function i4_log_10 ( i )

!*****************************************************************************80
!
!! i4_log_10() returns the integer part of the logarithm base 10 of an I4.
!
!  Discussion:
!
!    I4_LOG_10 ( I ) + 1 is the number of decimal digits in I.
!
!    An I4 is an integer value.
!
!  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
!
!  Licensing:
!
!    This code is distributed under the MIT license.
!
!  Modified:
!
!    08 June 2003
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer I, the number whose logarithm base 10
!    is desired.
!
!    Output, integer I4_LOG_10, the integer part of the
!    logarithm base 10 of the absolute value of X.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer i
  integer i_abs
  integer i4_log_10
  integer ten_pow

  if ( i == 0 ) then

    i4_log_10 = 0

  else

    i4_log_10 = 0
    ten_pow = 10

    i_abs = abs ( i )

    do while ( ten_pow <= i_abs )
      i4_log_10 = i4_log_10 + 1
      ten_pow = ten_pow * 10
    end do

  end if

  return
end
subroutine r8ge_print ( m, n, a, title )

!*****************************************************************************80
!
!! R8GE_PRINT prints an R8GE 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.
!
!    R8GE storage is used by LINPACK and LAPACK.
!
!  Licensing:
!
!    This code is distributed under the MIT license. 
!
!  Modified:
!
!    07 May 2000
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer M, the number of rows of the matrix.
!    M must be positive.
!
!    Input, integer N, the number of columns of the matrix.
!    N must be positive.
!
!    Input, real ( kind = rk ) A(M,N), the R8GE matrix.
!
!    Input, character ( len = * ) TITLE, a title.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer m
  integer n

  real ( kind = rk ) a(m,n)
  character ( len = * ) title

  call r8ge_print_some ( m, n, a, 1, 1, m, n, title )

  return
end
subroutine r8ge_print_some ( m, n, a, ilo, jlo, ihi, jhi, title )

!*****************************************************************************80
!
!! R8GE_PRINT_SOME prints some of an R8GE 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.
!
!    R8GE storage is used by LINPACK and LAPACK.
!
!  Licensing:
!
!    This code is distributed under the MIT license. 
!
!  Modified:
!
!    21 March 2001
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer M, the number of rows of the matrix.
!    M must be positive.
!
!    Input, integer N, the number of columns of the matrix.
!    N must be positive.
!
!    Input, real ( kind = rk ) A(M,N), the R8GE matrix.
!
!    Input, integer ILO, JLO, IHI, JHI, the first row and
!    column, and the last row and column to be printed.
!
!    Input, character ( len = * ) TITLE, a title.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer, parameter :: incx = 5
  integer m
  integer n

  real ( kind = rk ) a(m,n)
  character ( len = 14 ) ctemp(incx)
  integer i
  integer i2hi
  integer i2lo
  integer ihi
  integer ilo
  integer inc
  integer j
  integer j2
  integer j2hi
  integer j2lo
  integer jhi
  integer jlo
  character ( len = * ) title

  write ( *, '(a)' ) ' '
  write ( *, '(a)' ) trim ( title )
!
!  Print the columns of the matrix, in strips of 5.
!
  do j2lo = jlo, jhi, incx

    j2hi = j2lo + incx - 1
    j2hi = min ( j2hi, n )
    j2hi = min ( j2hi, jhi )

    inc = j2hi + 1 - j2lo

    write ( *, '(a)' ) ' '

    do j = j2lo, j2hi
      j2 = j + 1 - j2lo
      write ( ctemp(j2), '(i7,7x)' ) j
    end do

    write ( *, '(''  Col:  '',5a14)' ) ( ctemp(j2), j2 = 1, inc )
    write ( *, '(a)' ) '  Row'
    write ( *, '(a)' ) '  ---'
!
!  Determine the range of the rows in this strip.
!
    i2lo = max ( ilo, 1 )
    i2hi = min ( ihi, m )

    do i = i2lo, i2hi
!
!  Print out (up to) 5 entries in row I, that lie in the current strip.
!
      do j2 = 1, inc

        j = j2lo - 1 + j2

        write ( ctemp(j2), '(g14.6)' ) a(i,j)

      end do

      write ( *, '(i5,1x,5a14)' ) i, ( ctemp(j2), j2 = 1, inc )

    end do

  end do

  return
end
subroutine r8ge_random ( m, n, a )

