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

using namespace std;

# include "r8ge.hpp"

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

int i4_log_10 ( int i )

//****************************************************************************80
//
//  Purpose:
//
//    I4_LOG_10 returns the integer part of the logarithm base 10 of ABS(X).
//
//  Example:
//
//        I  I4_LOG_10
//    -----  --------
//        0    0
//        1    0
//        2    0
//        9    0
//       10    1
//       11    1
//       99    1
//      100    2
//      101    2
//      999    2
//     1000    3
//     1001    3
//     9999    3
//    10000    4
//
//  Discussion:
//
//    I4_LOG_10 ( I ) + 1 is the number of decimal digits in I.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    04 January 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int I, the number whose logarithm base 10 is desired.
//
//    Output, int I4_LOG_10, the integer part of the logarithm base 10 of
//    the absolute value of X.
//
{
  int i_abs;
  int ten_pow;
  int value;

  if ( i == 0 )
  {
    value = 0;
  }
  else
  {
    value = 0;
    ten_pow = 10;

    i_abs = abs ( i );

    while ( ten_pow <= i_abs )
    {
      value = value + 1;
      ten_pow = ten_pow * 10;
    }

  }

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

int i4_max ( int i1, int i2 )

//****************************************************************************80
//
//  Purpose:
//
//    I4_MAX returns the maximum of two I4's.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  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 is distributed under the MIT license.
//
//  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

int i4_power ( int i, int j )

//****************************************************************************80
//
//  Purpose:
//
//    I4_POWER returns the value of I^J.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    01 April 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int I, J, the base and the power.  J should be nonnegative.
//
//    Output, int I4_POWER, the value of I^J.
//
{
  int k;
  int value;

  if ( j < 0 )
  {
    if ( i == 1 )
    {
      value = 1;
    }
    else if ( i == 0 )
    {
      cerr << "\n";
      cerr << "I4_POWER - Fatal error!\n";
      cerr << "  I^J requested, with I = 0 and J negative.\n";
      exit ( 1 );
    }
    else
    {
      value = 0;
    }
  }
  else if ( j == 0 )
  {
    if ( i == 0 )
    {
      cerr << "\n";
      cerr << "I4_POWER - Fatal error!\n";
      cerr << "  I^J requested, with I = 0 and J = 0.\n";
      exit ( 1 );
    }
    else
    {
      value = 1;
    }
  }
  else if ( j == 1 )
  {
    value = i;
  }
  else
  {
    value = 1;
    for ( k = 1; k <= j; k++ )
    {
      value = value * i;
    }
  }
  return value;
}
//****************************************************************************80

int i4_uniform_ab ( int a, int b, int &seed )

//****************************************************************************80
//
//  Purpose:
//
//    I4_UNIFORM_AB returns a scaled pseudorandom I4 between A and B.
//
//  Discussion:
//
//    The pseudorandom number should be uniformly distributed
//    between A and B.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    02 October 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, int A, B, the limits of the interval.
//
//    Input/output, int &SEED, the "seed" value, which should NOT be 0.
//    On output, SEED has been updated.
//
//    Output, int I4_UNIFORM, a number between A and B.
//
{
  int c;
  const int i4_huge = 2147483647;
  int k;
  float r;
  int value;

  if ( seed == 0 )
  {
    cerr << "\n";
    cerr << "I4_UNIFORM_AB - Fatal error!\n";
    cerr << "  Input value of SEED = 0.\n";
    exit ( 1 );
  }
//
//  Guarantee A <= B.
//
  if ( b < a )
  {
    c = a;
    a = b;
    b = c;
  }

  k = seed / 127773;

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

  if ( seed < 0 )
  {
    seed = seed + i4_huge;
  }

  r = ( float ) ( seed ) * 4.656612875E-10;
//
//  Scale R to lie between A-0.5 and B+0.5.
//
  r = ( 1.0 - r ) * ( ( float ) a - 0.5 ) 
    +         r   * ( ( float ) b + 0.5 );
//
//  Use rounding to convert R to an integer between A and B.
//
  value = round ( r );
//
//  Guarantee A <= VALUE <= B.
//
  if ( value < a )
  {
    value = a;
  }
  if ( b < value )
  {
    value = b;
  }

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

void i4vec_print ( int n, int a[], string title )

//****************************************************************************80
//
//  Purpose:
//
//    I4VEC_PRINT prints an I4VEC.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    23 November 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of components of the vector.
//
//    Input, int 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-1; i++ ) 
  {
    cout << setw(6) << i + 1 << "  " 
         << setw(6) << a[i]  << "\n";
  }

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

double r8_sign ( double x )

//****************************************************************************80
//
//  Purpose:
//
//    R8_SIGN returns the sign of an R8.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    18 October 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, the number whose sign is desired.
//
//    Output, double R8_SIGN, the sign of X.
//
{
  double value;

  if ( x < 0.0 )
  {
    value = -1.0;
  }
  else
  {
    value = 1.0;
  }
  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 is distributed under the MIT license. 
//
//  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 r8_uniform_ab ( double a, double b, int &seed )

//****************************************************************************80
//
//  Purpose:
//
//    R8_UNIFORM_AB returns a scaled pseudorandom R8.
//
//  Discussion:
//
//    The pseudorandom number should be uniformly distributed
//    between A and B.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    09 April 2012
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double A, B, the limits of the interval.
//
//    Input/output, int &SEED, the "seed" value, which should NOT be 0.
//    On output, SEED has been updated.
//
//    Output, double R8_UNIFORM_AB, a number strictly between A and B.
//
{
  const int i4_huge = 2147483647;
  int k;
  double value;

  if ( seed == 0 )
  {
    cerr << "\n";
    cerr << "R8_UNIFORM_AB - 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;
  }

  value = ( double ) ( seed ) * 4.656612875E-10;

  value = a + ( b - a ) * value;

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

void r8ge_cg ( int n, double a[], double b[], double x[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_CG uses the conjugate gradient method on an R8GE system.
//
//  Discussion:
//
//    The R8GE storage format is used for a general M by N matrix.  A storage 
//    space is made for each entry.  The two dimensional logical
//    array can be thought of as a vector of M*N entries, starting with
//    the M entries in the column 1, then the M entries in column 2
//    and so on.  Considered as a vector, the entry A(I,J) is then stored
//    in vector location I+(J-1)*M.
//
//    The matrix A must be a positive definite symmetric band matrix.
//
//    The method is designed to reach the solution after N computational
//    steps.  However, roundoff may introduce unacceptably large errors for
//    some problems.  In such a case, calling the routine again, using
//    the computed solution as the new starting estimate, should improve
//    the results.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    05 June 2014
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Frank Beckman,
//    The Solution of Linear Equations by the Conjugate Gradient Method,
//    in Mathematical Methods for Digital Computers,
//    edited by John Ralston, Herbert Wilf,
//    Wiley, 1967,
//    ISBN: 0471706892,
//    LC: QA76.5.R3.
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//    N must be positive.
//
//    Input, double A[N*N], the matrix.
//
//    Input, double B[N], the right hand side vector.
//
//    Input/output, double X[N].
//    On input, an estimate for the solution, which may be 0.
//    On output, the approximate solution vector.
//
{
  double alpha;
  double *ap;
  double beta;
  int i;
  int it;
  double *p;
  double pap;
  double pr;
  double *r;
  double rap;
//
//  Initialize
//    AP = A * x,
//    R  = b - A * x,
//    P  = b - A * x.
//
  ap = r8ge_mv ( n, n, a, x );

  r = new double[n];
  for ( i = 0; i < n; i++ )
  {
    r[i] = b[i] - ap[i];
  }

  p = new double[n];
  for ( i = 0; i < n; i++ )
  {
    p[i] = b[i] - ap[i];
  }
//
//  Do the N steps of the conjugate gradient method.
//
  for ( it = 1; it <= n; it++ )
  {
//
//  Compute the matrix*vector product AP=A*P.
//
    delete [] ap;
    ap = r8ge_mv ( n, n, a, p );
//
//  Compute the dot products
//    PAP = P*AP,
//    PR  = P*R
//  Set
//    ALPHA = PR / PAP.
//
    pap = r8vec_dot_product ( n, p, ap );
    pr = r8vec_dot_product ( n, p, r );

    if ( pap == 0.0 )
    {
      delete [] ap;
      break;
    }

    alpha = pr / pap;
//
//  Set
//    X = X + ALPHA * P
//    R = R - ALPHA * AP.
//
    for ( i = 0; i < n; i++ )
    {
      x[i] = x[i] + alpha * p[i];
    }
    for ( i = 0; i < n; i++ )
    {
      r[i] = r[i] - alpha * ap[i];
    }
//
//  Compute the vector dot product
//    RAP = R*AP
//  Set
//    BETA = - RAP / PAP.
//
    rap = r8vec_dot_product ( n, r, ap );

    beta = - rap / pap;
//
//  Update the perturbation vector
//    P = R + BETA * P.
//
    for ( i = 0; i < n; i++ )
    {
      p[i] = r[i] + beta * p[i];
    }
  }

  delete [] p;
  delete [] r;

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

double r8ge_co ( int n, double a[], int pivot[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_CO factors an R8GE matrix and estimates its condition number.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//    For the system A * X = B, relative perturbations in A and B
//    of size EPSILON may cause relative perturbations in X of size
//    EPSILON/RCOND.
//
//    If RCOND is so small that the logical expression
//      1.0 + rcond == 1.0
//    is true, then A may be singular to working precision.  In particular,
//    RCOND is zero if exact singularity is detected or the estimate
//    underflows.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    08 October 2003
//
//  Author:
//
//    Original FORTRAN77 version by Dongarra, Bunch, Moler, Stewart.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,
//    LINPACK User's Guide,
//    SIAM, 1979,
//    ISBN13: 978-0-898711-72-1,
//    LC: QA214.L56.
//
//  Parameters:
//
//    Input, int N, the order of the matrix A.
//
//    Input/output, double A[N*N].  On input, a matrix to be factored.
//    On output, the LU factorization of the matrix.
//
//    Output, int PIVOT[N], the pivot indices.
//
//    Output, double R8GE_CO, an estimate of the reciprocal condition number of A.
//
{
  double anorm;
  double ek;
  int i;
  int info;
  int j;
  int k;
  int l;
  double rcond;
  double s;
  double sm;
  double t;
  double wk;
  double wkm;
  double ynorm;
  double *z;
//
//  Compute the L1 norm of A.
//
  anorm = 0.0;
  for ( j = 0; j < n; j++ )
  {
    s = 0.0;
    for ( i = 0; i < n; i++ )
    {
      s = s + fabs ( a[i+j*n] );
    }
    anorm = fmax ( anorm, s );
  }
//
//  Compute the LU factorization.
//
  info = r8ge_fa ( n, a, pivot );

  if ( info != 0 ) 
  {
    rcond = 0.0;
    return rcond;
  }
//
//  RCOND = 1 / ( norm(A) * (estimate of norm(inverse(A))) )
//
//  estimate of norm(inverse(A)) = norm(Z) / norm(Y)
//
//  where
//    A * Z = Y
//  and
//    A' * Y = E
//
//  The components of E are chosen to cause maximum local growth in the
//  elements of W, where U'*W = E.  The vectors are frequently rescaled
//  to avoid overflow.
//
//  Solve U' * W = E.
//
  ek = 1.0;
  z = new double[n];
  for ( i = 0; i < n; i++ )
  {
    z[i] = 0.0;
  }

  for ( k = 0; k < n; k++ )
  {
    if ( z[k] != 0.0 ) 
    {
      ek = - r8_sign ( z[k] ) * fabs ( ek );
    }

    if ( fabs ( a[k+k*n] ) < fabs ( ek - z[k] ) )
    {
      s = fabs ( a[k+k*n] ) / fabs ( ek - z[k] );
      for ( i = 0; i < n; i++ )
      {
        z[i] = s * z[i];
      }
      ek = s * ek;
    }

    wk = ek - z[k];
    wkm = -ek - z[k];
    s = fabs ( wk );
    sm = fabs ( wkm );

    if ( a[k+k*n] != 0.0 )
    {
      wk = wk / a[k+k*n];
      wkm = wkm / a[k+k*n];
    }
    else
    {
      wk = 1.0;
      wkm = 1.0;
    }

    if ( k + 2 <= n )
    {
      for ( j = k + 1; j < n; j++ )
      {
        sm = sm + fabs ( z[j] + wkm * a[k+j*n] );
        z[j] = z[j] + wk * a[k+j*n];
        s = s + fabs ( z[j] );
      }

      if ( s < sm )
      {
        t = wkm - wk;
        wk = wkm;
        for ( j = k+1; j < n; j++ )
        {
          z[j] = z[j] + t * a[k+j*n];
        }
      }
    }
    z[k] = wk;
  }

  s = 0.0;
  for ( i = 0; i < n; i++ )
  {
    s = s + fabs ( z[i] );
  }

  for ( i = 0; i < n; i++ )
  {
    z[i] = z[i] / s;
  }
//
//  Solve L' * Y = W
//
  for ( k = n - 1; 0 <= k; k-- )
  {
    for ( i = k + 1; i < n; i++ )
    {
      z[k] = z[k] + z[i] * a[i+k*n];
    }
    t = fabs ( z[k] );
    if ( 1.0 < t )
    {
      for ( i = 0; i < n; i++ )
      {
        z[i] = z[i] / t;
      }
    }

    l = pivot[k] - 1;

    t    = z[l];
    z[l] = z[k];
    z[k] = t;
  }

  t = 0.0;
  for ( i = 0; i < n; i++ )
  {
    t = t + fabs ( z[i] );
  }
  for ( i = 0; i < n; i++ )
  {
    z[i] = z[i] / t;
  }

  ynorm = 1.0;
//
//  Solve L * V = Y.
//
  for ( k = 0; k < n; k++ )
  {
    l = pivot[k] - 1;

    t    = z[l];
    z[l] = z[k];
    z[k] = t;

    for ( i = k + 1; i < n; i++ )
    {
      z[i] = z[i] + t * a[i+k*n];
    }

    t = fabs ( z[k] );

    if ( 1.0 < t )
    {
      ynorm = ynorm / t;
      for ( i = 0; i < n; i++ )
      {
        z[i] = z[i] / t;
      }
    }
  }
  s = 0.0;
  for ( i = 0; i < n; i++ )
  {
    s = s + fabs ( z[i] );
  }

  for ( i = 0; i < n; i++ )
  {
    z[i] = z[i] / s;
  }
  ynorm = ynorm / s;
//
//  Solve U * Z = V.
//
  for ( k = n - 1; 0 <= k; k-- )
  {
    if ( fabs ( a[k+k*n] ) < fabs ( z[k] ) )
    {
      s = fabs ( a[k+k*n] ) / fabs ( z[k] );
      for ( i = 0; i < n; i++ )
      {
        z[i] = s * z[i];
      }
      ynorm = s * ynorm;
    }

    if ( a[k+k*n] != 0.0 )
    {
      z[k] = z[k] / a[k+k*n];
    }
    else
    {
      z[k] = 1.0;
    }

    for ( i = 0; i < k; i++ )
    {
      z[i] = z[i] - a[i+k*n] * z[k];
    }
  }
//
//  Normalize Z in the L1 norm.
//
  s = 0.0;
  for ( i = 0; i < n; i++ )
  {
    s = s + fabs ( z[i] );
  }
  s = 1.0 / s;

  for ( i = 0; i < n; i++ )
  {
    z[i] = s * z[i];
  }
  ynorm = s * ynorm;

  if ( anorm != 0.0 )
  {
    rcond = ynorm / anorm;
  }
  else
  {
    rcond = 0.0;
  }

  delete [] z;

