# include <cstdlib>
# include <iostream>
# include <iomanip>
# include <cmath>
# include <omp.h>
# include <ctime>

using namespace std;

int main ( );
void test01 ( int n );
void test02 ( int n );
void test03 ( int n );

int isamax ( int n, float x[], int incx );
void matgen ( int lda, int n, float a[], float x[], float b[] );
void msaxpy ( int nr, int nc, float a[], int n, float x[], float y[] );
void msaxpy2 ( int nr, int nc, float a[], int n, float x[], float y[] );
int msgefa ( float a[], int lda, int n, int ipvt[] );
int msgefa2 ( float a[], int lda, int n, int ipvt[] );
void saxpy ( int n, float a, float x[], int incx, float y[], int incy );
float sdot ( int n, float x[], int incx, float y[], int incy );
int sgefa ( float a[], int lda, int n, int ipvt[] );
void sgesl ( float a[], int lda, int n, int ipvt[], float b[], int job );
void sscal ( int n, float a, float x[], int incx );
void sswap ( int n, float x[], int incx, float y[], int incy );
void timestamp ( );

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

int main ( )

//****************************************************************************80
//
//  Purpose:
//
//    MAIN is the main program for the SGEFA_OPENMP test program.
//
//  Discussion:
//
//    We want to compare methods of solving the linear system A*x=b.
//
//    The first way uses the standard sequential algorithm "SGEFA".
//
//    The second way uses a variant of SGEFA that has been modified to
//    take advantage of OpenMP.
//
//    The third way reruns the variant code, but with OpenMP turned off.
//
//  Modified:
//
//    07 April 2008
//
//  Author:
//
//    John Burkardt
//
{
  int n;

  timestamp ( );

  cout << "\n";
  cout << "SGEFA_OPENMP\n";
  cout << "  C++/OpenMP version\n";

  cout << "\n";
  cout << "  Number of processors available = " << omp_get_num_procs ( ) << "\n";
  cout << "  Number of threads =              " << omp_get_max_threads ( ) << "\n";

  cout << "\n";
  cout << " Algorithm        Mode          N    Error       Time\n";

  cout << "\n";
  n = 10;
  test01 ( n );
  test02 ( n );
  test03 ( n );

  cout << "\n";
  n = 100;
  test01 ( n );
  test02 ( n );
  test03 ( n );

  cout << "\n";
  n = 1000;
  test01 ( n );
  test02 ( n );
  test03 ( n );
//
//  Terminate.
//
  cout << "\n";
  cout << "SGEFA_OPENMP\n";
  cout << "  Normal end of execution.\n";

  cout << "\n";
  timestamp ( );

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

void test01 ( int n )

//****************************************************************************80
//
//  Purpose:
//
//    TEST01 runs the sequential version of SGEFA.
//
//  Modified:
//
//    07 April 2008
//
//  Author:
//
//    John Burkardt
//
{
  float *a;
  float *b;
  float err;
  int i;
  int info;
  int *ipvt;
  int job;
  int lda;
  double wtime;
  float *x;
//
//  Generate the linear system A * x = b.
//
  lda = n;
  a = new float[lda * n];
  b = new float[n];
  x = new float[n];

  matgen ( lda, n, a, x, b );
//
//  Factor the linear system.
//
  ipvt = new int[n];

  wtime = omp_get_wtime ( );
  info = sgefa ( a, lda, n, ipvt );
  wtime = omp_get_wtime ( ) - wtime;

  if ( info != 0 )
  {
    cout << "\n";
    cout << "TEST01 - Fatal error!\n";
    cout << "  SGEFA reports the matrix is singular.\n";
    exit ( 1 );
  }
//
//  Solve the linear system.
//
  job = 0;
  sgesl ( a, lda, n, ipvt, b, job );

  err = 0.0;
  for ( i = 0; i < n; i++ )
  {
    err = err + fabs ( x[i] - b[i] );
  }
  cout << "  Original  Sequential   "
       << "  " << setw(8) << n
       << "  " << setw(10) << err
       << "  " << setw(10) << wtime << "\n";

  delete [] a;
  delete [] b;
  delete [] ipvt;
  delete [] x;

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

void test02 ( int n )

