeded
//                    maxfev.
//
//         info = 3   xtol is too small. no further improvement in
//                    the approximate solution x is possible.
//
//         info = 4   iteration is not making good progress, as
//                    measured by the improvement from the last
//                    five jacobian evaluations.
//
//         info = 5   iteration is not making good progress, as
//                    measured by the improvement from the last
//                    ten iterations.
//
//       nfev is an integer output variable set to the number of
//         calls to fcn.
//
//       fjac is an output n by n array which contains the
//         orthogonal matrix q produced by the qr factorization
//         of the final approximate jacobian.
//
//       ldfjac is a positive integer input variable not less than n
//         which specifies the leading dimension of the array fjac.
//
//       r is an output array of length lr which contains the
//         upper triangular matrix produced by the qr factorization
//         of the final approximate jacobian, stored rowwise.
//
//       lr is a positive integer input variable not less than
//         (n*(n+1))/2.
//
//       qtf is an output array of length n which contains
//         the vector (q transpose)*fvec.
//
//       wa1, wa2, wa3, and wa4 are work arrays of length n.
//
{
    double actred;
    double delta;
    double epsmch;
    double fnorm;
    double fnorm1;
    int i;
    int iflag;
    int info;
    int iter;
    int iwa[1];
    int j;
    bool jeval;
    int l;
    int msum;
    int ncfail;
    int ncsuc;
    int nslow1;
    int nslow2;
    const double p001 = 0.001;
    const double p0001 = 0.0001;
    const double p1 = 0.1;
    const double p5 = 0.5;
    double pnorm;
    double prered;
    double ratio;
    bool sing;
    double sum;
    double temp;
    double xnorm;
//
//  Certain loops in this function were kept closer to their original FORTRAN77
//  format, to avoid confusing issues with the array index L.  These loops are
//  marked "DO NOT ADJUST", although they certainly could be adjusted (carefully)
//  once the initial translated code is tested.
//

//
//  EPSMCH is the machine precision.
//
    epsmch = r8_epsilon ( );

    info = 0;
    iflag = 0;
    nfev = 0;
//
//  Check the input parameters.
//
    if ( n <= 0 )
    {
        info = 0;
        return info;
    }
    if ( xtol < 0.0 )
    {
        info = 0;
        return info;
    }
    if ( maxfev <= 0 )
    {
        info = 0;
        return info;
    }
    if ( ml < 0 )
    {
        info = 0;
        return info;
    }
    if ( mu < 0 )
    {
        info = 0;
        return info;
    }
    if ( factor <= 0.0 )
    {
        info = 0;
        return info;
    }
    if ( ldfjac < n )
    {
        info = 0;
        return info;
    }
    if ( lr < ( n * ( n + 1 ) ) / 2 )
    {
        info = 0;
        return info;
    }
    if ( mode == 2 )
    {
        for ( j = 0; j < n; j++ )
        {
            if ( diag[j] <= 0.0 )
            {
                info = 0;
                return info;
            }
        }
    }
//
//  Evaluate the function at the starting point and calculate its norm.
//
    iflag = 1;
    fcn ( n, x, fvec, iflag );
    nfev = 1;
    if ( iflag < 0 )
    {
        info = iflag;
        return info;
    }

    fnorm = enorm ( n, fvec );
//
//  Determine the number of calls to FCN needed to compute the jacobian matrix.
//
    msum = i4_min ( ml + mu + 1, n );
//
//  Initialize iteration counter and monitors.
//
    iter = 1;
    ncsuc = 0;
    ncfail = 0;
    nslow1 = 0;
    nslow2 = 0;
//
//  Beginning of the outer loop.
//
    for ( ; ; )
    {
        jeval = true;
//
//  Calculate the jacobian matrix.
//
        iflag = 2;
        fdjac1 ( fcn, n, x, fvec, fjac, ldfjac, iflag, ml, mu, epsfcn, wa1, wa2 );

        nfev = nfev + msum;
        if ( iflag < 0 )
        {
            info = iflag;
            return info;
        }
//
//  Compute the QR factorization of the jacobian.
//
        qrfac ( n, n, fjac, ldfjac, false, iwa, 1, wa1, wa2 );
//
//  On the first iteration and if MODE is 1, scale according
//  to the norms of the columns of the initial jacobian.
//
        if ( iter == 1 )
        {
            if ( mode == 1 )
            {
                for ( j = 0; j < n; j++ )
                {
                    if ( wa2[j] != 0.0 )
                    {
                        diag[j] = wa2[j];
                    }
                    else
                    {
                        diag[j] = 1.0;
                    }
                }
            }
//
//  On the first iteration, calculate the norm of the scaled X
//  and initialize the step bound DELTA.
//
            for ( j = 0; j < n; j++ )
            {
                wa3[j] = diag[j] * x[j];
            }
            xnorm = enorm ( n, wa3 );