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

void r8ge_copy ( int m, int n, double a1[], double a2[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_COPY copies one R8GE to a "new" R8GE.
//
//  Discussion:
//
//    An R8GE is a doubly dimensioned array of R8's, which
//    may be stored as a vector in column-major order.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    03 July 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns.
//
//    Input, double A1[M*N], the matrix to be copied.
//
//    Output, double A2[M*N], the copy of A1.
//
{
  int i;
  int j;

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      a2[i+j*m] = a1[i+j*m];
    }
  }
  return;
}
//****************************************************************************80

double *r8ge_copy_new ( int m, int n, double a1[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_COPY_NEW copies one R8GE to a "new" R8GE.
//
//  Discussion:
//
//    An R8GE is a doubly dimensioned array of R8's, which
//    may be stored as a vector in column-major order.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    03 July 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns.
//
//    Input, double A1[M*N], the matrix to be copied.
//
//    Output, double R8GE_COPY_NEW[M*N], the copy of A1.
//
{
  double *a2;
  int i;
  int j;

  a2 = new double[m*n];

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      a2[i+j*m] = a1[i+j*m];
    }
  }
  return a2;
}
//****************************************************************************80

double r8ge_det ( int n, double a_lu[], int pivot[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_DET computes the determinant of a matrix factored by R8GE_FA or R8GE_TRF.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    25 March 2004
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,
//    LINPACK User's Guide,
//    SIAM, 1979,
//    ISBN13: 978-0-898711-72-1,
//    LC: QA214.L56.
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//    N must be positive.
//
//    Input, double A_LU[N*N], the LU factors from R8GE_FA or R8GE_TRF.
//
//    Input, int PIVOT[N], as computed by R8GE_FA or R8GE_TRF.
//
//    Output, double R8GE_DET, the determinant of the matrix.
//
{
  double det;
  int i;

  det = 1.0;

  for ( i = 1; i <= n; i++ )
  {
    det = det * a_lu[i-1+(i-1)*n];
    if ( pivot[i-1] != i )
    {
      det = -det;
    }
  }

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

double *r8ge_dif2 ( int m, int n )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_DIF2 returns the DIF2 matrix in R8GE format.
//
//  Example:
//
//    N = 5
//
//    2 -1  .  .  .
//   -1  2 -1  .  .
//    . -1  2 -1  .
//    .  . -1  2 -1
//    .  .  . -1  2
//
//  Properties:
//
//    A is banded, with bandwidth 3.
//
//    A is tridiagonal.
//
//    Because A is tridiagonal, it has property A (bipartite).
//
//    A is a special case of the TRIS or tridiagonal scalar matrix.
//
//    A is integral, therefore det ( A ) is integral, and 
//    det ( A ) * inverse ( A ) is integral.
//
//    A is Toeplitz: constant along diagonals.
//
//    A is symmetric: A' = A.
//
//    Because A is symmetric, it is normal.
//
//    Because A is normal, it is diagonalizable.
//
//    A is persymmetric: A(I,J) = A(N+1-J,N+1-I).
//
//    A is positive definite.
//
//    A is an M matrix.
//
//    A is weakly diagonally dominant, but not strictly diagonally dominant.
//
//    A has an LU factorization A = L * U, without pivoting.
//
//      The matrix L is lower bidiagonal with subdiagonal elements:
//
//        L(I+1,I) = -I/(I+1)
//
//      The matrix U is upper bidiagonal, with diagonal elements
//
//        U(I,I) = (I+1)/I
//
//      and superdiagonal elements which are all -1.
//
//    A has a Cholesky factorization A = L * L', with L lower bidiagonal.
//
//      L(I,I) =    sqrt ( (I+1) / I )
//      L(I,I-1) = -sqrt ( (I-1) / I )
//
//    The eigenvalues are
//
//      LAMBDA(I) = 2 + 2 * COS(I*PI/(N+1))
//                = 4 SIN^2(I*PI/(2*N+2))
//
//    The corresponding eigenvector X(I) has entries
//
//       X(I)(J) = sqrt(2/(N+1)) * sin ( I*J*PI/(N+1) ).
//
//    Simple linear systems:
//
//      x = (1,1,1,...,1,1),   A*x=(1,0,0,...,0,1)
//
//      x = (1,2,3,...,n-1,n), A*x=(0,0,0,...,0,n+1)
//
//    det ( A ) = N + 1.
//
//    The value of the determinant can be seen by induction,
//    and expanding the determinant across the first row:
//
//      det ( A(N) ) = 2 * det ( A(N-1) ) - (-1) * (-1) * det ( A(N-2) )
//                = 2 * N - (N-1)
//                = N + 1
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    06 June 2016
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Robert Gregory, David Karney,
//    A Collection of Matrices for Testing Computational Algorithms,
//    Wiley, 1969,
//    ISBN: 0882756494,
//    LC: QA263.68
//
//    Morris Newman, John Todd,
//    Example A8,
//    The evaluation of matrix inversion programs,
//    Journal of the Society for Industrial and Applied Mathematics,
//    Volume 6, Number 4, pages 466-476, 1958.
//
//    John Todd,
//    Basic Numerical Mathematics,
//    Volume 2: Numerical Algebra,
//    Birkhauser, 1980,
//    ISBN: 0817608117,
//    LC: QA297.T58.
//
//    Joan Westlake,
//    A Handbook of Numerical Matrix Inversion and Solution of 
//    Linear Equations,
//    John Wiley, 1968,
//    ISBN13: 978-0471936756,
//    LC: QA263.W47.
//
//  Parameters:
//
//    Input, int M, N, the order of the matrix.
//
//    Output, double R8GE_DIF2[M*N], the matrix.
//
{
  double *a;
  int i;
  int j;

  a = new double[m*n];

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      if ( j == i - 1 )
      {
        a[i+j*m] = -1.0;
      }
      else if ( j == i )
      {
        a[i+j*m] = 2.0;
      }
      else if ( j == i + 1 )
      {
        a[i+j*m] = -1.0;
      }
      else
      {
        a[i+j*m] = 0.0;
      }
    }
  }

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

double *r8ge_dilu ( int m, int n, double a[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_DILU produces the diagonal incomplete LU factor of an R8GE matrix.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    15 October 2003
//
//  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 R8GE matrix.
//
//    Output, double R8GE_DILU[M], the D-ILU factor.
//
{
  double *d;
  int i;
  int j;

  d = new double[m];

  for ( i = 0; i < m; i++ )
  {
    if ( i < n ) 
    {
      d[i] = a[i+i*m];
    }
    else
    {
      d[i] = 0.0;
    }
  }

  for ( i = 0; i < m && i < n; i++ )
  {
    d[i] = 1.0 / d[i];
    for ( j = i+1; j < m && j < n; j++ )
    {
      d[j] = d[j] - a[j+i*m] * d[i] * a[i+j*m];
    }
  }

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

int r8ge_fa ( int n, double a[], int pivot[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_FA performs a LINPACK-style PLU factorization of an R8GE matrix.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//    R8GE_FA is a simplified version of the LINPACK routine SGEFA.
//
//    The two dimensional array is stored by columns in a one dimensional
//    array.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    11 September 2003
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,
//    LINPACK User's Guide,
//    SIAM, 1979,
//    ISBN13: 978-0-898711-72-1,
//    LC: QA214.L56.
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//    N must be positive.
//
//    Input/output, double A[N*N], the matrix to be factored.
//    On output, A contains an upper triangular matrix and the multipliers
//    which were used to obtain it.  The factorization can be written
//    A = L * U, where L is a product of permutation and unit lower
//    triangular matrices and U is upper triangular.
//
//    Output, int PIVOT[N], a vector of pivot indices.
//
//    Output, int R8GE_FA, singularity flag.
//    0, no singularity detected.
//    nonzero, the factorization failed on the INFO-th step.
//
{
  int i;
  int j;
  int k;
  int l;
  double t;
//
  for ( k = 1; k <= n-1; k++ )
  {
//
//  Find L, the index of the pivot row.
//
    l = k;

    for ( i = k + 1; i <= n; i++ )
    {
      if ( fabs ( a[l-1+(k-1)*n] ) < fabs ( a[i-1+(k-1)*n] ) )
      {
        l = i;
      }
    }

    pivot[k-1] = l;
//
//  If the pivot index is zero, the algorithm has failed.
//
    if ( a[l-1+(k-1)*n] == 0.0 )
    {
      cerr << "\n";
      cerr << "R8GE_FA - Fatal error!\n";
      cerr << "  Zero pivot on step " << k << "\n";
      exit ( 1 );
    }
//
//  Interchange rows L and K if necessary.
//
    if ( l != k )
    {
      t              = a[l-1+(k-1)*n];
      a[l-1+(k-1)*n] = a[k-1+(k-1)*n];
      a[k-1+(k-1)*n] = t;
    }
//
//  Normalize the values that lie below the pivot entry A(K,K).
//
    for ( i = k+1; i <= n; i++ )
    {
      a[i-1+(k-1)*n] = -a[i-1+(k-1)*n] / a[k-1+(k-1)*n];
    }
//
//  Row elimination with column indexing.
//
    for ( j = k+1; j <= n; j++ )
    {
      if ( l != k )
      {
        t              = a[l-1+(j-1)*n];
        a[l-1+(j-1)*n] = a[k-1+(j-1)*n];
        a[k-1+(j-1)*n] = t;
      }

      for ( i = k+1; i <= n; i++ )
      {
        a[i-1+(j-1)*n] = a[i-1+(j-1)*n] + a[i-1+(k-1)*n] * a[k-1+(j-1)*n];
      }

    }

  }

  pivot[n-1] = n;

  if ( a[n-1+(n-1)*n] == 0.0 )
  {
    cerr << "\n";
    cerr << "R8GE_FA - Fatal error!\n";
    cerr << "  Zero pivot on step " << n << "\n";
    exit ( 1 );
  }

  return 0;
}
//****************************************************************************80

void r8ge_fs ( int n, double a[], double x[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_FS factors and solves an R8GE system.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//    The function does not save the LU factors of the matrix, and hence cannot
//    be used to efficiently solve multiple linear systems, or even to
//    factor A at one time, and solve a single linear system at a later time.
//
//    The function uses partial pivoting, but no pivot vector is required.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    04 December 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//    N must be positive.
//
//    Input/output, double A[N*N].
//    On input, A is the coefficient matrix of the linear system.
//    On output, A is in unit upper triangular form, and
//    represents the U factor of an LU factorization of the
//    original coefficient matrix.
//
//    Input/output, double X[N], on input, the right hand side of the linear system.
//    On output, the solution of the linear system.
//
{
  int i;
  int ipiv;
  int j;
  int jcol;
  double piv;
  double t;

  for ( jcol = 1; jcol <= n; jcol++ )
  {
//
//  Find the maximum element in column I.
//
    piv = fabs ( a[jcol-1+(jcol-1)*n] );
    ipiv = jcol;
    for ( i = jcol + 1; i <= n; i++ )
    {
      if ( piv < fabs ( a[i-1+(jcol-1)*n] ) )
      {
        piv = fabs ( a[i-1+(jcol-1)*n] );
        ipiv = i;
      }
    }

    if ( piv == 0.0 )
    {
      cerr << "\n";
      cerr << "R8GE_FS - Fatal error!\n";
      cerr << "  Zero pivot on step " << jcol << "\n";
      exit ( 1 );
    }
//
//  Switch rows JCOL and IPIV, and X.
//
    if ( jcol != ipiv )
    {
      for ( j = 1; j <= n; j++ )
      {
        t                 = a[jcol-1+(j-1)*n];
        a[jcol-1+(j-1)*n] = a[ipiv-1+(j-1)*n];
        a[ipiv-1+(j-1)*n] = t;
      }
      t         = x[jcol-1];
      x[jcol-1] = x[ipiv-1];
      x[ipiv-1] = t;
    }
//
//  Scale the pivot row.
//
    t = a[jcol-1+(jcol-1)*n];
    a[jcol-1+(jcol-1)*n] = 1.0;
    for ( j = jcol+1; j <= n; j++ )
    {
      a[jcol-1+(j-1)*n] = a[jcol-1+(j-1)*n] / t;
    }
    x[jcol-1] = x[jcol-1] / t;
//
//  Use the pivot row to eliminate lower entries in that column.
//
    for ( i = jcol+1; i <= n; i++ )
    {
      if ( a[i-1+(jcol-1)*n] != 0.0 )
      {
        t = - a[i-1+(jcol-1)*n];
        a[i-1+(jcol-1)*n] = 0.0;
        for ( j = jcol+1; j <= n; j++ )
        {
          a[i-1+(j-1)*n] = a[i-1+(j-1)*n] + t * a[jcol-1+(j-1)*n];
        }
        x[i-1] = x[i-1] + t * x[jcol-1];
      }
    }
  }
//
//  Back solve.
//
  for ( jcol = n; 2 <= jcol; jcol-- )
  {
    for ( i = 1; i < jcol; i++ )
    {
      x[i-1] = x[i-1] - a[i-1+(jcol-1)*n] * x[jcol-1];
    }
  }