///****************************************************************************80
//
//  Purpose:
//
//    TEST02 runs the revised version of SGEFA in parallel.
//
//  Modified:
//
//    07 April 2008
//
//  Author:
//
//    John Burkardt
//
{
  float *a;
  float *b;
  float err;
  int i;
  int info;
  int *ipvt;
  int job;
  int lda;
  double wtime;
  float *x;
//
//  Generate the linear system A * x = b.
//
  lda = n;
  a = new float[lda * n];
  b = new float[n];
  x = new float[n];

  matgen ( lda, n, a, x, b );
//
//  Factor the linear system.
//
  ipvt = new int[n];

  wtime = omp_get_wtime ( );
  info = msgefa ( a, lda, n, ipvt );
  wtime = omp_get_wtime ( ) - wtime;

  if ( info != 0 )
  {
    cout << "\n";
    cout << "TEST02 - Fatal error!\n";
    cout << "  MSGEFA reports the matrix is singular.\n";
    exit ( 1 );
  }
//
//  Solve the linear system.
//
  job = 0;
  sgesl ( a, lda, n, ipvt, b, job );

  err = 0.0;
  for ( i = 0; i < n; i++ )
  {
    err = err + fabs ( x[i] - b[i] );
  }

  cout << "  Revised     Parallel   "
       << "  " << setw(8) << n
       << "  " << setw(10) << err
       << "  " << setw(10) << wtime << "\n";

  delete [] a;
  delete [] b;
  delete [] ipvt;
  delete [] x;

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

void test03 ( int n )

//****************************************************************************80
//
//  Purpose:
//
//    TEST03 runs the revised version of SGEFA in sequential mode.
//
//  Modified:
//
//    07 April 2008
//
//  Author:
//
//    John Burkardt
//
{
  float *a;
  float *b;
  float err;
  int i;
  int info;
  int *ipvt;
  int job;
  int lda;
  double wtime;
  float *x;
//
//  Generate the linear system A * x = b.
//
  lda = n;
  a = new float[lda * n];
  b = new float[n];
  x = new float[n];

  matgen ( lda, n, a, x, b );
//
//  Factor the linear system.
//
  ipvt = new int[n];

  wtime = omp_get_wtime ( );
  info = msgefa2 ( a, lda, n, ipvt );
  wtime = omp_get_wtime ( ) - wtime;

  if ( info != 0 )
  {
    cout << "\n";
    cout << "TEST03 - Fatal error!\n";
    cout << "  MSGEFA2 reports the matrix is singular.\n";
    exit ( 1 );
  }
//
//  Solve the linear system.
//
  job = 0;
  sgesl ( a, lda, n, ipvt, b, job );

  err = 0.0;
  for ( i = 0; i < n; i++ )
  {
    err = err + fabs ( x[i] - b[i] );
  }

  cout << "  Revised   Sequential   "
       << "  " << setw(8) << n
       << "  " << setw(10) << err
       << "  " << setw(10) << wtime << "\n";

  delete [] a;
  delete [] b;
  delete [] ipvt;
  delete [] x;

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

int isamax ( int n, float x[], int incx )

//****************************************************************************80 
//
//  Purpose:
//
//    ISAMAX finds the index of the vector element of maximum absolute value.
//
//  Discussion:
//
//    WARNING: This index is a 1-based index, not a 0-based index!
//
//  Modified:
//
//    07 April 2008
//
//  Author:
//
//    FORTRAN77 original version by Lawson, Hanson, Kincaid, Krogh.
//    C++ version by John Burkardt
//
//  Reference:
//
//    Jack Dongarra, Cleve Moler, Jim Bunch, Pete Stewart,
//    LINPACK User's Guide,
//    SIAM, 1979,
//    ISBN13: 978-0-898711-72-1,
//    LC: QA214.L56.
//
//    Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh,
//    Algorithm 539: 
//    Basic Linear Algebra Subprograms for Fortran Usage,
//    ACM Transactions on Mathematical Software,
//    Volume 5, Number 3, September 1979, pages 308-323.
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Input, float X[*], the vector to be examined.
//
//    Input, int INCX, the increment between successive entries of SX.
//
//    Output, int ISAMAX, the index of the element of maximum
//    absolute value.
//
{
  float xmax;
  int i;
  int ix;
  int value;