            if ( xnorm == 0.0 )
            {
                delta = factor;
            }
            else
            {
                delta = factor * xnorm;
            }
        }
//
//  Form Q' * FVEC and store in QTF.
//
        for ( i = 0; i < n; i++ )
        {
            qtf[i] = fvec[i];
        }
        for ( j = 0; j < n; j++ )
        {
            if ( fjac[j+j*ldfjac] != 0.0 )
            {
                sum = 0.0;
                for ( i = j; i < n; i++ )
                {
                    sum = sum + fjac[i+j*ldfjac] * qtf[i];
                }
                temp = - sum / fjac[j+j*ldfjac];
                for ( i = j; i < n; i++ )
                {
                    qtf[i] = qtf[i] + fjac[i+j*ldfjac] * temp;
                }
            }
        }
//
//  Copy the triangular factor of the QR factorization into R.
//
//  DO NOT ADJUST THIS LOOP, BECAUSE OF L.
//
        sing = false;
        for ( j = 1; j <= n; j++ )
        {
            l = j;
            for ( i = 1; i <= j - 1; i++ )
            {
                r[l-1] = fjac[(i-1)+(j-1)*ldfjac];
                l = l + n - i;
            }
            r[l-1] = wa1[j-1];
            if ( wa1[j-1] == 0.0 )
            {
                sing = true;
            }
        }
//
//  Accumulate the orthogonal factor in FJAC.
//
        qform ( n, n, fjac, ldfjac );
//
//  Rescale if necessary.
//
        if ( mode == 1 )
        {
            for ( j = 0; j < n; j++ )
            {
                diag[j] = r8_max ( diag[j], wa2[j] );
            }
        }
//
//  Beginning of the inner loop.
//
        for ( ; ; )
        {
//
//  If requested, call FCN to enable printing of iterates.
//
            if ( 0 < nprint )
            {
                if ( ( iter - 1 ) % nprint == 0 )
                {
                    iflag = 0;
                    fcn ( n, x, fvec, iflag );
                    if ( iflag < 0 )
                    {
                        info = iflag;
                        return info;
                    }
                }
            }
//
//  Determine the direction P.
//
            dogleg ( n, r, lr, diag, qtf, delta, wa1, wa2, wa3 );
//
//  Store the direction P and X + P.  Calculate the norm of P.
//
            for ( j = 0; j < n; j++ )
            {
                wa1[j] = - wa1[j];
                wa2[j] = x[j] + wa1[j];
                wa3[j] = diag[j] * wa1[j];
            }
            pnorm = enorm ( n, wa3 );
//
//  On the first iteration, adjust the initial step bound.
//
            if ( iter == 1 )
            {
                delta = r8_min ( delta, pnorm );
            }
//
//  Evaluate the function at X + P and calculate its norm.
//
            iflag = 1;
            fcn ( n, wa2, wa4, iflag );
            nfev = nfev + 1;
            if ( iflag < 0 )
            {
                info = iflag;
                return info;
            }
            fnorm1 = enorm ( n, wa4 );
//
//  Compute the scaled actual reduction.
//
            if ( fnorm1 < fnorm )
            {
                actred = 1.0 - ( fnorm1 / fnorm ) * ( fnorm1 / fnorm );
            }
            else
            {
                actred = - 1.0;
            }
//
//  Compute the scaled predicted reduction.
//
//  DO NOT ADJUST THIS LOOP, BECAUSE OF L.
//
            l = 1;
            for ( i = 1; i <= n; i++ )
            {
                sum = 0.0;
                for ( j = i; j <= n; j++ )
                {
                    sum = sum + r[l-1] * wa1[j-1];
                    l = l + 1;
                }
                wa3[i-1] = qtf[i-1] + sum;
            }
            temp = enorm ( n, wa3 );

            if ( temp < fnorm )
            {
                prered = 1.0 - ( temp / fnorm ) * ( temp / fnorm );
            }
            else
            {
                prered = 0.0;
            }
//
//  Compute the ratio of the actual to the predicted reduction.
//
            if ( 0.0 < prered )
            {
                ratio = actred / prered;
            }
            else
            {
                ratio = 0.0;
            }
//
//  Update the step bound.
//
            if ( ratio < p1 )
            {
                ncsuc = 0;
                ncfail = ncfail + 1;
                delta = p5 * delta;
            }
            else
            {
                ncfail = 0;
                ncsuc = ncsuc + 1;
                if ( p5 <= ratio || 1 < ncsuc )
                {
                    delta = r8_max ( delta, pnorm / p5 );
                }
                if ( fabs ( ratio - 1.0 ) <= p1 )
                {
                    delta = pnorm / p5;
                }
            }
//
//  On successful iteration, update X, FVEC, and their norms.
//
            if ( p0001 <= ratio )
            {
                for ( j = 0; j < n; j++ )
                {
                    x[j] = wa2[j];
                    wa2[j] = diag[j] * x[j];
                    fvec[j] = wa4[j];
                }
                xnorm = enorm ( n, wa2 );
                fnorm = fnorm1;
                iter = iter + 1;
            }
//
//  Determine the progress of the iteration.
//
            nslow1 = nslow1 + 1;
            if ( p001 <= actred )
            {
                nslow1 = 0;
            }
            if ( jeval )
            {
                nslow2 = nslow2 + 1;
            }
            if ( p1 <= actred )
            {
                nslow2 = 0;
            }
//
//  Test for convergence.
//
            if ( delta <= xtol * xnorm || fnorm == 0.0 )
            {
                info = 1;
                return info;
            }
//
//  Tests for termination and stringent tolerances.
//
            if ( maxfev <= nfev )
            {
                info = 2;
                return info;
            }
            if ( p1 * r8_max ( p1 * delta, pnorm ) <= epsmch * xnorm )
            {
                info = 3;
                return info;
            }
            if ( nslow2 == 5 )
            {
                info = 4;
                return info;
            }
            if ( nslow1 == 10 )
            {
                info = 5;
                return info;
            }
//
//  Criterion for recalculating jacobian approximation by forward differences.
//
            if ( ncfail == 2 )
            {
                break;
            }
//
//  Calculate the rank one modification to the jacobian
//  and update QTF if necessary.
//
            for ( j = 0; j < n; j++ )
            {
                sum = 0.0;
                for ( i = 0; i < n; i++ )
                {
                    sum = sum + fjac[i+j*ldfjac] * wa4[i];
                }
                wa2[j] = ( sum - wa3[j] ) / pnorm;
                wa1[j] = diag[j] * ( ( diag[j] * wa1[j] ) / pnorm );
                if ( p0001 <= ratio )
                {
                    qtf[j] = sum;
                }
            }
//
//  Compute the QR factorization of the updated jacobian.
//
            sing = r1updt ( n, n, r, lr, wa1, wa2, wa3 );
            r1mpyq ( n, n, fjac, ldfjac, wa2, wa3 );
            r1mpyq ( 1, n, qtf, 1, wa2, wa3 );