!*****************************************************************************80
!
!! R8GE_RANDOM randomizes an R8GE 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.
!
!    R8GE storage is used by LINPACK and LAPACK.
!
!  Licensing:
!
!    This code is distributed under the MIT license.
!
!  Modified:
!
!    11 August 2004
!
!  Author:
!
!    John Burkardt
!
!  Reference:
!
!    Paul Bratley, Bennett Fox, Linus Schrage,
!    A Guide to Simulation,
!    Springer Verlag, pages 201-202, 1983.
!
!    Bennett Fox,
!    Algorithm 647:
!    Implementation and Relative Efficiency of Quasirandom
!    Sequence Generators,
!    ACM Transactions on Mathematical Software,
!    Volume 12, Number 4, pages 362-376, 1986.
!
!    Peter Lewis, Allen Goodman, James Miller,
!    A Pseudo-Random Number Generator for the System/360,
!    IBM Systems Journal,
!    Volume 8, pages 136-143, 1969.
!
!  Parameters:
!
!    Input, integer M, N, the number of rows and columns in
!    the array.
!
!    Output, real ( kind = rk ) A(M,N), the array.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer m
  integer n

  real ( kind = rk ) a(m,n)

  call random_number ( harvest = a(1:m,1:n) )

  return
end
subroutine r8ge_to_r8lt ( m, n, a_ge, a_lt )

!*****************************************************************************80
!
!! 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.  The format stores all M*N entries, even those which are zero.
!
!  Licensing:
!
!    This code is distributed under the MIT license. 
!
!  Modified:
!
!    21 August 2015
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer M, N, the number of rows and columns of 
!    the matrix.
!
!    Input, real ( kind = rk ) A_GE(N), the R8GE matrix.
!
!    Output, real ( kind = rk ) A_LT(N), the R8LT matrix.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer m
  integer n

  real ( kind = rk ) a_ge(m,n)
  real ( kind = rk ) a_lt(m,n)
  integer i
  integer j

  a_lt(1:m,1:n) = 0.0D+00
  do j = 1, n
    do i = j, m
      a_lt(i,j) = a_ge(i,j)
    end do
  end do

  return
end
subroutine r8lt_det ( n, a, det )

!*****************************************************************************80
!
!! R8LT_DET computes the determinant of an R8LT matrix.
!
!  Discussion:
!
!    The R8LT storage format is used for an M by N lower triangular matrix,
!    and sets aside storage even for the entries that must be zero.
!
!  Licensing:
!
!    This code is distributed under the MIT license. 
!
!  Modified:
!
!    06 August 2015
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer N, the order of the matrix.
!    N must be positive.
!
!    Input, real ( kind = rk ) A(N,N), the R8LT matrix.
!
!    Output, real ( kind = rk ) DET, the determinant of the matrix.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer n

  real ( kind = rk ) a(n,n)
  real ( kind = rk ) det
  integer i

  det = 1.0D+00
  do i = 1, n
    det = det * a(i,i)
  end do

  return
end
subroutine r8lt_indicator ( m, n, a )

!*****************************************************************************80
!
!! R8LT_INDICATOR sets up an R8LT indicator matrix.
!
!  Discussion:
!
!    The "indicator matrix" simply has a value like I*10+J at every
!    entry of a dense matrix, or at every entry of a compressed storage
!    matrix for which storage is allocated. 
!
!    The R8LT storage format is used for an M by N lower triangular matrix,
!    and sets aside storage even for the entries that must be zero.
!
!  Licensing:
!
!    This code is distributed under the MIT license. 
!
!  Modified:
!
!    1 February 2004
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer M, N, the number of rows and columns of 
!    the matrix.  M and N must be positive.
!
!    Output, real ( kind = rk ) A(M,N), the R8LT matrix.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer m
  integer n

  real ( kind = rk ) a(m,n)
  integer fac
  integer i
  integer i4_log_10
  integer j

  fac = 10 ** ( i4_log_10 ( n ) + 1 )

  do i = 1, m
    do j = 1, min ( i, n )
      a(i,j) = real ( fac * i + j, kind = rk )
    end do
    do j = i+1, n
      a(i,j) = 0.0D+00
    end do
  end do

  return
end
subroutine r8lt_inverse ( n, a )