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

double *r8ge_fs_new ( int n, double a[], double b[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_FS_NEW factors and solves an R8GE system.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//    The function does not save the LU factors of the matrix, and hence cannot
//    be used to efficiently solve multiple linear systems, or even to
//    factor A at one time, and solve a single linear system at a later time.
//
//    The function uses partial pivoting, but no pivot vector is required.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    04 December 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//    N must be positive.
//
//    Input/output, double A[N*N].
//    On input, A is the coefficient matrix of the linear system.
//    On output, A is in unit upper triangular form, and
//    represents the U factor of an LU factorization of the
//    original coefficient matrix.
//
//    Input, double B[N], the right hand side of the linear system.
//
//    Output, double R8GE_FS_NEW[N], the solution of the linear system.
//
{
  int i;
  int ipiv;
  int j;
  int jcol;
  double piv;
  double t;
  double *x;

  x = new double[n];

  for ( i = 0; i < n; i++ )
  {
    x[i] = b[i];
  }

  for ( jcol = 1; jcol <= n; jcol++ )
  {
//
//  Find the maximum element in column I.
//
    piv = fabs ( a[jcol-1+(jcol-1)*n] );
    ipiv = jcol;
    for ( i = jcol+1; i <= n; i++ )
    {
      if ( piv < fabs ( a[i-1+(jcol-1)*n] ) )
      {
        piv = fabs ( a[i-1+(jcol-1)*n] );
        ipiv = i;
      }
    }

    if ( piv == 0.0 )
    {
      cerr << "\n";
      cerr << "R8GE_FS_NEW - Fatal error!\n";
      cerr << "  Zero pivot on step " << jcol << "\n";
      exit ( 1 );
    }
//
//  Switch rows JCOL and IPIV, and X.
//
    if ( jcol != ipiv )
    {
      for ( j = 1; j <= n; j++ )
      {
        t                 = a[jcol-1+(j-1)*n];
        a[jcol-1+(j-1)*n] = a[ipiv-1+(j-1)*n];
        a[ipiv-1+(j-1)*n] = t;
      }
      t         = x[jcol-1];
      x[jcol-1] = x[ipiv-1];
      x[ipiv-1] = t;
    }
//
//  Scale the pivot row.
//
    t = a[jcol-1+(jcol-1)*n];
    a[jcol-1+(jcol-1)*n] = 1.0;
    for ( j = jcol+1; j <= n; j++ )
    {
      a[jcol-1+(j-1)*n] = a[jcol-1+(j-1)*n] / t;
    }
    x[jcol-1] = x[jcol-1] / t;
//
//  Use the pivot row to eliminate lower entries in that column.
//
    for ( i = jcol+1; i <= n; i++ )
    {
      if ( a[i-1+(jcol-1)*n] != 0.0 )
      {
        t = - a[i-1+(jcol-1)*n];
        a[i-1+(jcol-1)*n] = 0.0;
        for ( j = jcol+1; j <= n; j++ )
        {
          a[i-1+(j-1)*n] = a[i-1+(j-1)*n] + t * a[jcol-1+(j-1)*n];
        }
        x[i-1] = x[i-1] + t * x[jcol-1];
      }
    }
  }
//
//  Back solve.
//
  for ( jcol = n; 2 <= jcol; jcol-- )
  {
    for ( i = 1; i < jcol; i++ )
    {
      x[i-1] = x[i-1] - a[i-1+(jcol-1)*n] * x[jcol-1];
    }
  }

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

void r8ge_fss ( int n, double a[], int nb, double x[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_FSS factors and solves multiple R8GE systems.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//    This routine does not save the LU factors of the matrix, and hence cannot
//    be used to efficiently solve multiple linear systems, or even to
//    factor A at one time, and solve a single linear system at a later time.
//
//    This routine uses partial pivoting, but no pivot vector is required.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    11 August 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//    N must be positive.
//
//    Input/output, double A[N*N].
//    On input, A is the coefficient matrix of the linear system.
//    On output, A is in unit upper triangular form, and
//    represents the U factor of an LU factorization of the
//    original coefficient matrix.
//
//    Input, int NB, the number of right hand sides.
//
//    Input/output, double X[N*NB], on input, the right hand sides of the
//    linear systems.  On output, the solutions of the linear systems.
//
{
  int i;
  int ipiv;
  int j;
  int jcol;
  double piv;
  double t;

  for ( jcol = 1; jcol <= n; jcol++ )
  {
//
//  Find the maximum element in column I.
//
    piv = fabs ( a[jcol-1+(jcol-1)*n] );
    ipiv = jcol;
    for ( i = jcol+1; i <= n; i++ )
    {
      if ( piv < fabs ( a[i-1+(jcol-1)*n] ) )
      {
        piv = fabs ( a[i-1+(jcol-1)*n] );
        ipiv = i;
      }
    }

    if ( piv == 0.0 )
    {
      cerr << "\n";
      cerr << "R8GE_FSS - Fatal error!\n";
      cerr << "  Zero pivot on step " << jcol << "\n";
      exit ( 1 );
    }
//
//  Switch rows JCOL and IPIV, and X.
//
    if ( jcol != ipiv )
    {
      for ( j = 1; j <= n; j++ )
      {
        t                 = a[jcol-1+(j-1)*n];
        a[jcol-1+(j-1)*n] = a[ipiv-1+(j-1)*n];
        a[ipiv-1+(j-1)*n] = t;
      }
      for ( j = 0; j < nb; j++ )
      {
        t            = x[jcol-1+j*n];
        x[jcol-1+j*n] = x[ipiv-1+j*n];
        x[ipiv-1+j*n] = t;
      }
    }
//
//  Scale the pivot row.
//
    t = a[jcol-1+(jcol-1)*n];
    a[jcol-1+(jcol-1)*n] = 1.0;
    for ( j = jcol+1; j <= n; j++ )
    {
      a[jcol-1+(j-1)*n] = a[jcol-1+(j-1)*n] / t;
    }
    for ( j = 0; j < nb; j++ )
    {
      x[jcol-1+j*n] = x[jcol-1+j*n] / t;
    }
//
//  Use the pivot row to eliminate lower entries in that column.
//
    for ( i = jcol+1; i <= n; i++ )
    {
      if ( a[i-1+(jcol-1)*n] != 0.0 )
      {
        t = - a[i-1+(jcol-1)*n];
        a[i-1+(jcol-1)*n] = 0.0;
        for ( j = jcol+1; j <= n; j++ )
        {
          a[i-1+(j-1)*n] = a[i-1+(j-1)*n] + t * a[jcol-1+(j-1)*n];
        }
        for ( j = 0; j < nb; j++ )
        {
          x[i-1+j*n] = x[i-1+j*n] + t * x[jcol-1+j*n];
        }
      }
    }
  }
//
//  Back solve.
//
  for ( jcol = n; 2 <= jcol; jcol-- )
  {
    for ( i = 1; i < jcol; i++ )
    {
      for ( j = 0; j < nb; j++ )
      {
        x[i-1+j*n] = x[i-1+j*n] - a[i-1+(jcol-1)*n] * x[jcol-1+j*n];
      }
    }
  }

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

double *r8ge_fss_new ( int n, double a[], int nb, double b[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_FSS_NEW factors and solves multiple R8GE systems.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//    This routine does not save the LU factors of the matrix, and hence cannot
//    be used to efficiently solve multiple linear systems, or even to
//    factor A at one time, and solve a single linear system at a later time.
//
//    This routine uses partial pivoting, but no pivot vector is required.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    23 June 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//    N must be positive.
//
//    Input/output, double A[N*N].
//    On input, A is the coefficient matrix of the linear system.
//    On output, A is in unit upper triangular form, and
//    represents the U factor of an LU factorization of the
//    original coefficient matrix.
//
//    Input, int NB, the number of right hand sides.
//
//    Input, double B[N*NB], the right hand sides of the linear systems.
//
//    Output, double R8GE_FSS_NEW[N*NB], the solutions of the linear systems.
//
{
  int i;
  int ipiv;
  int j;
  int jcol;
  double piv;
  double t;
  double *x;

  x = new double[n*nb];

  for ( j = 0; j < nb; j++ )
  {
    for ( i = 0; i < n; i++ )
    {
      x[i+j*n] = b[i+j*n];
    }
  }
  for ( jcol = 1; jcol <= n; jcol++ )
  {
//
//  Find the maximum element in column I.
//
    piv = fabs ( a[jcol-1+(jcol-1)*n] );
    ipiv = jcol;
    for ( i = jcol+1; i <= n; i++ )
    {
      if ( piv < fabs ( a[i-1+(jcol-1)*n] ) )
      {
        piv = fabs ( a[i-1+(jcol-1)*n] );
        ipiv = i;
      }
    }

    if ( piv == 0.0 )
    {
      cerr << "\n";
      cerr << "R8GE_FSS_NEW - Fatal error!\n";
      cerr << "  Zero pivot on step " << jcol << "\n";
      exit ( 1 );
    }
//
//  Switch rows JCOL and IPIV, and X.
//
    if ( jcol != ipiv )
    {
      for ( j = 1; j <= n; j++ )
      {
        t                 = a[jcol-1+(j-1)*n];
        a[jcol-1+(j-1)*n] = a[ipiv-1+(j-1)*n];
        a[ipiv-1+(j-1)*n] = t;
      }
      for ( j = 0; j < nb; j++ )
      {
        t            = x[jcol-1+j*n];
        x[jcol-1+j*n] = x[ipiv-1+j*n];
        x[ipiv-1+j*n] = t;
      }
    }
//
//  Scale the pivot row.
//
    t = a[jcol-1+(jcol-1)*n];
    a[jcol-1+(jcol-1)*n] = 1.0;
    for ( j = jcol+1; j <= n; j++ )
    {
      a[jcol-1+(j-1)*n] = a[jcol-1+(j-1)*n] / t;
    }
    for ( j = 0; j < nb; j++ )
    {
      x[jcol-1+j*n] = x[jcol-1+j*n] / t;
    }
//
//  Use the pivot row to eliminate lower entries in that column.
//
    for ( i = jcol+1; i <= n; i++ )
    {
      if ( a[i-1+(jcol-1)*n] != 0.0 )
      {
        t = - a[i-1+(jcol-1)*n];
        a[i-1+(jcol-1)*n] = 0.0;
        for ( j = jcol+1; j <= n; j++ )
        {
          a[i-1+(j-1)*n] = a[i-1+(j-1)*n] + t * a[jcol-1+(j-1)*n];
        }
        for ( j = 0; j < nb; j++ )
        {
          x[i-1+j*n] = x[i-1+j*n] + t * x[jcol-1+j*n];
        }
      }
    }
  }
//
//  Back solve.
//
  for ( jcol = n; 2 <= jcol; jcol-- )
  {
    for ( i = 1; i < jcol; i++ )
    {
      for ( j = 0; j < nb; j++ )
      {
        x[i-1+j*n] = x[i-1+j*n] - a[i-1+(jcol-1)*n] * x[jcol-1+j*n];
      }
    }
  }