            jeval = false;
        }
//
//  End of the inner loop.
//
    }
//
//  End of the outer loop.
//
}
//****************************************************************************80

int hybrd1 ( void fcn ( int n, double x[], double fvec[], int &iflag ), int n,
             double x[], double fvec[], double tol, double wa[], int lwa )

//****************************************************************************80
//
//  Purpose:
//
//    HYBRD1 is a simplified interface to HYBRD.
//
//  Discussion:
//
//    The purpose of HYBRD1 is to find a zero of a system of
//    N nonlinear functions in N variables by a modification
//    of the Powell hybrid method.
//
//    This is done by using the more general nonlinear equation solver HYBRD.
//
//    The user must provide FCN, which calculates the functions.
//
//    The jacobian is calculated by a forward-difference approximation.
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    26 July 2014
//
//  Author:
//
//    Original FORTRAN77 version by Jorge More, Burt Garbow, Ken Hillstrom.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Jorge More, Burton Garbow, Kenneth Hillstrom,
//    User Guide for MINPACK-1,
//    Technical Report ANL-80-74,
//    Argonne National Laboratory, 1980.
//
//  Parameters:
//
//       fcn is the name of the user-supplied subroutine which
//         calculates the functions. fcn must be declared
//         in an external statement in the user calling
//         program, and should be written as follows.
//
//         subroutine fcn(n,x,fvec,iflag)
//         integer n,iflag
//         double precision x(n),fvec(n)
//         ----------
//         calculate the functions at x and
//         return this vector in fvec.
//         ---------
//         return
//         end
//
//         the value of iflag should not be changed by fcn unless
//         the user wants to terminate execution of hybrd1.
//         in this case set iflag to a negative integer.
//
//    Input, int N, the number of functions and variables.
//
//    Input/output, double X[N].  On input, an initial estimate of the solution.
//    On output, the final estimate of the solution.
//
//    Output, double FVEC[N], the functions evaluated at the output X.
//
//    Input, double TOL, a nonnegative variable. tTermination occurs when the
//    algorithm estimates that the relative error between X and the solution
//    is at most TOL.
//
//       info is an integer output variable. if the user has
//         terminated execution, info is set to the (negative)
//         value of iflag. see description of fcn. otherwise,
//         info is set as follows.
//         info = 0   improper input parameters.
//         info = 1   algorithm estimates that the relative error
//                    between x and the solution is at most tol.
//         info = 2   number of calls to fcn has reached or exceeded
//                    200*(n+1).
//         info = 3   tol is too small. no further improvement in
//                    the approximate solution x is possible.
//         info = 4   iteration is not making good progress.
//
//       wa is a work array of length lwa.
//
//       lwa is a positive integer input variable not less than
//         (n*(3*n+13))/2.
//
{
    double epsfcn;
    double factor;
    int index;
    int info;
    int j;
    int lr;
    int maxfev;
    int ml;
    int mode;
    int mu;
    int nfev;
    int nprint;
    double xtol;

    info = 0;
//
//  Check the input.
//
    if ( n <= 0 )
    {
        return info;
    }
    if ( tol <= 0.0 )
    {
        return info;
    }
    if ( lwa < ( n * ( 3 * n + 13 ) ) / 2 )
    {
        return info;
    }
//
//  Call HYBRD.
//
    xtol = tol;
    maxfev = 200 * ( n + 1 );
    ml = n - 1;
    mu = n - 1;
    epsfcn = 0.0;
    for ( j = 0; j < n; j++ )
    {
        wa[j] = 1.0;
    }
    mode = 2;
    factor = 100.0;
    nprint = 0;
    nfev = 0;
    lr = ( n * ( n + 1 ) ) / 2;
    index = 6 * n + lr;

    info = hybrd ( fcn, n, x, fvec, xtol, maxfev, ml, mu, epsfcn, wa, mode,
                   factor, nprint, nfev, wa+index, n, wa+6*n, lr,
                   wa+n, wa+2*n, wa+3*n, wa+4*n, wa+5*n );

    if ( info == 5 )
    {
        info = 4;
    }
    return info;
}
//****************************************************************************80

int i4_max ( int i1, int i2 )

//****************************************************************************80
//
//  Purpose:
//
//    I4_MAX returns the maximum of two I4's.
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    13 October 1998
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int I1, I2, are two integers to be compared.
//
//    Output, int I4_MAX, the larger of I1 and I2.
//
{
    int value;

    if ( i2 < i1 )
    {
        value = i1;
    }
    else
    {
        value = i2;
    }
    return value;
}
//****************************************************************************80

int i4_min ( int i1, int i2 )

//****************************************************************************80
//
//  Purpose:
//
//    I4_MIN returns the minimum of two I4's.
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    13 October 1998
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int I1, I2, two integers to be compared.
//
//    Output, int I4_MIN, the smaller of I1 and I2.
//
{
    int value;

    if ( i1 < i2 )
    {
        value = i1;
    }
    else
    {
        value = i2;
    }
    return value;
}
//****************************************************************************80

void qform ( int m, int n, double q[], int ldq )