!*****************************************************************************80
!
!! R8LT_INVERSE computes the inverse of an R8LT matrix.
!
!  Discussion:
!
!    The R8LT storage format is used for an M by N lower triangular matrix,
!    and sets aside storage even for the entries that must be zero.
!
!  Licensing:
!
!    This code is distributed under the MIT license. 
!
!  Modified:
!
!    03 May 2003
!
!  Author:
!
!    John Burkardt
!
!  Reference:
!
!    Albert Nijenhuis, Herbert Wilf,
!    Combinatorial Algorithms,
!    Second edition,
!    Academic Press, 1978,
!    ISBN 0-12-519260-6
!
!  Parameters:
!
!    Input, integer N, the order of the matrix.
!
!    Input/output, real ( kind = rk ) A(N,N).
!
!    On input, the lower triangular matrix to be inverted.
!    On output, the inverse of the lower triangular matrix.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer n

  real ( kind = rk ) a(n,n)
  integer i
  integer j

!  Check.
!
  do i = 1, n
    if ( a(i,i) == 0.0D+00 ) then
      write ( *, '(a)' ) ' '
      write ( *, '(a)' ) 'R8LT_INVERSE - Fatal error!'
      write ( *, '(a)' ) '  Zero diagonal element.'
      stop 1
    end if
  end do

  do j = 1, n

    do i = 1, n

      if ( i < j ) then

        a(i,j) = 0.0D+00

      else if ( i == j ) then

        a(i,j) = 1.0D+00 / a(i,j)

      else if ( j < i ) then

        a(i,j) = - sum ( a(i,j:i-1) * a(j:i-1,j) ) / a(i,i)

      end if

    end do
  end do

  return
end
subroutine r8lt_mm ( n, a, b, c )

!*****************************************************************************80
!
!! R8LT_MM multiplies two R8LT matrices.
!
!  Discussion:
!
!    The R8LT storage format is used for an M by N lower triangular matrix,
!    and sets aside storage even for the entries that must be zero.
!
!  Licensing:
!
!    This code is distributed under the MIT license. 
!
!  Modified:
!
!    03 May 2003
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer N, the order of the matrices.
!    N must be positive.
!
!    Input, real ( kind = rk ) A(N,N), B(N,N), the R8LT factor matrices.
!
!    Output, real ( kind = rk ) C(N,N), the R8LT product matrix.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer n

  real ( kind = rk ) a(n,n)
  real ( kind = rk ) b(n,n)
  real ( kind = rk ) c(n,n)
  integer i
  integer j

  c(1:n,1:n) = 0.0D+00

  do i = 1, n
    do j = 1, i
      c(i,j) = sum ( a(i,j:i) * b(j:i,j) )
    end do
  end do

  return
end
subroutine r8lt_mtm ( n, a, b, c )

!*****************************************************************************80
!
!! R8LT_MTM computes C=A'*B for R8LT matrices.
!
!  Discussion:
!
!    The R8LT storage format is used for an M by N lower triangular matrix,
!    and sets aside storage even for the entries that must be zero.
!
!  Licensing:
!
!    This code is distributed under the MIT license. 
!
!  Modified:
!
!    23 August 2015
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer N, the order of the matrices.
!    N must be positive.
!
!    Input, real ( kind = rk ) A(N,N), B(N,N), the R8LT factor matrices.
!
!    Output, real ( kind = rk ) C(N,N), the R8LT product matrix.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer n

  real ( kind = rk ) a(n,n)
  real ( kind = rk ) b(n,n)
  real ( kind = rk ) c(n,n)
  integer i
  integer j
  integer k

  c(1:n,1:n) = 0.0D+00

  do i = 1, n
    do j = 1, n
      k = min ( i, j )
      c(i,j) = sum ( a(k:n,i) * b(k:n,j) )
    end do
  end do

  return
end
subroutine r8lt_mtv ( m, n, a, x, b )

!*****************************************************************************80
!
!! R8LT_MTV multiplies an R8VEC by an R8LT matrix.
!
!  Discussion:
!
!    The R8LT storage format is used for an M by N lower triangular matrix,
!    and sets aside storage even for the entries that must be zero.
!
!  Licensing:
!
!    This code is distributed under the MIT license. 
!
!  Modified:
!
!    03 May 2003
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer M, the number of rows of the matrix.
!    M must be positive.
!
!    Input, integer N, the number of columns of the matrix.
!    N must be positive.
!
!    Input, real ( kind = rk ) A(M,N), the R8LT matrix.
!
!    Input, real ( kind = rk ) X(M), the vector to be multiplied by A.
!
!    Output, real ( kind = rk ) B(N), the product A * x.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer m
  integer n

  real ( kind = rk ) a(m,n)
  real ( kind = rk ) b(n)
  integer i
  real ( kind = rk ) x(m)

  do i = 1, n
    b(i) = sum ( x(i:m) * a(i:m,i) )
  end do

  return
end
subroutine r8lt_mv ( m, n, a, x, b )