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

double *r8ge_hilbert ( int m, int n )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_HILBERT returns the Hilbert matrix.
//
//  Formula:
//
//    A(I,J) = 1 / ( I + J - 1 )
//
//  Example:
//
//    N = 5
//
//    1/1 1/2 1/3 1/4 1/5
//    1/2 1/3 1/4 1/5 1/6
//    1/3 1/4 1/5 1/6 1/7
//    1/4 1/5 1/6 1/7 1/8
//    1/5 1/6 1/7 1/8 1/9
//
//  Properties:
//
//    A is a Hankel matrix: constant along anti-diagonals.
//
//    A is positive definite.
//
//    A is symmetric: A' = A.
//
//    Because A is symmetric, it is normal.
//
//    Because A is normal, it is diagonalizable.
//
//    A is totally positive.
//
//    A is a Cauchy matrix.
//
//    A is nonsingular.
//
//    A is very ill-conditioned.
//
//    The entries of the inverse of A are all integers.
//
//    The sum of the entries of the inverse of A is N*N.
//
//    The ratio of the absolute values of the maximum and minimum
//    eigenvalues is roughly EXP(3.5*N).
//
//    The determinant of the Hilbert matrix of order 10 is
//    2.16417... * 10^(-53).
//
//    If the (1,1) entry of the 5 by 5 Hilbert matrix is changed
//    from 1 to 24/25, the matrix is exactly singular.  And there
//    is a similar rule for larger Hilbert matrices.
//
//    The family of matrices is nested as a function of N.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    20 July 2008
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    MD Choi,
//    Tricks or treats with the Hilbert matrix,
//    American Mathematical Monthly,
//    Volume 90, 1983, pages 301-312.
//
//    Robert Gregory, David Karney,
//    A Collection of Matrices for Testing Computational Algorithms,
//    Wiley, 1969,
//    ISBN: 0882756494,
//    LC: QA263.68
//
//    Nicholas Higham,
//    Accuracy and Stability of Numerical Algorithms,
//    Society for Industrial and Applied Mathematics, Philadelphia, PA,
//    USA, 1996; section 26.1.
//
//    Donald Knuth,
//    The Art of Computer Programming,
//    Volume 1, Fundamental Algorithms, Second Edition
//    Addison-Wesley, Reading, Massachusetts, 1973, page 37.
//
//    Morris Newman, John Todd,
//    Example A13,
//    The evaluation of matrix inversion programs,
//    Journal of the Society for Industrial and Applied Mathematics,
//    Volume 6, 1958, pages 466-476.
//
//    Joan Westlake,
//    A Handbook of Numerical Matrix Inversion and Solution of 
//    Linear Equations,
//    John Wiley, 1968,
//    ISBN13: 978-0471936756,
//    LC: QA263.W47.
//
//  Parameters:
//
//    Input, int M, N, the order of the matrix.
//
//    Output, double R8GE_HILBERT[M*N], the matrix.
//
{
  double *a;
  int i;
  int j;

  a = new double[m*n];
 
  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      a[i+j*m] = 1.0 / ( double ) ( i + j + 1 );
    }
  }
  return a;
}
//****************************************************************************80

double *r8ge_hilbert_inverse ( int n )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_HILBERT_INVERSE returns the inverse of the HILBERT matrix.
//
//  Formula:
//
//    A(I,J) =  (-1)^(I+J) * (N+I-1)! * (N+J-1)! /
//           [ (I+J-1) * ((I-1)!*(J-1)!)^2 * (N-I)! * (N-J)! ]
//
//  Example:
//
//    N = 5
//
//       25    -300     1050    -1400     630
//     -300    4800   -18900    26880  -12600
//     1050  -18900    79380  -117600   56700
//    -1400   26880  -117600   179200  -88200
//      630  -12600    56700   -88200   44100
//
//  Properties:
//
//    A is symmetric: A' = A.
//
//    Because A is symmetric, it is normal.
//
//    Because A is normal, it is diagonalizable.
//
//    A is almost impossible to compute accurately by general routines
//    that compute the inverse.
//
//    A is the exact inverse of the Hilbert matrix; however, if the
//    Hilbert matrix is stored on a finite precision computer, and
//    hence rounded, A is actually a poor approximation
//    to the inverse of that rounded matrix.  Even though Gauss elimination
//    has difficulty with the Hilbert matrix, it can compute an approximate
//    inverse matrix whose residual is lower than that of the
//    "exact" inverse.
//
//    All entries of A are integers.
//
//    The sum of the entries of A is N^2.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    25 June 2011
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//
//    Output, double R8GE_HILBERT_INVERSE[N*N], the matrix.
//
{
  double *a;
  int i;
  int j;

  a = new double[n*n];
//
//  Set the (1,1) entry.
//
  a[0+0*n] = ( double ) ( n * n );
//
//  Define Row 1, Column J by recursion on Row 1 Column J-1
//
  i = 0;
  for ( j = 1; j < n; j++ )
  {
    a[i+j*n] = - a[i+(j-1)*n] 
      * ( double ) ( ( n + j ) * ( i + j ) * ( n - j ) ) 
      / ( double ) ( ( i + j + 1 ) * j * j );
  }
//
//  Define Row I by recursion on row I-1
//
  for ( i = 1; i < n; i++ )
  {
    for ( j = 0; j < n; j++ )
    {
      a[i+j*n] = - a[i-1+j*n] 
        * ( double ) ( ( n + i ) * ( i + j ) * ( n - i ) ) 
        / ( double ) ( ( i + j + 1 ) * ( i ) * ( i ) );
    }
  }
  return a;
}
//****************************************************************************80

void r8ge_identity ( int m, int n, double a[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_IDENTITY sets an R8GE matrix to the identity.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    15 September 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the order of A.
//
//    Output, double A[M*N], the identity matrix.
//
{
  int i;
  int j;

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      if ( i == j )
      {
        a[i+j*m] = 1.0;
      }
      else
      {
        a[i+j*m] = 0.0;
      }
    }
  }

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

double *r8ge_identity_new ( int m, int n )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_IDENTITY_NEW sets an R8GE matrix to the identity.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    09 february 2016
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the order of A.
//
//    Output, double R8GE_IDENTITY_NEW[M*N], the identity matrix.
//
{
  double *a;
  int i;
  int j;

  a = new double[m*n];

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      if ( i == j )
      {
        a[i+j*m] = 1.0;
      }
      else
      {
        a[i+j*m] = 0.0;
      }
    }
  }

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

void r8ge_ilu ( int m, int n, double a[], double l[], double u[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_ILU produces the incomplete LU factors of an R8GE matrix.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//    The incomplete LU factors of the M by N matrix A are:
//
//      L, an M by M unit lower triangular matrix,
//      U, an M by N upper triangular matrix
//
//    with the property that L and U are computed in the same way as
//    the usual LU factors, except that, whenever an off diagonal element
//    of the original matrix is zero, then the corresponding value of
//    U is forced to be zero.
//
//    This condition means that it is no longer the case that A = L*U.
//
//    On the other hand, L and U will have a simple sparsity structure
//    related to that of A.  The incomplete LU factorization is generally
//    used as a preconditioner in iterative schemes applied to sparse
//    matrices.  It is presented here merely for illustration.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    24 November 2003
//
//  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 R8GE matrix.
//
//    Output, double L[M*M], the M by M unit lower triangular factor.
//
//    Output, double U[M*N], the M by N upper triangular factor.
//
{
  int i;
  int j;
  int jhi;
  int k;
//
//  Initialize:
//
//    L := M by M Identity
//    U := A
//
  r8ge_identity ( m, m, l );

  r8ge_copy ( m, n, a, u );

  jhi = i4_min ( m - 1, n );

  for ( j = 0; j < jhi; j++ )
  {
//
//  Zero out the entries in column J, from row J+1 to M.
//
    for ( i = j+1; i < m; i++ )
    {
      if ( u[i+j*m] != 0.0 )
      {
        l[i+j*m] = u[i+j*m] / u[j+j*m];
        u[i+j*m] = 0.0;

        for ( k = j + 1; k < n; k++ )
        {
          if ( u[i+k*m] != 0.0 )
          {
            u[i+k*m] = u[i+k*m] - l[i+j*m] * u[j+k*m];
          }
        }
      }
    }
  }

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

double *r8ge_indicator ( int m, int n )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_INDICATOR sets up an R8GE indicator matrix.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    25 January 2005
//
//  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.
//
//    Output, double R8GE_INDICATOR[M*N], the R8GE matrix.
//
{
  double *a;
  int fac;
  int i;
  int j;

  a = new double[m*n];

  fac = i4_power ( 10, i4_log_10 ( n ) + 1 );

  for ( i = 1; i <= m; i++ )
  {
    for ( j = 1; j <= n; j++ )
    {
      a[i-1+(j-1)*m] = ( double ) ( fac * i + j );
    }
  }

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

double *r8ge_inverse ( int n, double a[], int pivot[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_INVERSE computes the inverse of an R8GE matrix factored by R8GE_FA.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//    R8GE_INVERSE is a simplified standalone version of the LINPACK routine
//    SGEDI.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    15 September 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the order of the matrix A.
//
//    Input, double A[N*N], the factor information computed by R8GE_FA.
//
//    Input, int PIVOT(N), the pivot vector from R8GE_FA.
//
//    Output, double R8GE_INVERSE[N*N], the inverse matrix.
//
{
  double *b;
  int i;
  int j;
  int k;
  double temp;

  b = new double[n*n];
//
//  Compute Inverse(U).
//
  for ( k = 1; k <= n; k++ )
  {
    for ( i = 1; i <= k-1; i++ )
    {
      b[i-1+(k-1)*n] = -b[i-1+(k-1)*n] / a[k-1+(k-1)*n];
    }
    b[k-1+(k-1)*n] = 1.0 / a[k-1+(k-1)*n];

    for ( j = k+1; j <= n; j++ )
    {
      b[k-1+(j-1)*n] = 0.0;
      for ( i = 1; i <= k; i++ )
      {
        b[i-1+(j-1)*n] = b[i-1+(j-1)*n] + b[i-1+(k-1)*n] * a[k-1+(j-1)*n];
      }
    }
  }
//
//  Multiply Inverse(U) by Inverse(L).
//
  for ( k = n-1; 1 <= k; k-- )
  {
    for ( i = k+1; i <= n; i++ )
    {
      b[i-1+(k-1)*n] = 0.0;
    }

    for ( j = k+1; j <= n; j++ )
    {
      for ( i = 1; i <= n; i++ )
      {
        b[i-1+(k-1)*n] = b[i-1+(k-1)*n] + b[i-1+(j-1)*n] * a[j-1+(k-1)*n];
      }
    }

    if ( pivot[k-1] != k )
    {
      for ( i = 1; i <= n; i++ )
      {
        temp = b[i-1+(k-1)*n];
        b[i-1+(k-1)*n] = b[i-1+(pivot[k-1]-1)*n];
        b[i-1+(pivot[k-1]-1)*n] = temp;
      }

    }

  }

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

double *r8ge_ml ( int n, double a_lu[], int pivot[], double x[], int job )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_ML computes A * x or A' * x, using R8GE_FA factors.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//    It is assumed that R8GE_FA has overwritten the original matrix
//    information by LU factors.  R8GE_ML is able to reconstruct the
//    original matrix from the LU factor data.
//
//    R8GE_ML allows the user to check that the solution of a linear
//    system is correct, without having to save an unfactored copy
//    of the matrix.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    11 September 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//    N must be positive.
//
//    Input, double A_LU[N*N], the LU factors from R8GE_FA.
//
//    Input, int PIVOT[N], the pivot vector computed by R8GE_FA.
//
//    Input, double X[N], the vector to be multiplied.
//
//    Input, int JOB, specifies the operation to be done:
//    JOB = 0, compute A * x.
//    JOB nonzero, compute A' * X.
//
//    Output, double R8GE_ML[N], the result of the multiplication.
//
{
  double *b;
  int i;
  int j;
  int k;
  double temp;
//
  b = new double[n];

  for ( i = 0; i < n; i++ )
  {
    b[i] = x[i];
  }

  if ( job == 0 )
  {
//
//  Y = U * X.
//
    for ( j = 1; j <= n; j++ )
    {
      for ( i = 1; i <= j-1; i++ )
      {
        b[i-1] = b[i-1] + a_lu[i-1+(j-1)*n] * b[j-1];
      }
      b[j-1] = a_lu[j-1+(j-1)*n] * b[j-1];
    }
//
//  B = PL * Y = PL * U * X = A * x.
//
    for ( j = n-1; 1 <= j; j-- )
    {
      for ( i = j+1; i <= n; i++ )
      {
        b[i-1] = b[i-1] - a_lu[i-1+(j-1)*n] * b[j-1];
      }

      k = pivot[j-1];

      if ( k != j )
      {
        temp   = b[k-1];
        b[k-1] = b[j-1];
        b[j-1] = temp;
      }
    }
  }
  else
  {
//
//  Y = (PL)' * X:
//
    for ( j = 1; j <= n-1; j++ )
    {
      k = pivot[j-1];

      if ( k != j )
      {
        temp   = b[k-1];
        b[k-1] = b[j-1];
        b[j-1] = temp;
      }

      temp = 0.0;
      for ( i = j+1; i <= n; i++ )
      {
        temp = temp + b[i-1] * a_lu[i-1+(j-1)*n];
      }
      b[j-1] = b[j-1] - temp;

    }
//
//  B = U' * Y = ( PL * U )' * X = A' * X.
//
    for ( i = n; 1 <= i; i-- )
    {
      for ( j = i+1; j <= n; j++ )
      {
        b[j-1] = b[j-1] + b[i-1] * a_lu[i-1+(j-1)*n];
      }
      b[i-1] = b[i-1] * a_lu[i-1+(i-1)*n];
    }

  }

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

double *r8ge_mm_new ( int n1, int n2, int n3, double a[], double b[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_MM_NEW multiplies two matrices.
//
//  Discussion:
//
//    An R8GE 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 MIT license.
//
//  Modified:
//
//    02 December 2011
//
//  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 R8GE_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

double *r8ge_mtm_new ( int n1, int n2, int n3, double a[], double b[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_MTM_NEW computes C = A' * B.
//
//  Discussion:
//
//    An R8GE 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 MIT license.
//
//  Modified:
//
//    07 September 2012
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N1, N2, N3, the order of the matrices.
//
//    Input, double A[N2*N1], double B[N2*N3], the matrices to multiply.
//
//    Output, double R8GE_MTM_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[k+i*n2] * b[k+j*n2];
      }
    }
  }

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

double *r8ge_mtv ( int m, int n, double a[], double x[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_MTV multiplies a vector times an R8GE matrix.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    13 September 2003
//
//  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 R8GE matrix.
//
//    Input, double X[M], the vector to be multiplied by A.
//
//    Output, double R8GE_MTV[N], the product A' * x.
//
{
  double *b;
  int i;
  int j;

  b = r8vec_zeros_new ( n );

  for ( i = 0; i < n; i++ )
  {
    for ( j = 0; j < m; j++ )
    {
      b[i] = b[i] + a[j+i*m] * x[j];
    }
  }