//****************************************************************************80
//
//  Purpose:
//
//    QFORM constructs the standard form of Q from its factored form.
//
//  Discussion:
//
//    This function proceeds from the computed QR factorization of
//    an M by N matrix A to accumulate the M by M orthogonal matrix
//    Q from its factored form.
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    02 January 2018
//
//  Author:
//
//    Original FORTRAN77 version by Jorge More, Burt Garbow, Ken Hillstrom.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Jorge More, Burton Garbow, Kenneth Hillstrom,
//    User Guide for MINPACK-1,
//    Technical Report ANL-80-74,
//    Argonne National Laboratory, 1980.
//
//  Parameters:
//
//    Input, int M, the number of rows of A, and the order of Q.
//
//    Input, int N, the number of columns of A.
//
//    Input/output, double Q[LDQ*N].  On input, the full lower trapezoid in
//    the first min(M,N) columns of Q contains the factored form.
//    On output Q has been accumulated into a square matrix.
//
//    Input, int LDQ, the leading dimension of the array Q.
//
{
    int i;
    int j;
    int k;
    int minmn;
    double sum;
    double temp;
    double *wa;
//
//  Zero out the upper triangle of Q in the first min(M,N) columns.
//
    minmn = i4_min ( m, n );

    for ( j = 1; j < minmn; j++ )
    {
        for ( i = 0; i <= j - 1; i++ )
        {
            q[i+j*ldq] = 0.0;
        }
    }
//
//  Initialize remaining columns to those of the identity matrix.
//
    for ( j = n; j < m; j++ )
    {
        for ( i = 0; i < m; i++ )
        {
            q[i+j*ldq] = 0.0;
        }
        q[j+j*ldq] = 1.0;
    }
//
//  Accumulate Q from its factored form.
//
    wa = new double[m];

    for ( k = minmn - 1; 0 <= k; k-- )
    {
        for ( i = k; i < m; i++ )
        {
            wa[i] = q[i+k*ldq];
            q[i+k*ldq] = 0.0;
        }
        q[k+k*ldq] = 1.0;

        if ( wa[k] != 0.0 )
        {
            for ( j = k; j < m; j++ )
            {
                sum = 0.0;
                for ( i = k; i < m; i++ )
                {
                    sum = sum + q[i+j*ldq] * wa[i];
                }
                temp = sum / wa[k];
                for ( i = k; i < m; i++ )
                {
                    q[i+j*ldq] = q[i+j*ldq] - temp * wa[i];
                }
            }
        }
    }
//
//  Free memory.
//
    delete [] wa;

    return;
}
//****************************************************************************80

void qrfac ( int m, int n, double a[], int lda, bool pivot, int ipvt[],
             int lipvt, double rdiag[], double acnorm[] )

//****************************************************************************80
//
//  Purpose:
//
//    QRFAC computes the QR factorization of an M by N matrix.
//
//  Discussion:
//
//    This function uses Householder transformations with optional column
//    pivoting to compute a QR factorization of the M by N matrix A.
//
//    That is, QRFAC determines an orthogonal
//    matrix Q, a permutation matrix P, and an upper trapezoidal
//    matrix R with diagonal elements of nonincreasing magnitude,
//    such that A*P = Q*R.
//
//    The Householder transformation for
//    column k, k = 1,2,...,min(m,n), is of the form
//
//      i - (1/u(k))*u*u'
//
//    where U has zeros in the first K-1 positions.
//
//    The form of this transformation and the method of pivoting first
//    appeared in the corresponding LINPACK function.
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    02 January 2017
//
//  Author:
//
//    Original FORTRAN77 version by Jorge More, Burt Garbow, Ken Hillstrom.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Jorge More, Burton Garbow, Kenneth Hillstrom,
//    User Guide for MINPACK-1,
//    Technical Report ANL-80-74,
//    Argonne National Laboratory, 1980.
//
//  Parameters:
//
//    Input, int M, the number of rows of A.
//
//    Input, int N, the number of columns of A.
//
//    Input/output, double A[M*N].  On input, the matrix for which the QR
//    factorization is to be computed.  On output, the strict upper trapezoidal
//    part contains the strict upper trapezoidal part of the R factor, and
//    the lower trapezoidal part contains a factored form of Q, the non-trivial
//    elements of the U vectors described above.
//
//    Input, int LDA, a positive value not less than M which specifies the
//    leading dimension of the array A.
//
//    Input, bool PIVOT.  If true, then column pivoting is enforced.
//
//       ipvt is an integer output array of length lipvt. ipvt
//         defines the permutation matrix p such that a*p = q*r.
//         column j of p is column ipvt(j) of the identity matrix.
//         if pivot is false, ipvt is not referenced.
//
//       lipvt is a positive integer input variable. if pivot is false,
//         then lipvt may be as small as 1. if pivot is true, then
//         lipvt must be at least n.
//
//       rdiag is an output array of length n which contains the
//         diagonal elements of r.
//
//       acnorm is an output array of length n which contains the
//         norms of the corresponding columns of the input matrix a.
//         if this information is not needed, then acnorm can coincide
//         with rdiag.
//
{
    double ajnorm;
    double epsmch;
    int i;
    int j;
    int jp1;
    int k;
    int kmax;
    int minmn;
    const double p05 = 0.05;
    double sum;
    double temp;
    double *wa;
//
//  EPSMCH is the machine precision.
//
    epsmch = r8_epsilon ( );
//
//  Compute the initial column norms and initialize several arrays.
//
    wa = new double[n];