  value = 0;

  if ( n < 1 || incx <= 0 )
  {
    return value;
  }

  value = 1;

  if ( n == 1 )
  {
    return value;
  }

  if ( incx == 1 )
  {
    xmax = fabs ( x[0] );

    for ( i = 1; i < n; i++ )
    {
      if ( xmax < fabs ( x[i] ) )
      {
        value = i + 1;
        xmax = fabs ( x[i] );
      }
    }
  }
  else
  {
    ix = 0;
    xmax = fabs ( x[0] );
    ix = ix + incx;

    for ( i = 1; i < n; i++ )
    {
      if ( xmax < fabs ( x[ix] ) )
      {
        value = i + 1;
        xmax = fabs ( x[ix] );
      }
      ix = ix + incx;
    }
  }

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

void matgen ( int lda, int n, float a[], float x[], float b[] )

//****************************************************************************80
// 
//  Purpose:
//
//    MATGEN generates a "random" matrix for testing.
//
//  Modified:
//
//    27 April 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int LDA, the leading dimension of the matrix.
//
//    Input, int N, the order of the matrix, and the length of the vector.
//
//    Output, float A[LDA*N], the matrix.
//
//    Output, float X[N], the solution vector.
//
//    Output, float B[N], the right hand side vector.
//
{
  int i;
  int j;
  int seed;
  float value;

  seed = 1325;
//
//  Set the matrix A.
//
  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < n; i++ )
    {
      seed = ( 3125 * seed ) % 65536;
      value = ( ( float ) seed - 32768.0 ) / 16384.0;
      a[i+j*lda] = value;
    }
  }
//
//  Set x.
//
  for ( i = 0; i < n; i++ )
  {
    x[i] = ( float ) ( i + 1 ) / ( ( float ) n );
  }
//
//  Set b = A * x.
//
  for ( i = 0; i < n; i++ ) 
  {
    b[i] = 0.0;
    for ( j = 0; j < n; j++ )
    {
      b[i] = b[i] + a[i+j*lda] * x[j];
    }
  }
  return;
}
//****************************************************************************80

void msaxpy ( int nr, int nc, float a[], int n, float x[], float y[] )

//****************************************************************************80
//
//  Purpose:
//
//    MSAXPY carries out multiple "SAXPY" operations.
//
//  Discussion:
//
//    This routine carries out the step of Gaussian elimination where multiples
//    of the pivot row are added to the rows below the pivot row.
//
//    A single call to MSAXPY replaces multiple calls to SAXPY.
//
//  Modified:
//
//    07 April 2008
//
//  Author:
//
//    C original version by Wesley Petersen
//
//  Parameters:
//
//    Input, int NR, NC, ???
//
//    Input, float A[*], ...
//
//    Input, int N, ...
//
//    Input, float X[*], ...
//
//    Output, float Y[*], ...
//

{
  int i,j;

# pragma omp parallel \
  shared ( a, nc, nr, x, y ) \
  private ( i, j ) 

# pragma omp for 

  for ( j = 0; j < nc; j++)
  {
    for ( i = 0; i < nr; i++ )
    {
      y[i+j*n] += a[j*n] * x[i];
    }
  }
  return;
}
//****************************************************************************80

void msaxpy2 ( int nr, int nc, float a[], int n, float x[], float y[] )

//****************************************************************************80
//
//  Purpose:
//
//    MSAXPY2 carries out multiple "SAXPY" operations.
//
//  Discussion:
//
//    This routine carries out the step of Gaussian elimination where multiples
//    of the pivot row are added to the rows below the pivot row.
//
//    A single call to MSAXPY replaces multiple calls to SAXPY.
//
//  Modified:
//
//    07 April 2008
//
//  Author:
//
//    C original version by Wesley Petersen
//
//  Parameters:
//
//    Input, int NR, NC, ???
//
//    Input, float A[*], ...
//
//    Input, int N, ...
//
//    Input, float X[*], ...
//
//    Output, float Y[*], ...
//
{
  int i,j;

  for ( j = 0; j < nc; j++)
  {
    for ( i = 0; i < nr; i++ )
    {
      y[i+j*n] += a[j*n] * x[i];
    }
  }
  return;
}
//****************************************************************************80

int msgefa ( float a[], int lda, int n, int ipvt[] )