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

double *r8ge_mu ( int m, int n, double a_lu[], char trans, int pivot[], 
  double x[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_MU computes A * x or A' * x, using R8GE_TRF factors.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//    It is assumed that R8GE_TRF has overwritten the original matrix
//    information by PLU factors.  R8GE_MU is able to reconstruct the
//    original matrix from the PLU factor data.
//
//    R8GE_MU allows the user to check that the solution of a linear
//    system is correct, without having to save an unfactored copy
//    of the matrix.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    01 September 2006
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Edward Anderson, Zhaojun Bai, Christian Bischof, Susan Blackford, 
//    James Demmel, Jack Dongarra, Jeremy Du Croz, Anne Greenbaum, 
//    Sven Hammarling, Alan McKenney, Danny Sorensen,
//    LAPACK User's Guide,
//    Second Edition,
//    SIAM, 1995.
//
//  Parameters:
//
//    Input, int M, the number of rows in the matrix.
//
//    Input, int N, the number of columns in the matrix.
//
//    Input, double A_LU[M*N], the LU factors from R8GE_TRF.
//
//    Input, char TRANS, specifies the form of the system of equations:
//    'N':  A * x = b  (No transpose)
//    'T':  A'* X = B  (Transpose)
//    'C':  A'* X = B  (Conjugate transpose = Transpose)
//
//    Input, int PIVOT[*], the pivot vector computed by R8GE_TRF.
//
//    Input, double X[*], the vector to be multiplied.
//    For the untransposed case, X should have N entries.
//    For the transposed case, X should have M entries.
//
//    Output, double R8GE_MU[*], the result of the multiplication.
//    For the untransposed case, the result should have M entries.
//    For the transposed case, the result should have N entries.
//
{
  double *b;
  int i;
  int j;
  int k;
  int mn_max;
  int npiv;
  double temp;
  double *y;

  npiv = i4_min ( m - 1, n );
  mn_max = i4_max ( m, n );
  y = new double[mn_max];

  if ( trans == 'n' || trans == 'N' )
  {
    b = new double[m];
//
//  Y[MN] = U[MNxN] * X[N].
//
    for ( i = 0; i < n; i++ )
    {
      y[i] = 0.0;
    }

    for ( j = 1; j <= n; j++ )
    {
      for ( i = 1; i <= i4_min ( j, m ); i++ )
      {
        y[i-1] = y[i-1] + a_lu[i-1+(j-1)*m] * x[j-1];
      }
    }
//
//  Z[M] = L[MxMN] * Y[MN] = L[MxMN] * U[MNxN] * X[N].
//
    for ( i = 0; i < m; i++ )
    {
      if ( i < n ) 
      {
        b[i] = y[i];
      }
      else
      {
        b[i] = 0.0;
      }
    }

    for ( j = i4_min ( m-1, n ); 1 <= j; j-- )
    {
      for ( i = j+1; i <= m; i++ )
      {
        b[i-1] = b[i-1] + a_lu[i-1+(j-1)*m] * y[j-1];
      }
    }
//
//  B = P * Z = P * L * Y = P * L * U * X = A * x.
//
    for ( j = npiv; 1 <= j; j-- )
    {
      k = pivot[j-1];

      if ( k != j )
      {
        temp = b[k-1];
        b[k-1] = b[j-1];
        b[j-1] = temp;
      }
    }
  }
  else if ( trans == 't' || trans == 'T' ||
            trans == 'c' || trans == 'C' )
  {
    b = new double[n];
//
//  Y = P' * X:
//
    for ( i = 1; i <= npiv; i++ )
    {
      k = pivot[i-1];

      if ( k != i )
      {
        temp = x[k-1];
        x[k-1] = x[i-1];
        x[i-1] = temp;
      }
    }

    for ( i = 0; i < n; i++ )
    {
      if ( i < m ) 
      {
        b[i] = x[i];
      }
      else
      {
        b[i] = 0.0;
      }
    }
//
//  Z = L' * Y:
//
    for ( j = 1; j <= i4_min ( m - 1, n ); j++ )
    {
      for ( i = j+1; i <= m; i++ )
      {
        b[j-1] = b[j-1] + x[i-1] * a_lu[i-1+(j-1)*m];
      }
    }
//
//  B = U' * Z.
//
    for ( i = m; 1 <= i; i-- )
    {
      for ( j = i+1; j <= n; j++ )
      {
        b[j-1] = b[j-1] + b[i-1] * a_lu[i-1+(j-1)*m];
      }
      if ( i <= n )
      {
        b[i-1] = b[i-1] * a_lu[i-1+(i-1)*m];
      }
    }
//
//  Now restore X.
//
    for ( i = npiv; 1 <= i; i-- )
    {
      k = pivot[i-1];

      if ( k != i )
      {
        temp = x[k-1];
        x[k-1] = x[i-1];
        x[i-1] = temp;
      }
    }
  }
//
//  Illegal value of TRANS.
//
  else
  {
    cerr << "\n";
    cerr << "R8GE_MU - Fatal error!\n";
    cerr << "  Illegal value of TRANS = \"" << trans << "\"\n";
    exit ( 1 );
  }

  delete [] y;

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

double *r8ge_mv ( int m, int n, double a[], double x[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_MV multiplies an R8GE matrix times a vector.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    11 September 2003
//
//  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 R8GE matrix.
//
//    Input, double X[N], the vector to be multiplied by A.
//
//    Output, double R8GE_MV[M], the product A * x.
//
{
  double *b;
  int i;
  int j;

  b = r8vec_zeros_new ( m );

  for ( i = 0; i < m; i++ )
  {
    for ( j = 0; j < n; j++ )
    {
      b[i] = b[i] + a[i+j*m] * x[j];
    }
  }

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

void r8ge_plu ( int m, int n, double a[], double p[], double l[], double u[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_PLU produces the PLU factors of an R8GE matrix.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//    The PLU factors of the M by N matrix A are:
//
//      P, an M by M permutation matrix P,
//      L, an M by M unit lower triangular matrix,
//      U, an M by N upper triangular matrix.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    07 December 2003
//
//  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 to be factored.
//
//    Output, double P[M*M], the M by M permutation factor.
//
//    Output, double L[M*M], the M by M unit lower triangular factor.
//
//    Output, double U[M*N], the M by N upper triangular factor.
//
{
  int i;
  int j;
  int k;
  int pivot_row;
  double pivot_value;
  double t;
//
//  Initialize:
//
//    P: = M by M Identity
//    L: = M by M Identity
//    U: = A
//

  r8ge_identity ( m, m, p );
  r8ge_identity ( m, m, l );
  r8ge_copy ( m, n, a, u );
//
//  On step J, find the pivot row and the pivot value.
//
  for ( j = 0; j < i4_min ( m-2, n-1 ); j++ )
  {
    pivot_value = 0.0;
    pivot_row = -1;

    for ( i = j; i < m; i++ )
    {
      if ( pivot_value < fabs ( u[i+j*m] ) )
      {
        pivot_value = fabs ( u[i+j*m] );
        pivot_row = i;
      }
    }
//
//  If the pivot row is nonzero swap:
//  * rows J and PIVOT_ROW of U;
//  * rows J and PIVOT_ROW of L and cols J and PIVOT_ROW of L;
//  * cols J and PIVOT_ROW of P.
//
    if ( pivot_row != -1 )
    {
      for ( k = 0; k < n; k++ )
      {
        t                = u[j+k*m];
        u[j+k*m]         = u[pivot_row+k*m];
        u[pivot_row+k*m] = t;
      }

      for ( k = 0; k < m; k++ )
      {
        t                = l[j+k*m];
        l[j+k*m]         = l[pivot_row+k*m];
        l[pivot_row+k*m] = t;
      }
      for ( k = 0; k < m; k++ )
      {
        t                = l[k+j*m];
        l[k+j*m]         = l[k+pivot_row*m];
        l[k+pivot_row*m] = t;
      }

      for ( k = 0; k < m; k++ )
      {
        t                = p[k+j*m];
        p[k+j*m]         = p[k+pivot_row*m];
        p[k+pivot_row*m] = t;
      }
//
//  Zero out the entries in column J, from row J+1 to M.
//
      for ( i = j+1; i < m; i++ )
      {
        if ( u[i+j*m] != 0.0 )
        {
          l[i+j*m] = u[i+j*m] / u[j+j*m];
          u[i+j*m] = 0.0;
          for ( k = j+1; k < n; k++ )
          {
            u[i+k*m] = u[i+k*m] - l[i+j*m] * u[j+k*m];
          }
        }
      }
    }
  }

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

double *r8ge_poly ( int n, double a[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_POLY computes the characteristic polynomial of an R8GE matrix.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    03 November 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//    N must be positive.
//
//    Input, double A[N*N], the R8GE matrix.
//
//    Output, double R8GE_POLY[N+1], the coefficients of the characteristic
//    polynomial of A.  P(I) contains the coefficient of X**I.
//
{
  int i;
  int j;
  int k;
  int order;
  double *p;
  double trace;
  double *work1;
  double *work2;

  p = new double[n+1];
  work2 = new double[n*n];
//
//  Initialize WORK1 to the identity matrix.
//
  work1 = r8ge_identity_new ( n, n );

  p[n] = 1.0;

  for ( order = n-1; 0 <= order; order-- )
  {
//
//  Work2 = A * WORK1.
//
    for ( j = 0; j < n; j++ )
    {
      for ( i = 0; i < n; i++ )
      {
        work2[i+j*n] = 0.0;
        for ( k = 0; k < n; k++ )
        {
          work2[i+j*n] = work2[i+j*n] + a[i+k*n] * work1[k+j*n];
        }
      }
    }
//
//  Take the trace.
//
    trace = 0.0;
    for ( i = 0; i < n; i++ )
    {
      trace = trace + work2[i+i*n];
    }
//
//  P(ORDER) = - Trace ( WORK2 ) / ( N - ORDER )
//
    p[order] = -trace / ( double ) ( n - order );
//
//  WORK1 := WORK2 + P(ORDER) * Identity.
//
    for ( j = 0; j < n; j++ )
    {
      for ( i = 0; i < n; i++ )
      {
        work1[i+j*n] = work2[i+j*n];
      }
    }
    for ( j = 0; j < n; j++ )
    {
      work1[j+j*n] = work1[j+j*n] + p[order];
    }
  }

  delete [] work1;
  delete [] work2;

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

void r8ge_print ( int m, int n, double a[], string title )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_PRINT prints an R8GE matrix.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    06 April 2006
//
//  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 R8GE matrix.
//
//    Input, string TITLE, a title.
//
{
  r8ge_print_some ( m, n, a, 1, 1, m, n, title );

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

void r8ge_print_some ( int m, int n, double a[], int ilo, int jlo, int ihi, 
  int jhi, string title )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_PRINT_SOME prints some of an R8GE matrix.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    06 April 2006
//
//  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 R8GE 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

double *r8ge_random ( int m, int n, int &seed )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_RANDOM randomizes an R8GE matrix.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    15 January 2004
//
//  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/output, int &SEED, a seed for the random number generator.
//
//    Output, double R8GE_RANDOM[M*N], the randomized M by N matrix, 
//    with entries between 0 and 1.
//
{
  double *a;
  int i;
  int j;

  a = new double[m*n];

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      a[i+j*m] = r8_uniform_01 ( seed );
    }
  }
  return a;
}
//****************************************************************************80

double *r8ge_res ( int m, int n, double a[], double x[], double b[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_RES computes the residual for an R8GE system.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    04 February 2016
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//    N must be positive.
//
//    Input, double A[M*N], the original, UNFACTORED matrix.
//
//    Input, double X[N], an estimate of the solution the linear system.
//
//    Input, double B[M], the right hand side vector.
//
//    Output, double R8GE_RES[M], the residual vector: b - A * x.
//
{
  double *r;
  int i;

  r = r8ge_mv ( m, n, a, x );

  for ( i = 0; i < m; i++ )
  {
    r[i] = r[i] - b[i];
  }

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

void r8ge_sl ( int n, double a_lu[], int pivot[], double x[], int job )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_SL solves an R8GE system factored by R8GE_FA.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//    R8GE_SL is a simplified version of the LINPACK routine SGESL.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    09 April 2012
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//    N must be positive.
//
//    Input, double A_LU[N*N], the LU factors from R8GE_FA.
//
//    Input, int PIVOT[N], the pivot vector from R8GE_FA.
//
//    Input/output, double X[N], on input, the right hand side vector.
//    On output, the solution vector.
//
//    Input, int JOB, specifies the operation.
//    0, solve A * x = b.
//    nonzero, solve A' * x = b.
//
{
  int i;
  int k;
  int l;
  double t;
//
//  Solve A * x = b.
//
  if ( job == 0 )
  {
//
//  Solve PL * Y = B.
//
    for ( k = 1; k <= n-1; k++ )
    {
      l = pivot[k-1];

      if ( l != k )
      {
        t      = x[l-1];
        x[l-1] = x[k-1];
        x[k-1] = t;
      }
      for ( i = k+1; i <= n; i++ )
      {
        x[i-1] = x[i-1] + a_lu[i-1+(k-1)*n] * x[k-1];
      }
    }
//
//  Solve U * X = Y.
//
    for ( k = n; 1 <= k; k-- )
    {
      x[k-1] = x[k-1] / a_lu[k-1+(k-1)*n];
      for ( i = 1; i <= k-1; i++ )
      {
        x[i-1] = x[i-1] - a_lu[i-1+(k-1)*n] * x[k-1];
      }
    }
  }
//
//  Solve A' * X = B.
//
  else
  {
//
//  Solve U' * Y = B.
//
    for ( k = 1; k <= n; k++ )
    {
      t = 0.0;
      for ( i = 1; i <= k-1; i++ )
      {
        t = t + x[i-1] * a_lu[i-1+(k-1)*n];
      }
      x[k-1] = ( x[k-1] - t ) / a_lu[k-1+(k-1)*n];
    }
//
//  Solve ( PL )' * X = Y.
//
    for ( k = n-1; 1 <= k; k-- )
    {
      t = 0.0;
      for ( i = k+1; i <= n; i++ )
      {
        t = t + x[i-1] * a_lu[i-1+(k-1)*n];
      }
      x[k-1] = x[k-1] + t;

      l = pivot[k-1];

      if ( l != k )
      {
        t      = x[l-1];
        x[l-1] = x[k-1];
        x[k-1] = t;
      }
    }
  }