    for ( j = 0; j < n; j++ )
    {
        acnorm[j] = enorm ( m, a+j*lda );
        rdiag[j] = acnorm[j];
        wa[j] = rdiag[j];
        if ( pivot )
        {
            ipvt[j] = j;
        }
    }
//
//  Reduce A to R with Householder transformations.
//
    minmn = i4_min ( m, n );

    for ( j = 0; j < minmn; j++ )
    {
        if ( pivot )
        {
//
//  Bring the column of largest norm into the pivot position.
//
            kmax = j;
            for ( k = j; k < n; k++ )
            {
                if ( rdiag[kmax] < rdiag[k] )
                {
                    kmax = k;
                }
            }
            if ( kmax != j )
            {
                for ( i = 0; i < m; i++ )
                {
                    temp          = a[i+j*lda];
                    a[i+j*lda]    = a[i+kmax*lda];
                    a[i+kmax*lda] = temp;
                }
                rdiag[kmax] = rdiag[j];
                wa[kmax]    = wa[j];
                k          = ipvt[j];
                ipvt[j]    = ipvt[kmax];
                ipvt[kmax] = k;
            }
        }
//
//  Compute the Householder transformation to reduce the
//  J-th column of A to a multiple of the J-th unit vector.
//
        ajnorm = enorm ( m - j, a+j+j*lda );

        if ( ajnorm != 0.0 )
        {
            if ( a[j+j*lda] < 0.0 )
            {
                ajnorm = - ajnorm;
            }
            for ( i = j; i < m; i++ )
            {
                a[i+j*lda] = a[i+j*lda] / ajnorm;
            }
            a[j+j*lda] = a[j+j*lda] + 1.0;
//
//  Apply the transformation to the remaining columns and update the norms.
//
            jp1 = j + 1;
            for ( k = j + 1; k < n; k++ )
            {
                sum = 0.0;
                for ( i = j; i < m; i++ )
                {
                    sum = sum + a[i+j*lda] * a[i+k*lda];
                }
                temp = sum / a[j+j*lda];
                for ( i = j; i < m; i++ )
                {
                    a[i+k*lda] = a[i+k*lda] - temp * a[i+j*lda];
                }
                if ( pivot && rdiag[k] != 0.0 )
                {
                    temp = a[j+k*lda] / rdiag[k];
                    rdiag[k] = rdiag[k] * sqrt ( r8_max ( 0.0, 1.0 - temp * temp ) );
                    if ( p05 * ( rdiag[k] / wa[k] ) * ( rdiag[k] / wa[k] ) <= epsmch )
                    {
                        rdiag[k] = enorm ( m - 1 - j, a+(j+1)+k*lda );
                        wa[k] = rdiag[k];
                    }
                }
            }
        }
        rdiag[j] = - ajnorm;
    }
//
//  Free memory.
//
    delete [] wa;

    return;
}
//****************************************************************************80

void r1mpyq ( int m, int n, double a[], int lda, double v[], double w[] )

//****************************************************************************80
//
//  Purpose:
//
//    R1MPYQ multiplies an M by N matrix A by the Q factor.
//
//  Discussion:
//
//    Given an M by N matrix A, this function computes a*q where
//    q is the product of 2*(n - 1) transformations
//
//      gv(n-1)*...*gv(1)*gw(1)*...*gw(n-1)
//
//    and gv(i), gw(i) are givens rotations in the (i,n) plane which
//    eliminate elements in the i-th and n-th planes, respectively.
//
//    Q itself is not given, rather the information to recover the
//    GV and GW rotations is supplied.
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    05 April 2010
//
//  Author:
//
//    Original FORTRAN77 version by Jorge More, Burt Garbow, Ken Hillstrom.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Jorge More, Burton Garbow, Kenneth Hillstrom,
//    User Guide for MINPACK-1,
//    Technical Report ANL-80-74,
//    Argonne National Laboratory, 1980.
//
//  Parameters:
//
//    Input, int M, the number of rows of A.
//
//    Input, int N, the number of columns of A.
//
//    Input/output, double A[M*N].  On input, the matrix to be postmultiplied
//    by the orthogonal matrix Q described above.  On output, the value of A*Q.
//
//    Input, int LDA, a positive value not less than M
//    which specifies the leading dimension of the array A.
//
//    Input, double V[N].  V(I) must contain the information necessary to
//    recover the givens rotation GV(I) described above.
//
//       w is an input array of length n. w(i) must contain the
//         information necessary to recover the givens rotation gw(i)
//         described above.
//
{
    double c;
    int i;
    int j;
    double s;
    double temp;
//
//  Apply the first set of Givens rotations to A.
//
    for ( j = n - 2; 0 <= j; j-- )
    {
        if ( 1.0 < fabs ( v[j] ) )
        {
            c = 1.0 / v[j];
            s = sqrt ( 1.0 - c * c );
        }
        else
        {
            s = v[j];
            c = sqrt ( 1.0 - s * s );
        }
        for ( i = 0; i < m; i++ )
        {
            temp           = c * a[i+j*lda] - s * a[i+(n-1)*lda];
            a[i+(n-1)*lda] = s * a[i+j*lda] + c * a[i+(n-1)*lda];
            a[i+j*lda]     = temp;
        }
    }
//
//  Apply the second set of Givens rotations to A.
//
    for ( j = 0; j < n - 1; j++ )
    {
        if ( 1.0 < fabs ( w[j] ) )
        {
            c = 1.0 / w[j];
            s = sqrt ( 1.0 - c * c );
        }
        else
        {
            s = w[j];
            c = sqrt ( 1.0 - s * s );
        }
        for ( i = 0; i < m; i++ )
        {
            temp           =   c * a[i+j*lda] + s * a[i+(n-1)*lda];
            a[i+(n-1)*lda] = - s * a[i+j*lda] + c * a[i+(n-1)*lda];
            a[i+j*lda]     = temp;
        }
    }

    return;
}
//****************************************************************************80

bool r1updt ( int m, int n, double s[], int ls, double u[], double v[],
              double w[] )