!*****************************************************************************80
!
!! R8LT_MV multiplies an R8LT matrix by an R8VEC.
!
!  Discussion:
!
!    The R8LT storage format is used for an M by N lower triangular matrix,
!    and sets aside storage even for the entries that must be zero.
!
!  Licensing:
!
!    This code is distributed under the MIT license. 
!
!  Modified:
!
!    03 May 2003
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer M, the number of rows of the matrix.
!    M must be positive.
!
!    Input, integer N, the number of columns of the matrix.
!    N must be positive.
!
!    Input, real ( kind = rk ) A(M,N), the R8LT matrix.
!
!    Input, real ( kind = rk ) X(N), the vector to be multiplied by A.
!
!    Output, real ( kind = rk ) B(M), the product A * x.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer m
  integer n

  real ( kind = rk ) a(m,n)
  real ( kind = rk ) b(m)
  integer i
  integer jmax
  real ( kind = rk ) x(n)

  do i = 1, m
    jmax = min ( i, n )
    b(i) = sum ( a(i,1:jmax) * x(1:jmax) )
  end do

  return
end
subroutine r8lt_print ( m, n, a, title )

!*****************************************************************************80
!
!! R8LT_PRINT prints an R8LT matrix.
!
!  Discussion:
!
!    The R8LT storage format is used for an M by N lower triangular matrix,
!    and sets aside storage even for the entries that must be zero.
!
!  Licensing:
!
!    This code is distributed under the MIT license. 
!
!  Modified:
!
!    03 May 2003
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer M, the number of rows of the matrix.
!    M must be positive.
!
!    Input, integer N, the number of columns of the matrix.
!    N must be positive.
!
!    Input, real ( kind = rk ) A(M,N), the R8LT matrix.
!
!    Input, character ( len = * ) TITLE, a title.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer m
  integer n

  real ( kind = rk ) a(m,n)
  character ( len = * ) title

  call r8lt_print_some ( m, n, a, 1, 1, m, n, title )

  return
end
subroutine r8lt_print_some ( m, n, a, ilo, jlo, ihi, jhi, title )

!*****************************************************************************80
!
!! R8LT_PRINT_SOME prints some of an R8LT matrix.
!
!  Discussion:
!
!    The R8LT storage format is used for an M by N lower triangular matrix,
!    and sets aside storage even for the entries that must be zero.
!
!  Licensing:
!
!    This code is distributed under the MIT license. 
!
!  Modified:
!
!    03 May 2003
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer M, the number of rows of the matrix.
!    M must be positive.
!
!    Input, integer N, the number of columns of the matrix.
!    N must be positive.
!
!    Input, real ( kind = rk ) A(M,N), the R8LT matrix.
!
!    Input, integer ILO, JLO, IHI, JHI, the first row and
!    column, and the last row and column to be printed.
!
!    Input, character ( len = * ) TITLE, a title.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer, parameter :: incx = 5
  integer m
  integer n

  real ( kind = rk ) a(m,n)
  character ( len = 14 ) ctemp(incx)
  integer i
  integer i2hi
  integer i2lo
  integer ihi
  integer ilo
  integer inc
  integer j
  integer j2
  integer j2hi
  integer j2lo
  integer jhi
  integer jlo
  character ( len = * ) title

  write ( *, '(a)' ) ' '
  write ( *, '(a)' ) trim ( title )
!
!  Print the columns of the matrix, in strips of 5.
!
  do j2lo = jlo, jhi, incx

    j2hi = j2lo + incx - 1
    j2hi = min ( j2hi, n )
    j2hi = min ( j2hi, jhi )

    inc = j2hi + 1 - j2lo

    write ( *, '(a)' ) ' '

    do j = j2lo, j2hi
      j2 = j + 1 - j2lo
      write ( ctemp(j2), '(i7,7x)' ) j
    end do

    write ( *, '(a,5a14)' ) '  Col: ', ( ctemp(j2), j2 = 1, inc )
    write ( *, '(a)' ) '  Row'
    write ( *, '(a)' ) '  ---'
!
!  Determine the range of the rows in this strip.
!
    i2lo = max ( ilo, 1 )
    i2lo = max ( i2lo, j2lo )
    i2hi = min ( ihi, m )

    do i = i2lo, i2hi
!
!  Print out (up to) 5 entries in row I, that lie in the current strip.
!
      do j2 = 1, inc

        j = j2lo - 1 + j2

        if ( i < j ) then
          ctemp(j2) = '              '
        else
          write ( ctemp(j2), '(g14.6)' ) a(i,j)
        end if

      end do

      write ( *, '(i5,1x,5a14)' ) i, ( ctemp(j2), j2 = 1, inc )

    end do

  end do

  return
end
subroutine r8lt_random ( m, n, a )