//****************************************************************************80
// 
//  Purpose:
//
//    MSGEFA factors a matrix by gaussian elimination.
//
//  Discussion:
//
//    Matrix references which would, mathematically, be written A(I,J)
//    must be written here as:
//    * A[I+J*LDA], when the value is needed, or
//    * A+I+J*LDA, when the address is needed.
//
//    This variant of SGEFA uses OpenMP for improved parallel execution.
//    The step in which multiples of the pivot row are added to individual
//    rows has been replaced by a single call which updates the entire
//    matrix sub-block.
//
//  Modified:
//
//    07 March 2008
//
//  Author:
//
//    Wesley Petersen.
//
//  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/output, float A[LDA*N].  On input, the matrix to be factored.
//    On output, 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.
//
//    Input, int LDA, the leading dimension of the matrix.
//
//    Input, int N, the order of the matrix.
//
//    Output, int IPVT[N], the pivot indices.
//
//    Output, int MSGEFA, indicates singularity.
//    If 0, this is the normal value, and the algorithm succeeded.
//    If K, then on the K-th elimination step, a zero pivot was encountered.
//    The matrix is numerically not invertible.
//
{
  int info;
  int k,kp1,l,nm1;
  float t;

  info = 0;
  nm1 = n - 1;
  for ( k = 0; k < nm1; k++ )
  {
    kp1 = k + 1;
    l = isamax ( n-k, a+k+k*lda, 1 ) + k - 1;
    ipvt[k] = l + 1;

    if ( a[l+k*lda] == 0.0 )
    {
      info = k + 1;
      return info;
    }

    if ( l != k )
    {
      t          = a[l+k*lda];
      a[l+k*lda] = a[k+k*lda];
      a[k+k*lda] = t;
    }
    t = -1.0 / a[k+k*lda]; 
    sscal ( n-k-1, t, a+kp1+k*lda, 1 );
//
//  Interchange the pivot row and the K-th row.
//
    if ( l != k )
    {
      sswap ( n-k-1, a+l+kp1*lda, lda, a+k+kp1*lda, lda );
    }
//
//  Add multiples of the K-th row to rows K+1 through N.
//
    msaxpy ( n-k-1, n-k-1, a+k+kp1*lda, n, a+kp1+k*lda, a+kp1+kp1*lda );
  }

  ipvt[n-1] = n;

  if ( a[n-1+(n-1)*lda] == 0.0 )
  {
    info = n;
  }

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

int msgefa2 ( float a[], int lda, int n, int ipvt[] )

//****************************************************************************80
// 
//  Purpose:
//
//    MSGEFA2 factors a matrix by gaussian elimination.
//
//  Discussion:
//
//    Matrix references which would, mathematically, be written A(I,J)
//    must be written here as:
//    * A[I+J*LDA], when the value is needed, or
//    * A+I+J*LDA, when the address is needed.
//
//    This variant of SGEFA uses OpenMP for improved parallel execution.
//    The step in which multiples of the pivot row are added to individual
//    rows has been replaced by a single call which updates the entire
//    matrix sub-block.
//
//  Modified:
//
//    07 March 2008
//
//  Author:
//
//    Wesley Petersen.
//
//  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/output, float A[LDA*N].  On input, the matrix to be factored.
//    On output, 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.
//
//    Input, int LDA, the leading dimension of the matrix.
//
//    Input, int N, the order of the matrix.
//
//    Output, int IPVT[N], the pivot indices.
//
//    Output, int MSGEFA, indicates singularity.
//    If 0, this is the normal value, and the algorithm succeeded.
//    If K, then on the K-th elimination step, a zero pivot was encountered.
//    The matrix is numerically not invertible.
//
{
  int info;
  int k,kp1,l,nm1;
  float t;

  info = 0;
  nm1 = n - 1;
  for ( k = 0; k < nm1; k++ )
  {
    kp1 = k + 1;
    l = isamax ( n-k, a+k+k*lda, 1 ) + k - 1;
    ipvt[k] = l + 1;