//****************************************************************************80
//
//  Purpose:
//
//    R1UPDT updates the Q factor after a rank one update of the matrix.
//
//  Discussion:
//
//    Given an M by N lower trapezoidal matrix S, an M-vector U,
//    and an N-vector V, the problem is to determine an
//    orthogonal matrix Q such that
//
//      (S + U*V') * Q
//
//    is again lower trapezoidal.
//
//    This function determines q as the product of 2*(n - 1) transformations
//
//      gv(n-1)*...*gv(1)*gw(1)*...*gw(n-1)
//
//    where gv(i), gw(i) are givens rotations in the (i,n) plane
//    which eliminate elements in the i-th and n-th planes,
//    respectively.
//
//    Q itself is not accumulated, rather the
//    information to recover the gv, gw rotations is returned.
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    08 April 2010
//
//  Author:
//
//    Original FORTRAN77 version by Jorge More, Burt Garbow, Ken Hillstrom.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Jorge More, Burton Garbow, Kenneth Hillstrom,
//    User Guide for MINPACK-1,
//    Technical Report ANL-80-74,
//    Argonne National Laboratory, 1980.
//
//  Parameters:
//
//    Input, int M, the number of rows of S.
//
//    Input, int N, the number of columns of S.  N must not exceed M.
//
//    Input/output, double S[LS].  On input, the lower trapezoidal matrix S
//    stored by columns.  On output, the lower trapezoidal matrix produced
//    as described above.
//
//    Input, int LS, a positive value not less than (N*(2*M-N+1))/2.
//
//    Input, double U[M], contains the vector U.
//
//       v is an array of length n. on input v must contain the vector
//         v. on output v(i) contains the information necessary to
//         recover the givens rotation gv(i) described above.
//
//       w is an output array of length m. w(i) contains information
//         necessary to recover the givens rotation gw(i) described
//         above.
//
//       sing is a logical output variable. sing is set true if any
//         of the diagonal elements of the output s are zero. otherwise
//         sing is set false.
//
{
    double cotan;
    double cs;
    double giant;
    int i;
    int j;
    int jj;
    int l;
    int nm1;
    const double p25 = 0.25;
    const double p5 = 0.5;
    double sn;
    bool sing;
    double tan;
    double tau;
    double temp;
//
//  Because of the computation of the pointer JJ, this function was
//  converted from FORTRAN77 to C++ in a conservative way.  All computations
//  are the same, and only array indexing is adjusted.
//
//  GIANT is the largest magnitude.
//
    giant = r8_huge ( );
//
//  Initialize the diagonal element pointer.
//
    jj = ( n * ( 2 * m - n + 1 ) ) / 2 - ( m - n );
//
//  Move the nontrivial part of the last column of S into W.
//
    l = jj;
    for ( i = n; i <= m; i++ )
    {
        w[i-1] = s[l-1];
        l = l + 1;
    }
//
//  Rotate the vector V into a multiple of the N-th unit vector
//  in such a way that a spike is introduced into W.
//
    nm1 = n - 1;

    for ( j = n - 1; 1 <= j; j-- )
    {
        jj = jj - ( m - j + 1 );
        w[j-1] = 0.0;

        if ( v[j-1] != 0.0 )
        {
//
//  Determine a Givens rotation which eliminates the J-th element of V.
//
            if ( fabs ( v[n-1] ) < fabs ( v[j-1] ) )
            {
                cotan = v[n-1] / v[j-1];
                sn = p5 / sqrt ( p25 + p25 * cotan * cotan );
                cs = sn * cotan;
                tau = 1.0;
                if ( 1.0 < fabs ( cs ) * giant )
                {
                    tau = 1.0 / cs;
                }
            }
            else
            {
                tan = v[j-1] / v[n-1];
                cs = p5 / sqrt ( p25 + p25 * tan * tan );
                sn = cs * tan;
                tau = sn;
            }
//
//  Apply the transformation to V and store the information
//  necessary to recover the Givens rotation.
//
            v[n-1] = sn * v[j-1] + cs * v[n-1];
            v[j-1] = tau;
//
//  Apply the transformation to S and extend the spike in W.
//
            l = jj;
            for ( i = j; i <= m; i++ )
            {
                temp   = cs * s[l-1] - sn * w[i-1];
                w[i-1] = sn * s[l-1] + cs * w[i-1];
                s[l-1] = temp;
                l = l + 1;
            }
        }
    }
//
//  Add the spike from the rank 1 update to W.
//
    for ( i = 1; i <= m; i++ )
    {
        w[i-1] = w[i-1] + v[n-1] * u[i-1];
    }
//
//  Eliminate the spike.
//
    sing = false;

    for ( j = 1; j <= nm1; j++ )
    {
//
//  Determine a Givens rotation which eliminates the
//  J-th element of the spike.
//
        if ( w[j-1] != 0.0 )
        {