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

double *r8ge_sl_it ( int n, double a[], double a_lu[], int pivot[], double b[], 
  int job, double x[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_SL_IT applies one step of iterative refinement following R8GE_SL.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//    It is assumed that:
//
//    * the original matrix A has been factored by R8GE_FA;
//    * the linear system A * x = b has been solved once by R8GE_SL.
//
//    (Actually, it is not necessary to solve the system once using R8GE_SL.
//    You may simply supply the initial estimated solution X = 0.)
//
//    Each time this routine is called, it will compute the residual in
//    the linear system, apply one step of iterative refinement, and
//    add the computed correction to the current solution.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    16 October 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//    N must be positive.
//
//    Input, double A[N*N], the original, UNFACTORED R8GE matrix.
//
//    Input, double A_LU[N*N], the LU factors from R8GE_FA.
//
//    Input, int PIVOT[N], the pivot vector from R8GE_FA.
//
//    Input, double B[N], the right hand side vector.
//
//    Input, int JOB, specifies the operation.
//    0, solve A*X=B.
//    nonzero, solve A'*X=B.
//
//    Input, double X[N], an estimate of the solution of A * x = b.
//
//    Output, double R8GE_SL_IT[N], the solution after one step of 
//    iterative refinement.
//
{
  double *dx;
  int i;
  double *r;
  double *x_new;
//
//  Compute the residual vector.
//
  r = r8ge_res ( n, n, a, x, b );
//
//  Solve A * dx = r
//
  dx = r8ge_sl_new ( n, a_lu, pivot, r, job );
//
//  Add dx to x.
//
  x_new = new double[n];

  for ( i = 0; i < n; i++ )
  {
    x_new[i] = x[i] + dx[i];
  }

  delete [] r;
  delete [] dx;

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

double *r8ge_sl_new ( int n, double a_lu[], int pivot[], double b[], int job )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_SL_NEW solves an R8GE system factored by R8GE_FA.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//    R8GE_SL is a simplified version of the LINPACK routine SGESL.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    11 September 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//    N must be positive.
//
//    Input, double A_LU[N*N], the LU factors from R8GE_FA.
//
//    Input, int PIVOT[N], the pivot vector from R8GE_FA.
//
//    Input, double B[N], the right hand side vector.
//
//    Input, int JOB, specifies the operation.
//    0, solve A * x = b.
//    nonzero, solve A' * x = b.
//
//    Output, double R8GE_SL[N], the solution vector.
//
{
  int i;
  int k;
  int l;
  double t;
  double *x;
//
  x = new double[n];

  for ( i = 0; i < n; i++ )
  {
    x[i] = b[i];
  }
//
//  Solve A * x = b.
//
  if ( job == 0 )
  {
//
//  Solve PL * Y = B.
//
    for ( k = 1; k <= n-1; k++ )
    {
      l = pivot[k-1];

      if ( l != k )
      {
        t      = x[l-1];
        x[l-1] = x[k-1];
        x[k-1] = t;
      }
      for ( i = k+1; i <= n; i++ )
      {
        x[i-1] = x[i-1] + a_lu[i-1+(k-1)*n] * x[k-1];
      }
    }
//
//  Solve U * X = Y.
//
    for ( k = n; 1 <= k; k-- )
    {
      x[k-1] = x[k-1] / a_lu[k-1+(k-1)*n];
      for ( i = 1; i <= k-1; i++ )
      {
        x[i-1] = x[i-1] - a_lu[i-1+(k-1)*n] * x[k-1];
      }
    }
  }
//
//  Solve A' * X = B.
//
  else
  {
//
//  Solve U' * Y = B.
//
    for ( k = 1; k <= n; k++ )
    {
      t = 0.0;
      for ( i = 1; i <= k-1; i++ )
      {
        t = t + x[i-1] * a_lu[i-1+(k-1)*n];
      }
      x[k-1] = ( x[k-1] - t ) / a_lu[k-1+(k-1)*n];
    }
//
//  Solve ( PL )' * X = Y.
//
    for ( k = n-1; 1 <= k; k-- )
    {
      t = 0.0;
      for ( i = k+1; i <= n; i++ )
      {
        t = t + x[i-1] * a_lu[i-1+(k-1)*n];
      }
      x[k-1] = x[k-1] + t;

      l = pivot[k-1];

      if ( l != k )
      {
        t      = x[l-1];
        x[l-1] = x[k-1];
        x[k-1] = t;
      }

    }

  }

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

double *r8ge_to_r8gb ( int m, int n, int ml, int mu, double a[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_TO_R8GB copies an R8GE matrix to an R8GB matrix.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//    The R8GB storage format is for an M by N banded matrix, with lower
//    bandwidth ML and upper bandwidth MU.  Storage includes room for ML
//    extra superdiagonals, which may be required to store nonzero entries
//    generated during Gaussian elimination.
//
//    It usually doesn't make sense to try to store a general matrix
//    in a band matrix format.  You can always do it, but it will take
//    more space, unless the general matrix is actually banded.
//
//    The purpose of this routine is to allow a user to set up a
//    banded matrix in the easy-to-use general format, and have this
//    routine take care of the compression of the data into general
//    format.  All the user has to do is specify the bandwidths.
//
//    Note that this routine "believes" what the user says about the
//    bandwidth.  It will assume that all entries in the general matrix
//    outside of the bandwidth are zero.
//
//    The original M by N matrix is "collapsed" downward, so that diagonals
//    become rows of the storage array, while columns are preserved.  The
//    collapsed array is logically 2*ML+MU+1 by N.
//
//    LINPACK and LAPACK band storage requires that an extra ML
//    superdiagonals be supplied to allow for fillin during Gauss
//    elimination.  Even though a band matrix is described as
//    having an upper bandwidth of MU, it effectively has an
//    upper bandwidth of MU+ML.  This routine will copy nonzero
//    values it finds in these extra bands, so that both unfactored
//    and factored matrices can be handled.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    16 March 2005
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Edward Anderson, Zhaojun Bai, Christian Bischof, Susan Blackford, 
//    James Demmel, Jack Dongarra, Jeremy Du Croz, Anne Greenbaum, 
//    Sven Hammarling, Alan McKenney, Danny Sorensen,
//    LAPACK User's Guide,
//    Second Edition,
//    SIAM, 1995.
//
//    Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,
//    LINPACK User's Guide,
//    SIAM, 1979,
//    ISBN13: 978-0-898711-72-1,
//    LC: QA214.L56.
//
//  Parameters:
//
//    Input, int M, the number of rows of the matrices.
//    M must be positive.
//
//    Input, int N, the number of columns of the matrices.
//    N must be positive.
//
//    Input, int ML, MU, the lower and upper bandwidths of A1.
//    ML and MU must be nonnegative, and no greater than min(M,N)-1.
//
//    Output, double A[M*N], the R8GE matrix.
//
//    Input, double R8GE_TO_R8GB[(2*ML+MU+1)*N], the R8GB matrix.
//
{
  double *b;
  int i;
  int j;
  int jhi;
  int jlo;
  int k;

  b = new double[(2*ml+mu+1)*n];

  for ( k = 0; k < (2*ml+mu+1)*n; k++ )
  {
    b[k] = 0.0;
  }

  for ( i = 1; i <= m; i++ )
  {
    jlo = i4_max ( i - ml, 1 );
    jhi = i4_min ( i + mu, n );

    for ( j = jlo; j <= jhi; j++ )
    {
      b[ml+mu+i-j+(j-1)*(2*ml+mu+1)] = a[i-1+(j-1)*m];
    }
  }

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

double *r8ge_to_r8lt ( int m, int n, double a_ge[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_TO_R8LT copies an R8GE matrix to an R8LT matrix.
//
//  Discussion:
//
//    The R8GE storage format is used for a general M by N matrix.  A storage 
//    space is made for each entry.  The two dimensional logical
//    array can be thought of as a vector of M*N entries, starting with
//    the M entries in the column 1, then the M entries in column 2
//    and so on.  Considered as a vector, the entry A(I,J) is then stored
//    in vector location I+(J-1)*M.
//
//    The R8LT storage format is used for an M by N lower triangular matrix,
//    and allocates space even for the zero entries.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    21 August 2015
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the order of the matrix.
//
//    Input, double A_GE[M,N], the R8GE matrix.
//
//    Output, double R8GE_TO_R8LT[M,N], the R8LT matrix.
//
{
  double *a_lt;
  int i;
  int j;

  a_lt = new double[m*n];

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      if ( j <= i ) 
      {
        a_lt[i+j*m] = a_ge[i+j*m];
      }
      else
      {
        a_lt[i+j*m] = 0.0;
      }
    }
  }

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

double *r8ge_to_r8po ( int n, double a[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_TO_R8PO copies an R8GE matrix to an R8PO matrix.
//
//  Discussion:
//
//    The R8PO format assumes the matrix is square and symmetric; it is also 
//    typically assumed that the matrix is positive definite.  These are not
//    required here.  The copied R8PO matrix simply zeros out the lower triangle
//    of the R8GE matrix.
//
//    The R8GE storage format is used for a general M by N matrix.  A storage 
//    space is made for each entry.  The two dimensional logical
//    array can be thought of as a vector of M*N entries, starting with
//    the M entries in the column 1, then the M entries in column 2
//    and so on.  Considered as a vector, the entry A(I,J) is then stored
//    in vector location I+(J-1)*M.
//
//    The R8PO storage format is used for a symmetric positive definite 
//    matrix and its inverse.  (The Cholesky factor of an R8PO matrix is an
//    upper triangular matrix, so it will be in R8GE storage format.)
//
//    Only the diagonal and upper triangle of the square array are used.
//    This same storage scheme is used when the matrix is factored by
//    R8PO_FA, or inverted by R8PO_INVERSE.  For clarity, the lower triangle
//    is set to zero.
//
//    R8PO storage is used by LINPACK and LAPACK.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    02 August 2015
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//
//    Input, double A[N,N], the R8GE matrix.
//
//    Output, double R8GE_TO_R8PO[N,N], the R8PO matrix.
//
{
  double *b;
  int i;
  int j;

  b = new double[n*n];

  for ( i = 0; i < n; i++ )
  {
    for ( j = 0; j < n; j++ )
    {
      if ( i <= j ) 
      {
        b[i+j*n] = a[i+j*n];
      }
      else
      {
        b[i+j*n] = 0.0;
      }
    }
  }

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

void r8ge_to_r8ri ( int n, double a[], int nz, int ija[], double sa[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_TO_R8RI converts an R8GE matrix to R8RI form.
//
//  Discussion:
//
//    A R8GE matrix is in general storage.
//
//    An R8RI matrix is in row indexed sparse storage form.
//
//    The size of the arrays IJA and SA can be determined by calling
//    R8GE_TO_R8RI_SIZE.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    20 January 2013
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    William Press, Brian Flannery, Saul Teukolsky, William Vetterling,
//    Numerical Recipes in FORTRAN: The Art of Scientific Computing,
//    Third Edition,
//    Cambridge University Press, 2007,
//    ISBN13: 978-0-521-88068-8,
//    LC: QA297.N866.
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//
//    Input, double A[N*N], the matrix stored in GE 
//    or "general" format.
//
//    Input, int NZ, the size required for the RI
//    or "row indexed" sparse storage.
//
//    Output, int IJA[NZ], the index vector.
//
//    Output, double SA[NZ], the value vector.
//
{
  int i;
  int im;
  int j;
  int k;
  int l;

  for ( k = 0; k < n; k++ )
  {
    i = k;
    j = k;
    sa[k] = a[i+j*n];
  }

  k = n;
  sa[k] = 0.0;

  for ( i = 0; i <= n; i++ )
  {
    ija[i] = 0;
  }

  im = 0;

  for ( i = 0; i < n; i++ )
  {
    for ( j = 0; j < n; j++ )
    {
      if ( i != j )
      {
        if ( a[i+j*n] != 0.0 )
        {
          k = k + 1;
          if ( ija[i] == 0 )
          {
            for ( l = im; l <= i; l++ )
            {
              ija[l] = k;
            }
            im = i + 1;
          }
          ija[k] = j;
          sa[k] = a[i+j*n];
        }
      }
    }
  }

  ija[n] = k + 1;