    if ( a[l+k*lda] == 0.0 )
    {
      info = k + 1;
      return info;
    }

    if ( l != k )
    {
      t          = a[l+k*lda];
      a[l+k*lda] = a[k+k*lda];
      a[k+k*lda] = t;
    }
    t = -1.0 / a[k+k*lda]; 
    sscal ( n-k-1, t, a+kp1+k*lda, 1 );
//
//  Interchange the pivot row and the K-th row.
//
    if ( l != k )
    {
      sswap ( n-k-1, a+l+kp1*lda, lda, a+k+kp1*lda, lda );
    }
//
//  Add multiples of the K-th row to rows K+1 through N.
//
    msaxpy2 ( n-k-1, n-k-1, a+k+kp1*lda, n, a+kp1+k*lda, a+kp1+kp1*lda );
  }

  ipvt[n-1] = n;

  if ( a[n-1+(n-1)*lda] == 0.0 )
  {
    info = n;
  }

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

void saxpy ( int n, float a, float x[], int incx, float y[], int incy )

//****************************************************************************80
//
//  Purpose:
//
//    SAXPY computes float constant times a vector plus a vector.
//
//  Discussion:
//
//    This routine uses unrolled loops for increments equal to one.
//
//  Modified:
//
//    23 February 2006
//
//  Author:
//
//    FORTRAN77 original version by Dongarra, Moler, Bunch, Stewart.
//    C++ version by John Burkardt
//
//  Reference:
//
//    Jack Dongarra, Cleve Moler, Jim Bunch, Pete Stewart,
//    LINPACK User's Guide,
//    SIAM, 1979,
//    ISBN13: 978-0-898711-72-1,
//    LC: QA214.L56.
//
//    Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh,
//    Basic Linear Algebra Subprograms for Fortran Usage,
//    Algorithm 539, 
//    ACM Transactions on Mathematical Software, 
//    Volume 5, Number 3, September 1979, pages 308-323.
//
//  Parameters:
//
//    Input, int N, the number of elements in X and Y.
//
//    Input, float A, the multiplier of X.
//
//    Input, float X[*], the first vector.
//
//    Input, int INCX, the increment between successive entries of X.
//
//    Input/output, float Y[*], the second vector.
//    On output, Y[*] has been replaced by Y[*] + A * X[*].
//
//    Input, int INCY, the increment between successive entries of Y.
//
{
  int i;
  int ix;
  int iy;
  int m;

  if ( n <= 0 )
  {
    return;
  }

  if ( a == 0.0 )
  {
    return;
  }
//
//  Code for unequal increments or equal increments
//  not equal to 1.
//
  if ( incx != 1 || incy != 1 )
  {
    if ( 0 <= incx )
    {
      ix = 0;
    }
    else
    {
      ix = ( - n + 1 ) * incx;
    }

    if ( 0 <= incy )
    {
      iy = 0;
    }
    else
    {
      iy = ( - n + 1 ) * incy;
    }

    for ( i = 0; i < n; i++ )
    {
      y[iy] = y[iy] + a * x[ix];
      ix = ix + incx;
      iy = iy + incy;
    }
  }
//
//  Code for both increments equal to 1.
//
  else
  {
    m = n % 4;

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

    for ( i = m; i < n; i = i + 4 )
    {
      y[i  ] = y[i  ] + a * x[i  ];
      y[i+1] = y[i+1] + a * x[i+1];
      y[i+2] = y[i+2] + a * x[i+2];
      y[i+3] = y[i+3] + a * x[i+3];
    }
  }

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

float sdot ( int n, float x[], int incx, float y[], int incy )