            if ( fabs ( s[jj-1] ) < fabs ( w[j-1] ) )
            {
                cotan = s[jj-1] / w[j-1];
                sn = p5 / sqrt ( p25 + p25 * cotan * cotan );
                cs = sn * cotan;
                tau = 1.0;
                if ( 1.0 < fabs ( cs ) * giant )
                {
                    tau = 1.0 / cs;
                }
            }
            else
            {
                tan = w[j-1] / s[jj-1];
                cs = p5 / sqrt ( p25 + p25 * tan * tan );
                sn = cs * tan;
                tau = sn;
            }
//
//  Apply the transformation to s and reduce the spike in w.
//
            l = jj;

            for ( i = j; i <= m; i++ )
            {
                temp   =   cs * s[l-1] + sn * w[i-1];
                w[i-1] = - sn * s[l-1] + cs * w[i-1];
                s[l-1] = temp;
                l = l + 1;
            }
//
//  Store the information necessary to recover the givens rotation.
//
            w[j-1] = tau;
        }
//
//  Test for zero diagonal elements in the output s.
//
        if ( s[jj-1] == 0.0 )
        {
            sing = true;
        }
        jj = jj + ( m - j + 1 );
    }
//
//  Move W back into the last column of the output S.
//
    l = jj;
    for ( i = n; i <= m; i++ )
    {
        s[l-1] = w[i-1];
        l = l + 1;
    }
    if ( s[jj-1] == 0.0 )
    {
        sing = true;
    }
    return sing;
}
//****************************************************************************80

double r8_epsilon ( )

//****************************************************************************80
//
//  Purpose:
//
//    R8_EPSILON returns the R8 roundoff unit.
//
//  Discussion:
//
//    The roundoff unit is a number R which is a power of 2 with the
//    property that, to the precision of the computer's arithmetic,
//      1 < 1 + R
//    but
//      1 = ( 1 + R / 2 )
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    01 September 2012
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Output, double R8_EPSILON, the R8 round-off unit.
//
{
    const double value = 2.220446049250313E-016;

    return value;
}
//****************************************************************************80

double r8_huge ( )

//****************************************************************************80
//
//  Purpose:
//
//    R8_HUGE returns a "huge" R8.
//
//  Discussion:
//
//    The value returned by this function is NOT required to be the
//    maximum representable R8.  This value varies from machine to machine,
//    from compiler to compiler, and may cause problems when being printed.
//    We simply want a "very large" but non-infinite number.
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    06 October 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Output, double R8_HUGE, a "huge" R8 value.
//
{
    double value;

    value = 1.0E+30;

    return value;
}
//****************************************************************************80

double r8_max ( double x, double y )

//****************************************************************************80
//
//  Purpose:
//
//    R8_MAX returns the maximum of two R8's.
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    18 August 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, Y, the quantities to compare.
//
//    Output, double R8_MAX, the maximum of X and Y.
//
{
    double value;

    if ( y < x )
    {
        value = x;
    }
    else
    {
        value = y;
    }
    return value;
}
//****************************************************************************80

double r8_min ( double x, double y )

//****************************************************************************80
//
//  Purpose:
//
//    R8_MIN returns the minimum of two R8's.
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    31 August 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, Y, the quantities to compare.
//
//    Output, double R8_MIN, the minimum of X and Y.
//
{
    double value;

    if ( y < x )
    {
        value = y;
    }
    else
    {
        value = x;
    }
    return value;
}
//****************************************************************************80

double r8_tiny ( )

//****************************************************************************80
//
//  Purpose:
//
//    R8_TINY returns a "tiny" R8.
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    08 March 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Output, double R8_TINY, a "tiny" R8 value.
//
{
    double value;

    value = 0.4450147717014E-307;

    return value;
}
//****************************************************************************80

double r8_uniform_01 ( int &seed )

//****************************************************************************80
//
//  Purpose:
//
//    R8_UNIFORM_01 returns a unit pseudorandom R8.
//
//  Discussion:
//
//    This routine implements the recursion
//
//      seed = ( 16807 * seed ) mod ( 2^31 - 1 )
//      u = seed / ( 2^31 - 1 )
//
//    The integer arithmetic never requires more than 32 bits,
//    including a sign bit.
//
//    If the initial seed is 12345, then the first three computations are
//
//      Input     Output      R8_UNIFORM_01
//      SEED      SEED
//
//         12345   207482415  0.096616
//     207482415  1790989824  0.833995
//    1790989824  2035175616  0.947702
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    09 April 2012
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Paul Bratley, Bennett Fox, Linus Schrage,
//    A Guide to Simulation,
//    Second Edition,
//    Springer, 1987,
//    ISBN: 0387964673,
//    LC: QA76.9.C65.B73.
//
//    Bennett Fox,
//    Algorithm 647:
//    Implementation and Relative Efficiency of Quasirandom
//    Sequence Generators,
//    ACM Transactions on Mathematical Software,
//    Volume 12, Number 4, December 1986, pages 362-376.
//
//    Pierre L'Ecuyer,
//    Random Number Generation,
//    in Handbook of Simulation,
//    edited by Jerry Banks,
//    Wiley, 1998,
//    ISBN: 0471134031,
//    LC: T57.62.H37.
//
//    Peter Lewis, Allen Goodman, James Miller,
//    A Pseudo-Random Number Generator for the System/360,
//    IBM Systems Journal,
//    Volume 8, Number 2, 1969, pages 136-143.
//
//  Parameters:
//
//    Input/output, int &SEED, the "seed" value.  Normally, this
//    value should not be 0.  On output, SEED has been updated.
//
//    Output, double R8_UNIFORM_01, a new pseudorandom variate,
//    strictly between 0 and 1.
//
{
    const int i4_huge = 2147483647;
    int k;
    double r;