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

int r8ge_to_r8ri_size ( int n, double a[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_TO_R8RI_SIZE determines the size of an R8RI matrix.
//
//  Discussion:
//
//    N spaces are always used for the diagonal entries, plus a dummy.
//    The remaining spaces store off-diagonal nonzeros.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    19 January 2013
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    William Press, Brian Flannery, Saul Teukolsky, William Vetterling,
//    Numerical Recipes in FORTRAN: The Art of Scientific Computing,
//    Third Edition,
//    Cambridge University Press, 2007,
//    ISBN13: 978-0-521-88068-8,
//    LC: QA297.N866.
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//
//    Input, double A[N*N], the matrix stored in GE or "general" format.
//
//    Output, int R8GE_TO_R8RI_SIZE, the size required for the RI
//    or "row indexed" sparse storage.
//
{
  int i;
  int j;
  int nz;

  nz = n + 1;

  for ( i = 0; i < n; i++ )
  {
    for ( j = 0; j < n; j++ )
    {
      if ( i != j )
      {
        if ( a[i+j*n] != 0.0 )
        {
          nz = nz + 1;
        }
      }
    }
  }

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

double *r8ge_to_r8ut ( int m, int n, double a_ge[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_TO_R8UT copies an R8GE matrix to an R8UT matrix.
//
//  Discussion:
//
//    The R8GE storage format is used for a general M by N matrix.  A storage 
//    space is made for each entry.  The two dimensional logical
//    array can be thought of as a vector of M*N entries, starting with
//    the M entries in the column 1, then the M entries in column 2
//    and so on.  Considered as a vector, the entry A(I,J) is then stored
//    in vector location I+(J-1)*M.
//
//    The R8UT storage format is used for an M by N upper triangular matrix,
//    and allocates space even for the zero entries.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    19 August 2015
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the order of the matrix.
//
//    Input, double A_GE[M,N], the R8GE matrix.
//
//    Output, double R8GE_TO_R8UT[M,N], the R8UT matrix.
//
{
  double *a_ut;
  int i;
  int j;

  a_ut = new double[m*n];

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      if ( i <= j ) 
      {
        a_ut[i+j*m] = a_ge[i+j*m];
      }
      else
      {
        a_ut[i+j*m] = 0.0;
      }
    }
  }

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

double *r8ge_to_r8vec ( int m, int n, double *a )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_TO_R8VEC copies an R8GE matrix to a real vector.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//    In C++  and FORTRAN, this routine is not really needed.  In MATLAB,
//    a data item carries its dimensionality implicitly, and so cannot be
//    regarded sometimes as a vector and sometimes as an array.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    17 March 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns in the array.
//
//    Input, double R8VEC_TO_R8GE[M*N], the array to be copied.
//
//    Output, double X[M*N], the vector.
//
{
  int i;
  int j;
  int k;
  double *x;

  x = new double[m*n];

  k = 0;
  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      x[k] = a[i+j*m];
      k = k + 1;
    }
  }

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

double *r8ge_to_r8vm ( int m, int n, double a_ge[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_TO_R8VM converts an R8GE matrix to an R8VM matrix.
//
//  Discussion:
//
//    The R8VM storage format is used for an M by N Vandermonde matrix.
//    An M by N Vandermonde matrix is defined by the values in its second
//    row, which will be written here as X(1:N).  The matrix has a first 
//    row of 1's, a second row equal to X(1:N), a third row whose entries
//    are the squares of the X values, up to the M-th row whose entries
//    are the (M-1)th powers of the X values.  The matrix can be stored
//    compactly by listing just the values X(1:N).
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    18 August 2015
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns of the matrix.
//
//    Input, double A_GE[M*N], the R8GE matrix.
//
//    Output, double R8GE_TO_R8VM[N], the R8VM matrix.
//
{
  double *a_vm;
  int i;
  int j;

  a_vm = new double[n];

  i = 1;
  for ( j = 0; j < n; j++ )
  {
    a_vm[j] = a_ge[i+j*m];
  }

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

double *r8ge_transpose_new ( int m, int n, double a[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_TRANSPOSE_NEW returns the transpose of an R8GE.
//
//  Discussion:
//
//    An R8GE is a doubly dimensioned array of R8 values, stored as a vector
//    in column-major order.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    12 October 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns of the matrix A.
//
//    Input, double A[M*N], the matrix whose transpose is desired.
//
//    Output, double R8GE_TRANSPOSE_NEW[N*M], the transposed matrix.
//
{
  double *b;
  int i;
  int j;

  b = new double[n*m];

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      b[j+i*n] = a[i+j*m];
    }
  }
  return b;
}
//****************************************************************************80

void r8ge_transpose_print ( int m, int n, double a[], string title )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_TRANSPOSE_PRINT prints an R8GE, transposed.
//
//  Discussion:
//
//    An R8GE is a doubly dimensioned array of R8 values, stored as a vector
//    in column-major order.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    10 September 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns.
//
//    Input, double A[M*N], an M by N matrix to be printed.
//
//    Input, string TITLE, a title.
//
{
  r8ge_transpose_print_some ( m, n, a, 1, 1, m, n, title );

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

void r8ge_transpose_print_some ( int m, int n, double a[], int ilo, int jlo,
  int ihi, int jhi, string title )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_TRANSPOSE_PRINT_SOME prints some of an R8GE, transposed.
//
//  Discussion:
//
//    An R8GE is a doubly dimensioned array of R8 values, stored as a vector
//    in column-major order.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    07 April 2014
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns.
//
//    Input, double A[M*N], an M by N matrix to be printed.
//
//    Input, int ILO, JLO, the first row and column to print.
//
//    Input, int IHI, JHI, the last row and column to print.
//
//    Input, string TITLE, a title.
//
{
# define INCX 5

  int i;
  int i2;
  int i2hi;
  int i2lo;
  int i2lo_hi;
  int i2lo_lo;
  int inc;
  int j;
  int j2hi;
  int j2lo;

  cout << "\n";
  cout << title << "\n";

  if ( m <= 0 || n <= 0 )
  {
    cout << "\n";
    cout << "  (None)\n";
    return;
  }

  if ( ilo < 1 )
  {
    i2lo_lo = 1;
  }
  else
  {
    i2lo_lo = ilo;
  }

  if ( ihi < m )
  {
    i2lo_hi = m;
  }
  else
  {
    i2lo_hi = ihi;
  }

  for ( i2lo = i2lo_lo; i2lo <= i2lo_hi; i2lo = i2lo + INCX )
  {
    i2hi = i2lo + INCX - 1;

    if ( m < i2hi )
    {
      i2hi = m;
    }
    if ( ihi < i2hi )
    {
      i2hi = ihi;
    }

    inc = i2hi + 1 - i2lo;

    cout << "\n";
    cout << "  Row: ";
    for ( i = i2lo; i <= i2hi; i++ )
    {
      cout << setw(7) << i - 1 << "       ";
    }
    cout << "\n";
    cout << "  Col\n";
    cout << "\n";

    if ( jlo < 1 )
    {
      j2lo = 1;
    }
    else
    {
      j2lo = jlo;
    }
    if ( n < jhi )
    {
      j2hi = n;
    }
    else
    {
      j2hi = jhi;
    }

    for ( j = j2lo; j <= j2hi; j++ )
    {
      cout << setw(5) << j - 1 << ":";
      for ( i2 = 1; i2 <= inc; i2++ )
      {
        i = i2lo - 1 + i2;
        cout << setw(14) << a[(i-1)+(j-1)*m];
      }
      cout << "\n";
    }
  }

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

int r8ge_trf ( int m, int n, double a[], int pivot[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_TRF performs a LAPACK-style PLU factorization of an R8GE matrix.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//    R8GE_TRF is a standalone version of the LAPACK routine SGETRF.
//
//    The factorization uses partial pivoting with row interchanges,
//    and has the form
//      A = P * L * U
//    where P is a permutation matrix, L is lower triangular with unit
//    diagonal elements (lower trapezoidal if N < M), and U is upper
//    triangular (upper trapezoidal if M < N).
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    09 November 2003
//
//  Author:
//
//    Original FORTRAN77 version by Anderson, Bai, Bischof, Blackford,
//    Demmel, Dongarra, DuCroz, Greenbaum, Hammarling, McKenney, Sorensen.
//    C++ version by John Burkardt
//
//  Reference:
//
//    Edward Anderson, Zhaojun Bai, Christian Bischof, Susan Blackford, 
//    James Demmel, Jack Dongarra, Jeremy Du Croz, Anne Greenbaum, 
//    Sven Hammarling, Alan McKenney, Danny Sorensen,
//    LAPACK User's Guide,
//    Second Edition,
//    SIAM, 1995.
//
//  Parameters:
//
//    Input, int M, the number of rows of the matrix A.  0 <= M.
//
//    Input, int N, the number of columns of the matrix A.  0 <= N.
//
//    Input/output, double A[M*N].
//    On entry, the M by N matrix to be factored.
//    On exit, the factors L and U from the factorization
//    A = P*L*U; the unit diagonal elements of L are not stored.
//
//    Output, int PIVOT[min(M,N)], the pivot indices.
//
//    Output, int R8GE_TRF.
//    = 0: successful exit
//    = -K, the K-th argument had an illegal value
//    = K: U(K,K) is exactly zero. The factorization
//         has been completed, but the factor U is exactly
//         singular, and division by zero will occur if it is used
//         to solve a system of equations.
//
{
  int i;
  int ii;
  int info;
  int j;
  int jj;
  int jp;
  double temp;
//
//  Test the input parameters.
//
  info = 0;

  if ( m < 0 )
  {
    return (-1);
  }
  else if ( n < 0 )
  {
    return (-2);
  }

  if ( m == 0 || n == 0 )
  {
    return 0;
  }

  for ( j = 1; j <= i4_min ( m, n ); j++ )
  {
//
//  Find the pivot.
//
    temp = fabs ( a[j-1+(j-1)*m] );
    jp = j;
    for ( i = j+1; i <= m; i++ )
    {
      if ( temp < fabs ( a[i-1+(j-1)*m] ) )
      {
        temp = fabs ( a[i-1+(j-1)*m] );
        jp = i;
      }
    }

    pivot[j-1] = jp;
//
//  Apply the interchange to columns 1:N.
//  Compute elements J+1:M of the J-th column.
//
    if ( a[jp-1+(j-1)*m] != 0.0 )
    {
      if ( jp != j )
      {
        for ( jj = 1; jj <= n; jj++ )
        {
          temp             = a[j-1+(jj-1)*m];
          a[j-1+(jj-1)*m]  = a[jp-1+(jj-1)*m];
          a[jp-1+(jj-1)*m] = temp;
        }
      }

      if ( j < m )
      {
        for ( i = j+1; i <= m; i++ )
        {
          a[i-1+(j-1)*m] = a[i-1+(j-1)*m] / a[j-1+(j-1)*m];
        }
      }
    }
    else if ( info == 0 )
    {
      info = j;
    }
//
//  Update the trailing submatrix.
//
    if ( j < i4_min ( m, n ) )
    {
      for ( ii = j+1; ii <= m; ii++ )
      {
        for ( i = j+1; i <= n; i++ )
        {
          a[ii-1+(i-1)*m] = a[ii-1+(i-1)*m] - a[ii-1+(j-1)*m] * a[j-1+(i-1)*m];
        }
      }
    }
  }

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

double *r8ge_trs ( int n, int nrhs, char trans, double a[], int pivot[], 
  double b[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_TRS solves a system of linear equations factored by R8GE_TRF.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//    R8GE_TRS is a standalone version of the LAPACK routine SGETRS.
//
//    R8GE_TRS solves a system of linear equations
//      A * x = b  or  A' * X = B
//    with a general N by N matrix A using the PLU factorization computed
//    by R8GE_TRF.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    09 November 2003
//
//  Author:
//
//    Original FORTRAN77 version by Anderson, Bai, Bischof, Blackford,
//    Demmel, Dongarra, DuCroz, Greenbaum, Hammarling, McKenney, Sorensen.
//    C++ version by John Burkardt
//
//  Reference:
//
//    Edward Anderson, Zhaojun Bai, Christian Bischof, Susan Blackford, 
//    James Demmel, Jack Dongarra, Jeremy Du Croz, Anne Greenbaum, 
//    Sven Hammarling, Alan McKenney, Danny Sorensen,
//    LAPACK User's Guide,
//    Second Edition,
//    SIAM, 1995.
//
//  Parameters:
//
//    Input, int N, the order of the matrix A.  0 <= N.
//
//    Input, int NRHS, the number of right hand sides.  0 <= NRHS.
//
//    Input, char TRANS, specifies the form of the system of equations:
//    'N':  A * x = b  (No transpose)
//    'T':  A'* X = B  (Transpose)
//    'C':  A'* X = B  (Conjugate transpose = Transpose)
//
//    Input, double A[N*N], the factors L and U from the factorization
//    A = P*L*U as computed by R8GE_TRF.
//
//    Input, int PIVOT[N], the pivot indices from R8GE_TRF.
//
//    Input, double B[N*NRHS], the right hand side matrix.
//
//    Output, double R8GE_TRS[N*NRHS], the solution matrix X.
//
{
  int i;
  int j;
  int k;
  double temp;
  double *x;

  if ( trans != 'n' && trans != 'N' && 
       trans != 't' && trans != 'T' && 
       trans != 'c' && trans != 'C' )
  {
    return NULL;
  }

  if ( n < 0 )
  {
    return NULL;
  }

  if ( nrhs < 0 )
  {
    return NULL;
  }

  if ( n == 0 )
  {
    return NULL;
  }
  if ( nrhs == 0 )
  {
    return NULL;
  }

  x = new double[n*nrhs];
  for ( k = 0; k < nrhs; k++ )
  {
    for ( i = 0; i < n; i++ )
    {
      x[i+k*n] = b[i+k*n];
    }
  }