//****************************************************************************80
//
//  Purpose:
//
//    SDOT forms the dot product of two vectors.
//
//  Discussion:
//
//    This routine uses unrolled loops for increments equal to one.
//
//  Modified:
//
//    23 February 2006
//
//  Author:
//
//    FORTRAN77 original version by Dongarra, Moler, Bunch, Stewart
//    C++ version by John Burkardt
//
//  Reference:
//
//    Jack Dongarra, Cleve Moler, Jim Bunch, Pete Stewart,
//    LINPACK User's Guide,
//    SIAM, 1979.
//
//    Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh,
//    Basic Linear Algebra Subprograms for Fortran Usage,
//    Algorithm 539, 
//    ACM Transactions on Mathematical Software, 
//    Volume 5, Number 3, September 1979, pages 308-323.
//
//  Parameters:
//
//    Input, int N, the number of entries in the vectors.
//
//    Input, float X[*], the first vector.
//
//    Input, int INCX, the increment between successive entries in X.
//
//    Input, float Y[*], the second vector.
//
//    Input, int INCY, the increment between successive entries in Y.
//
//    Output, float SDOT, the sum of the product of the corresponding
//    entries of X and Y.
//
{
  int i;
  int ix;
  int iy;
  int m;
  float temp;

  temp = 0.0;

  if ( n <= 0 )
  {
    return temp;
  }
//
//  Code for unequal increments or equal increments
//  not equal to 1.
//
  if ( incx != 1 || incy != 1 )
  {
    if ( 0 <= incx )
    {
      ix = 0;
    }
    else
    {
      ix = ( - n + 1 ) * incx;
    }

    if ( 0 <= incy )
    {
      iy = 0;
    }
    else
    {
      iy = ( - n + 1 ) * incy;
    }

    for ( i = 0; i < n; i++ )
    {
      temp = temp + x[ix] * y[iy];
      ix = ix + incx;
      iy = iy + incy;
    }
  }
//
//  Code for both increments equal to 1.
//
  else
  {
    m = n % 5;

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

    for ( i = m; i < n; i = i + 5 )
    {
      temp = temp + x[i  ] * y[i  ] 
                  + x[i+1] * y[i+1] 
                  + x[i+2] * y[i+2] 
                  + x[i+3] * y[i+3] 
                  + x[i+4] * y[i+4];
    }
  }

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

int sgefa ( float a[], int lda, int n, int ipvt[] )

//****************************************************************************80
//
//  Purpose:
//
//    SGEFA factors a double precision matrix by gaussian elimination.
//
//  Discussion:
//
//    Matrix references which would, mathematically, be written A(I,J)
//    must be written here as:
//    * A[I+J*LDA], when the value is needed, or
//    * A+I+J*LDA, when the address is needed.
//
//  Modified:
//
//    07 March 2008
//
//  Author:
//
//    FORTRAN77 original version by Cleve Moler.
//    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/output, float A[LDA*N].  On input, the matrix to be factored.
//    On output, 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.
//
//    Input, int LDA, the leading dimension of the matrix.
//
//    Input, int N, the order of the matrix.
//
//    Output, int IPVT[N], the pivot indices.
//
//    Output, int SGEFA, indicates singularity.
//    If 0, this is the normal value, and the algorithm succeeded.
//    If K, then on the K-th elimination step, a zero pivot was encountered.
//    The matrix is numerically not invertible.
//
{
  int j;
  int info;
  int k;
  int l;
  float t;

  info = 0;

  for ( k = 1; k <= n - 1; k++ ) 
  {
// 
//  Find l = pivot index.
//
    l = isamax ( n-k+1, &a[k-1+(k-1)*lda], 1 ) + k - 1;
    ipvt[k-1] = l;
// 
//  Zero pivot implies this column already triangularized.
//
    if ( a[l-1+(k-1)*lda] != 0.0 ) 
    {
// 
//  Interchange if necessary.
//
      if ( l != k ) 
      {
        t                = a[l-1+(k-1)*lda];
        a[l-1+(k-1)*lda] = a[k-1+(k-1)*lda];
        a[k-1+(k-1)*lda] = t; 
      }
// 
//  Compute multipliers.
//
      t = - 1.0 / a[k-1+(k-1)*lda];
      sscal ( n-k, t, &a[k+(k-1)*lda], 1 );
// 
//  Row elimination with column indexing.
//
      for ( j = k + 1; j <= n; j++ ) 
      {
        t = a[l-1+(j-1)*lda];
        if (l != k) 
        {
          a[l-1+(j-1)*lda] = a[k-1+(j-1)*lda];
          a[k-1+(j-1)*lda] = t;
        }
        saxpy ( n-k, t, &a[k+(k-1)*lda], 1, &a[k+(j-1)*lda], 1 );
      } 
    }
    else
    { 
      info = k;
    }
  } 
  ipvt[n-1] = n;

  if (a[n-1+(n-1)*lda] == 0.0 ) 
  {
    info = n - 1;
  }
  return info;
}
//****************************************************************************80

void sgesl ( float a[], int lda, int n, int ipvt[], float b[], int job )