    if ( seed == 0 )
    {
        cerr << "\n";
        cerr << "R8_UNIFORM_01 - Fatal error!\n";
        cerr << "  Input value of SEED = 0.\n";
        exit ( 1 );
    }

    k = seed / 127773;

    seed = 16807 * ( seed - k * 127773 ) - k * 2836;

    if ( seed < 0 )
    {
        seed = seed + i4_huge;
    }
    r = ( double ) ( seed ) * 4.656612875E-10;

    return r;
}
//****************************************************************************80

double *r8mat_mm_new ( int n1, int n2, int n3, double a[], double b[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_MM_NEW multiplies two matrices.
//
//  Discussion:
//
//    An R8MAT is a doubly dimensioned array of R8 values, stored as a vector
//    in column-major order.
//
//    For this routine, the result is returned as the function value.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    18 October 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N1, N2, N3, the order of the matrices.
//
//    Input, double A[N1*N2], double B[N2*N3], the matrices to multiply.
//
//    Output, double R8MAT_MM_NEW[N1*N3], the product matrix C = A * B.
//
{
    double *c;
    int i;
    int j;
    int k;

    c = new double[n1*n3];

    for ( i = 0; i < n1; i++ )
    {
        for ( j = 0; j < n3; j++ )
        {
            c[i+j*n1] = 0.0;
            for ( k = 0; k < n2; k++ )
            {
                c[i+j*n1] = c[i+j*n1] + a[i+k*n1] * b[k+j*n2];
            }
        }
    }

    return c;
}
//****************************************************************************80

void r8mat_print ( int m, int n, double a[], string title )

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_PRINT prints an R8MAT.
//
//  Discussion:
//
//    An R8MAT is a doubly dimensioned array of R8 values, stored as a vector
//    in column-major order.
//
//    Entry A(I,J) is stored as A[I+J*M]
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    10 September 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, the number of rows in A.
//
//    Input, int N, the number of columns in A.
//
//    Input, double A[M*N], the M by N matrix.
//
//    Input, string TITLE, a title.
//
{
    r8mat_print_some ( m, n, a, 1, 1, m, n, title );

    return;
}
//****************************************************************************80

void r8mat_print_some ( int m, int n, double a[], int ilo, int jlo, int ihi,
                        int jhi, string title )

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_PRINT_SOME prints some of an R8MAT.
//
//  Discussion:
//
//    An R8MAT is a doubly dimensioned array of R8 values, stored as a vector
//    in column-major order.
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    10 September 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, the number of rows of the matrix.
//    M must be positive.
//
//    Input, int N, the number of columns of the matrix.
//    N must be positive.
//
//    Input, double A[M*N], the matrix.
//
//    Input, int ILO, JLO, IHI, JHI, designate the first row and
//    column, and the last row and column to be printed.
//
//    Input, string TITLE, a title.
//
{
# define INCX 5

    int i;
    int i2hi;
    int i2lo;
    int j;
    int j2hi;
    int j2lo;

    cout << "\n";
    cout << title << "\n";
//
//  Print the columns of the matrix, in strips of 5.
//
    for ( j2lo = jlo; j2lo <= jhi; j2lo = j2lo + INCX )
    {
        j2hi = j2lo + INCX - 1;
        j2hi = i4_min ( j2hi, n );
        j2hi = i4_min ( j2hi, jhi );

        cout << "\n";
//
//  For each column J in the current range...
//
//  Write the header.
//
        cout << "  Col:    ";
        for ( j = j2lo; j <= j2hi; j++ )
        {
            cout << setw(7) << j << "       ";
        }
        cout << "\n";
        cout << "  Row\n";
        cout << "\n";
//
//  Determine the range of the rows in this strip.
//
        i2lo = i4_max ( ilo, 1 );
        i2hi = i4_min ( ihi, m );

        for ( i = i2lo; i <= i2hi; i++ )
        {
//
//  Print out (up to) 5 entries in row I, that lie in the current strip.
//
            cout << setw(5) << i << "  ";
            for ( j = j2lo; j <= j2hi; j++ )
            {
                cout << setw(12) << a[i-1+(j-1)*m] << "  ";
            }
            cout << "\n";
        }
    }

    return;
# undef INCX
}
//****************************************************************************80

void r8vec_print ( int n, double a[], string title )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_PRINT prints an R8VEC.
//
//  Discussion:
//
//    An R8VEC is a vector of R8's.
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    16 August 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of components of the vector.
//
//    Input, double A[N], the vector to be printed.
//
//    Input, string TITLE, a title.
//
{
    int i;

    cout << "\n";
    cout << title << "\n";
    cout << "\n";
    for ( i = 0; i < n; i++ )
    {
        cout << "  " << setw(8)  << i
             << "  " << setw(14) << a[i]  << "\n";
    }

    return;
}
//****************************************************************************80

void timestamp ( )

//****************************************************************************80
//
//  Purpose:
//
//    TIMESTAMP prints the current YMDHMS date as a time stamp.
//
//  Example:
//
//    31 May 2001 09:45:54 AM
//
//  Licensing:
//
//    This code may freely be copied, modified, and used for any purpose.
//
//  Modified:
//
//    08 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    None
//
{
# define TIME_SIZE 40

    static char time_buffer[TIME_SIZE];
    const struct std::tm *tm_ptr;
    size_t len;
    std::time_t now;

    now = std::time ( NULL );
    tm_ptr = std::localtime ( &now );

    len = std::strftime ( time_buffer, TIME_SIZE, "%d %B %Y %I:%M:%S %p", tm_ptr );

    std::cout << time_buffer << "\n";

    return;
# undef TIME_SIZE
}