  if ( trans == 'n' || trans == 'N' )
  {
//
//  Apply row interchanges to the right hand sides.
//
    for ( i = 1; i <= n; i++ )
    {
      if ( pivot[i-1] != i )
      {
        for ( k = 0; k < nrhs; k++ )
        {
          temp                = x[i-1+k*n];
          x[i-1+k*n]          = x[pivot[i-1]-1+k*n];
          x[pivot[i-1]-1+k*n] = temp;
        }
      }
    }
//
//  Solve L * x = b, overwriting b with x.
//
    for ( k = 0; k < nrhs; k++ )
    {
      for ( j = 1; j <= n-1; j++ )
      {
        for ( i = j+1; i <= n; i++ )
        {
          x[i-1+k*n] = x[i-1+k*n] - a[i-1+(j-1)*n] * x[j-1+k*n];
        }
      }
    }
//
//  Solve U * x = b, overwriting b with x.
//
    for ( k = 0; k < nrhs; k++ )
    {
      for ( j = n; 1 <= j; j-- )
      {
        x[j-1+k*n] = x[j-1+k*n] / a[j-1+(j-1)*n];
        for ( i = 1; i < j; i++ )
        {
          x[i-1+k*n] = x[i-1+k*n] - a[i-1+(j-1)*n] * x[j-1+k*n];
        }
      }
    }
  }
  else
  {
//
//  Solve U' * x = b, overwriting b with x.
//
    for ( k = 0; k < nrhs; k++ )
    {
      for ( j = 1; j <= n; j++ )
      {
        x[j-1+k*n] = x[j-1+k*n] / a[j-1+(j-1)*n];
        for ( i = j+1; i <= n; i++ )
        {
          x[i-1+k*n] = x[i-1+k*n] - a[j-1+(i-1)*n] * x[j-1+k*n];
        }
      }
    }
//
//  Solve L' * x = b, overwriting b with x.
//
    for ( k = 0; k < nrhs; k++ )
    {
      for ( j = n; 2 <= j; j-- )
      {
        for ( i = 1; i < j; i++ )
        {
          x[i-1+k*n] = x[i-1+k*n] - a[j-1+(i-1)*n] * x[j-1+k*n];
        }
      }
    }
//
//  Apply row interchanges to the solution vectors.
//
    for ( i = n; 1 <= i; i-- )
    {
      if ( pivot[i-1] != i )
      {
        for ( k = 0; k < nrhs; k++ )
        {
          temp                = x[i-1+k*n];
          x[i-1+k*n]          = x[pivot[i-1]-1+k*n];
          x[pivot[i-1]-1+k*n] = temp;
        }
      }
    }
  }

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

double *r8ge_zeros ( int m, int n )

//****************************************************************************80
//
//  Purpose:
//
//    R8GE_ZEROS zeros an R8GE matrix.
//
//  Discussion:
//
//    The R8GE storage format is used for a "general" M by N matrix.  
//    A physical storage space is made for each logical entry.  The two 
//    dimensional logical array is mapped to a vector, in which storage is 
//    by columns.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    13 September 2003
//
//  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.
//
//    Output, double R8GE_ZERO[M*N], the M by N matrix.
//
{
  double *a;
  int i;
  int j;

  a = new double[m*n];

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      a[i+j*m] = 0.0;
    }
  }

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

double r8vec_dot_product ( int n, double x[], double y[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_DOT_PRODUCT computes the dot product of two R8VEC's.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    22 October 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of elements in the vectors.
//
//    Input, double X[N], Y[N], the two vectors.
//
//    Output, double R8VEC_DOT_PRODUCT, the dot product of the vectors.
//
{
  double dot;
  int i;

  dot = 0.0;

  for ( i = 0; i < n; i++ ) 
  {
    dot = dot + x[i] * y[i];
  }

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

double *r8vec_indicator1_new ( int n )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_INDICATOR1_NEW sets an R8VEC to the indicator1 vector {1,2,3...}.
//
//  Discussion:
//
//    An R8VEC is a vector of R8's.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    20 September 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of elements of A.
//
//    Output, double R8VEC_INDICATOR1_NEW[N], the array to be initialized.
//
{
  double *a;
  int i;

  a = new double[n];

  for ( i = 0; i <= n-1; i++ ) 
  {
    a[i] = ( double ) ( i + 1 );
  }

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

double r8vec_norm ( int n, double a[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_NORM returns the L2 norm of an R8VEC.
//
//  Discussion:
//
//    An R8VEC is a vector of R8's.
//
//    The vector L2 norm is defined as:
//
//      R8VEC_NORM = sqrt ( sum ( 1 <= I <= N ) A(I)^2 ).
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    01 March 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in A.
//
//    Input, double A[N], the vector whose L2 norm is desired.
//
//    Output, double R8VEC_NORM, the L2 norm of A.
//
{
  int i;
  double v;

  v = 0.0;

  for ( i = 0; i < n; i++ )
  {
    v = v + a[i] * a[i];
  }
  v = sqrt ( v );

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

double r8vec_norm_affine ( int n, double v0[], double v1[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_NORM_AFFINE returns the affine L2 norm of an R8VEC.
//
//  Discussion:
//
//    The affine vector L2 norm is defined as:
//
//      R8VEC_NORM_AFFINE(V0,V1)
//        = sqrt ( sum ( 1 <= I <= N ) ( V1(I) - V0(I) )^2 )
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    27 October 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the dimension of the vectors.
//
//    Input, double V0[N], the base vector.
//
//    Input, double V1[N], the vector.
//
//    Output, double R8VEC_NORM_AFFINE, the affine L2 norm.
//
{
  int i;
  double value;

  value = 0.0;

  for ( i = 0; i < n; i++ )
  {
    value = value + ( v1[i] - v0[i] ) * ( v1[i] - v0[i] );
  }
  value = sqrt ( value );

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

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

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_PRINT prints an R8VEC.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    14 November 2003
//
//  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(6)  << i + 1 << "  " 
         << setw(14) << a[i]  << "\n";
  }

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

void r8vec_print_some ( int n, double a[], int max_print, string title )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_PRINT_SOME prints "some" of an R8VEC.
//
//  Discussion:
//
//    The user specifies MAX_PRINT, the maximum number of lines to print.
//
//    If N, the size of the vector, is no more than MAX_PRINT, then
//    the entire vector is printed, one entry per line.
//
//    Otherwise, if possible, the first MAX_PRINT-2 entries are printed,
//    followed by a line of periods suggesting an omission,
//    and the last entry.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    27 February 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries of the vector.
//
//    Input, double A[N], the vector to be printed.
//
//    Input, int MAX_PRINT, the maximum number of lines
//    to print.
//
//    Input, string TITLE, a title.
//
{
  int i;

  if ( max_print <= 0 )
  {
    return;
  }

  if ( n <= 0 )
  {
    return;
  }

  cout << "\n";
  cout << title << "\n";
  cout << "\n";

  if ( n <= max_print )
  {
    for ( i = 0; i < n; i++ )
    {
      cout << "  " << setw(8) << i
           << "  " << setw(14) << a[i] << "\n";
    }
  }
  else if ( 3 <= max_print )
  {
    for ( i = 0; i < max_print - 2; i++ )
    {
      cout << "  " << setw(8) << i
           << ": " << setw(14) << a[i] << "\n";
    }
    cout << "  ........  ..............\n";
    i = n - 1;
    cout << "  " << setw(8) << i
         << ": " << setw(14) << a[i] << "\n";
  }
  else
  {
    for ( i = 0; i < max_print - 1; i++ )
    {
      cout << "  " << setw(8) << i
           << ": " << setw(14) << a[i] << "\n";
    }
    i = max_print - 1;
    cout << "  " << setw(8) << i
         << ": " << setw(14) << a[i]
         << "  " << "...more entries...\n";
  }

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

double *r8vec_to_r8ge ( int m, int n, double *x )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_TO_R8GE copies an R8VEC into an R8GE matrix.
//
//  Discussion:
//
//    In C++  and FORTRAN, this routine is not really needed.  In MATLAB,
//    a data item carries its dimensionality implicitly, and so cannot be
//    regarded sometimes as a vector and sometimes as an array.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    17 March 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns in the array.
//
//    Input, double X[M*N], the vector to be copied into the array.
//
//    Output, double R8VEC_TO_R8GE[M*N], the array.
//
{
  double *a;
  int i;
  int j;
  int k;

  a = new double[m*n];

  k = 0;
  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      a[i+j*m] = x[k];
      k = k + 1;
    }
  }

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

double *r8vec_uniform_01_new ( int n, int &seed )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_UNIFORM_01_NEW returns a new unit pseudorandom R8VEC.
//
//  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.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    19 August 2004
//
//  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, int N, the number of entries in the vector.
//
//    Input/output, int &SEED, a seed for the random number generator.
//
//    Output, double R8VEC_UNIFORM_01_NEW[N], the vector of pseudorandom values.
//
{
  int i;
  const int i4_huge = 2147483647;
  int k;
  double *r;

  if ( seed == 0 )
  {
    cerr << "\n";
    cerr << "R8VEC_UNIFORM_01_NEW - Fatal error!\n";
    cerr << "  Input value of SEED = 0.\n";
    exit ( 1 );
  }

  r = new double[n];

  for ( i = 0; i < n; i++ )
  {
    k = seed / 127773;

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

    if ( seed < 0 )
    {
      seed = seed + i4_huge;
    }

    r[i] = ( double ) ( seed ) * 4.656612875E-10;
  }

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

double *r8vec_uniform_ab_new ( int n, double a, double b, int &seed )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_UNIFORM_AB_NEW returns a scaled pseudorandom R8VEC.
//
//  Discussion:
//
//    Each dimension ranges from A to B.
//
//    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.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  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, int N, the number of entries in the vector.
//
//    Input, double A, B, the lower and upper limits of the pseudorandom values.
//
//    Input/output, int &SEED, a seed for the random number generator.
//
//    Output, double R8VEC_UNIFORM_AB_NEW[N], the vector of pseudorandom values.
//
{
  int i;
  const int i4_huge = 2147483647;
  int k;
  double *r;

  if ( seed == 0 )
  {
    cerr << "\n";
    cerr << "R8VEC_UNIFORM_AB_NEW - Fatal error!\n";
    cerr << "  Input value of SEED = 0.\n";
    exit ( 1 );
  }

  r = new double[n];

  for ( i = 0; i < n; i++ )
  {
    k = seed / 127773;

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

    if ( seed < 0 )
    {
      seed = seed + i4_huge;
    }

    r[i] = a + ( b - a ) * ( double ) ( seed ) * 4.656612875E-10;
  }

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

double *r8vec_zeros_new ( int n )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_ZEROS_NEW creates and zeroes an R8VEC.
//
//  Discussion:
//
//    An R8VEC is a vector of R8's.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    10 July 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Output, double R8VEC_ZEROS_NEW[N], a vector of zeroes.
//
{
  double *a;
  int i;

  a = new double[n];

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

void r8vec2_print ( int n, double a1[], double a2[], string title )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC2_PRINT prints an R8VEC2.
//
//  Discussion:
//
//    An R8VEC2 is a dataset consisting of N pairs of real values, stored
//    as two separate vectors A1 and A2.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    19 November 2002
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of components of the vector.
//
//    Input, double A1[N], double A2[N], the vectors to be printed.
//
//    Input, string TITLE, a title.
//
{
  int i;

  cout << "\n";
  cout << title << "\n";
  cout << "\n";
  for ( i = 0; i <= n - 1; i++ )
  {
    cout << setw(6)  << i
         << ": " << setw(14) << a1[i]
         << "  " << setw(14) << a2[i] << "\n";
  }

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

void r8vec2_print_some ( int n, double x1[], double x2[], int max_print, 
  string title )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC2_PRINT_SOME prints "some" of two real vectors.
//
//  Discussion:
//
//    The user specifies MAX_PRINT, the maximum number of lines to print.
//
//    If N, the size of the vectors, is no more than MAX_PRINT, then
//    the entire vectors are printed, one entry of each per line.
//
//    Otherwise, if possible, the first MAX_PRINT-2 entries are printed,
//    followed by a line of periods suggesting an omission,
//    and the last entry.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    13 November 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters: 
//
//    Input, int N, the number of entries of the vectors.
//
//    Input, double X1[N], X2[N], the vector to be printed.
//
//    Input, int MAX_PRINT, the maximum number of lines to print.
//
//    Input, string TITLE, a title.
//
{
  int i;

  if ( max_print <= 0 )
  {
    return;
  }

  if ( n <= 0 )
  {
    return;
  }

  cout << "\n";
  cout << title << "\n";
  cout << "\n";

  if ( n <= max_print )
  {
    for ( i = 0; i < n; i++ )
    {
      cout << setw(6)  << i + 1 << "  "
           << setw(14) << x1[i] << "  "
           << setw(14) << x2[i] << "\n";
    }
  }
  else if ( 3 <= max_print )
  {
    for ( i = 0; i < max_print-2; i++ )
    {
      cout << setw(6)  << i + 1 << "  "
           << setw(14) << x1[i] << "  "
           << setw(14) << x2[i] << "\n";
    }
    cout << "......  ..............  ..............\n";
    i = n - 1;
    cout << setw(6)  << i + 1 << "  "
         << setw(14) << x1[i] << "  "
         << setw(14) << x2[i] << "\n";
  }
  else
  {
    for ( i = 0; i < max_print - 1; i++ )
    {
      cout << setw(6)  << i + 1 << "  "
           << setw(14) << x1[i] << "  "
           << setw(14) << x2[i] << "\n";
    }
    i = max_print - 1;
    cout << setw(6)  << i + 1 << "  "
         << setw(14) << x1[i] << "  "
         << setw(14) << x2[i] << "...more entries...\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 is distributed under the MIT license. 
//
//  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;
  std::time_t now;

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

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

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

  return;
# undef TIME_SIZE
}