//****************************************************************************80
//
//  Purpose:
//
//    SGESL solves a real general linear system A * X = B.
//
//  Discussion:
//
//    SGESL can solve either of the systems A * X = B or A' * X = B.
//
//    The system matrix must have been factored by SGECO or SGEFA.
//
//    A division by zero will occur if the input factor contains a
//    zero on the diagonal.  Technically this indicates singularity
//    but it is often caused by improper arguments or improper
//    setting of LDA.  It will not occur if the subroutines are
//    called correctly and if SGECO has set 0.0 < RCOND
//    or SGEFA has set INFO == 0.
//
//  Modified:
//
//    04 April 2006
//
//  Author:
//
//    FORTRAN77 original by Dongarra, Moler, Bunch and Stewart.
//    C++ translation by John Burkardt.
//
//  Reference:
//
//    Jack Dongarra, Cleve Moler, Jim Bunch, Pete Stewart,
//    LINPACK User's Guide,
//    SIAM, (Society for Industrial and Applied Mathematics),
//    3600 University City Science Center,
//    Philadelphia, PA, 19104-2688.
//    ISBN: 0-89871-172-X
//
//  Parameters:
//
//    Input, float A[LDA*N], the output from SGECO or SGEFA.
//
//    Input, int LDA, the leading dimension of A.
//
//    Input, int N, the order of the matrix A.
//
//    Input, int IPVT[N], the pivot vector from SGECO or SGEFA.
//
//    Input/output, float B[N].
//    On input, the right hand side vector.
//    On output, the solution vector.
//
//    Input, int JOB.
//    0, solve A * X = B;
//    nonzero, solve A' * X = B.
//
{
  int k;
  int l;
  float t;
//
//  Solve A * X = B.
//

  if ( job == 0 )
  {
    for ( k = 1; k <= n-1; k++ )
    {
      l = ipvt[k-1];
      t = b[l-1];

      if ( l != k )
      {
        b[l-1] = b[k-1];
        b[k-1] = t;
      }
      saxpy ( n-k, t, a+k+(k-1)*lda, 1, b+k, 1 );
    }

    for ( k = n; 1 <= k; k-- )
    {
      b[k-1] = b[k-1] / a[k-1+(k-1)*lda];
      t = -b[k-1];
      saxpy ( k-1, t, a+0+(k-1)*lda, 1, b, 1 );
    }
  }
//
//  Solve A' * X = B.
//
  else
  {
    for ( k = 1; k <= n; k++ )
    {
      t = sdot ( k-1, a+0+(k-1)*lda, 1, b, 1 );
      b[k-1] = ( b[k-1] - t ) / a[k-1+(k-1)*lda];
    }

    for ( k = n-1; 1 <= k; k-- )
    {
      b[k-1] = b[k-1] + sdot ( n-k, a+k+(k-1)*lda, 1, b+k, 1 );
      l = ipvt[k-1];

      if ( l != k )
      {
        t = b[l-1];
        b[l-1] = b[k-1];
        b[k-1] = t;
      }
    }
  }
  return;
}
//****************************************************************************80

void sscal ( int n, float sa, float x[], int incx )

//****************************************************************************80
//
//  Purpose:
//
//    SSCAL scales a float vector by a constant.
//
//  Modified:
//
//    23 February 2006
//
//  Author:
//
//    FORTRAN77 original version by Lawson, Hanson, Kincaid, Krogh.
//    C++ version by John Burkardt
//
//  Reference:
//
//    Jack Dongarra, Cleve Moler, Jim Bunch, Pete Stewart,
//    LINPACK User's Guide,
//    SIAM, 1979,
//    ISBN13: 978-0-898711-72-1,
//    LC: QA214.L56.
//
//    Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh,
//    Basic Linear Algebra Subprograms for Fortran Usage,
//    Algorithm 539,
//    ACM Transactions on Mathematical Software,
//    Volume 5, Number 3, September 1979, pages 308-323.
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Input, float SA, the multiplier.
//
//    Input/output, float X[*], the vector to be scaled.
//
//    Input, int INCX, the increment between successive entries of X.
//
{
  int i;
  int ix;
  int m;

  if ( n <= 0 )
  {
  }
  else if ( incx == 1 )
  {
    m = n % 5;

    for ( i = 0; i < m; i++ )
    {
      x[i] = sa * x[i];
    }

    for ( i = m; i < n; i = i + 5 )
    {
      x[i]   = sa * x[i];
      x[i+1] = sa * x[i+1];
      x[i+2] = sa * x[i+2];
      x[i+3] = sa * x[i+3];
      x[i+4] = sa * x[i+4];
    }
  }
  else
  {
    if ( 0 <= incx )
    {
      ix = 0;
    }
    else
    {
      ix = ( - n + 1 ) * incx;
    }

    for ( i = 0; i < n; i++ )
    {
      x[ix] = sa * x[ix];
      ix = ix + incx;
    }

  }

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

void sswap ( int n, float x[], int incx, float y[], int incy )

//****************************************************************************80
//
//  Purpose:
//
//    SSWAP interchanges two float vectors.
//
//  Modified:
//
//    23 February 2006
//
//  Author:
//
//    FORTRAN77 original version by Lawson, Hanson, Kincaid, Krogh.
//    C++ version by John Burkardt
//
//  Reference:
//
//    Jack Dongarra, Cleve Moler, Jim Bunch, Pete Stewart,
//    LINPACK User's Guide,
//    SIAM, 1979,
//    ISBN13: 978-0-898711-72-1,
//    LC: QA214.L56.
//
//    Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh,
//    Basic Linear Algebra Subprograms for Fortran Usage,
//    Algorithm 539, 
//    ACM Transactions on Mathematical Software, 
//    Volume 5, Number 3, September 1979, pages 308-323.
//
//  Parameters:
//
//    Input, int N, the number of entries in the vectors.
//
//    Input/output, float X[*], one of the vectors to swap.
//
//    Input, int INCX, the increment between successive entries of X.
//
//    Input/output, float Y[*], one of the vectors to swap.
//
//    Input, int INCY, the increment between successive elements of Y.
//
{
  int i;
  int ix;
  int iy;
  int m;
  float temp;

  if ( n <= 0 )
  {
  }
  else if ( incx == 1 && incy == 1 )
  {
    m = n % 3;

    for ( i = 0; i < m; i++ )
    {
      temp = x[i];
      x[i] = y[i];
      y[i] = temp;
    }

    for ( i = m; i < n; i = i + 3 )
    {
      temp = x[i];
      x[i] = y[i];
      y[i] = temp;

      temp = x[i+1];
      x[i+1] = y[i+1];
      y[i+1] = temp;

      temp = x[i+2];
      x[i+2] = y[i+2];
      y[i+2] = temp;
    }
  }
  else
  {
    if ( 0 <= incx )
    {
      ix = 0;
    }
    else
    {
      ix = ( - n + 1 ) * incx;
    }

    if ( 0 <= incy )
    {
      iy = 0;
    }
    else
    {
      iy = ( - n + 1 ) * incy;
    }

    for ( i = 0; i < n; i++ )
    {
      temp = x[ix];
      x[ix] = y[iy];
      y[iy] = temp;
      ix = ix + incx;
      iy = iy + incy;
    }
  }
  return;
}
//****************************************************************************80

void timestamp ( )

//****************************************************************************80
//
//  Purpose:
//
//    TIMESTAMP prints the current YMDHMS date as a time stamp.
//
//  Example:
//
//    31 May 2001 09:45:54 AM
//
//  Modified:
//
//    24 September 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    None
//
{
# define TIME_SIZE 40

  static char time_buffer[TIME_SIZE];
  const struct tm *tm;
  time_t now;

  now = time ( NULL );
  tm = localtime ( &now );

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

  cout << time_buffer << "\n";

  return;
# undef TIME_SIZE
}