!*****************************************************************************80
!
!! R8LT_RANDOM randomizes an R8LT matrix.
!
!  Discussion:
!
!    The R8LT storage format is used for an M by N lower triangular matrix,
!    and sets aside storage even for the entries that must be zero.
!
!  Licensing:
!
!    This code is distributed under the MIT license. 
!
!  Modified:
!
!    25 August 2015
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer M, N, the number of rows and columns of
!    the matrix.  M and N must be positive.
!
!    Output, real ( kind = rk ) A(M,N), the R8LT matrix.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer m
  integer n

  real ( kind = rk ) a(m,n)
  integer i
  integer j

  a(1:m,1:n) = 0.0D+00

  do j = 1, n
    do i = j, m
      call random_number ( harvest = a(i,j) )
    end do
  end do

  return
end
subroutine r8lt_sl ( n, a, b )

!*****************************************************************************80
!
!! R8LT_SL solves A*x=b, where A is an R8LT matrix.
!
!  Discussion:
!
!    The R8LT storage format is used for an M by N lower triangular matrix,
!    and sets aside storage even for the entries that must be zero.
!
!    No factorization of the lower triangular matrix is required.
!
!  Licensing:
!
!    This code is distributed under the MIT license. 
!
!  Modified:
!
!    22 August 2015
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer N, the order of the matrix.
!
!    Input, real ( kind = rk ) A(N,N), the R8LT matrix.
!
!    Input/output, real ( kind = rk ) B(N).
!    On input, the right hand side.
!    On output, the solution vector.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer n

  real ( kind = rk ) a(n,n)
  real ( kind = rk ) b(n)
  integer j

  do j = 1, n
    b(j) = b(j) / a(j,j)
    b(j+1:n) = b(j+1:n) - a(j+1:n,j) * b(j)
  end do

  return
end
subroutine r8lt_slt ( n, a, b )

!*****************************************************************************80
!
!! R8LT_SLT solves A'*x=b where A is an R8LT matrix.
!
!  Discussion:
!
!    The R8LT storage format is used for an M by N lower triangular matrix,
!    and sets aside storage even for the entries that must be zero.
!
!    No factorization of the lower triangular matrix is required.
!
!  Licensing:
!
!    This code is distributed under the MIT license. 
!
!  Modified:
!
!    22 August 2015
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer N, the order of the matrix.
!
!    Input, real ( kind = rk ) A(N,N), the R8LT matrix.
!
!    Input/output, real ( kind = rk ) B(N).
!    On input, the right hand side.
!    On output, the solution vector.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer n

  real ( kind = rk ) a(n,n)
  real ( kind = rk ) b(n)
  integer j

  do j = n, 1, -1
    b(j) = b(j) / a(j,j)
    b(1:j-1) = b(1:j-1) - a(j,1:j-1) * b(j)
  end do

  return
end
subroutine r8lt_to_r8ge ( m, n, a_lt, a_ge )

!*****************************************************************************80
!
!! R8LT_TO_R8GE copies an R8LT matrix to an R8GE matrix.
!
!  Discussion:
!
!    The R8LT storage format is used for an M by N lower triangular 
!    matrix.  The format stores all M*N entries, even those which are zero.
!
!    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.
!
!  Licensing:
!
!    This code is distributed under the MIT license. 
!
!  Modified:
!
!    21 August 2015
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer M, N, the number of rows and columns of 
!    the matrix.
!
!    Input, real ( kind = rk ) A_LT(N), the R8LT matrix.
!
!    Output, real ( kind = rk ) A_GE(N), the R8GE matrix.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer m
  integer n

  real ( kind = rk ) a_ge(m,n)
  real ( kind = rk ) a_lt(m,n)
  integer i
  integer j

  a_ge(1:m,1:n) = 0.0D+00
  do j = 1, n
    do i = j, m
      a_ge(i,j) = a_lt(i,j)
    end do
  end do

  return
end
subroutine r8lt_zeros ( m, n, a )

!*****************************************************************************80
!
!! R8LT_ZEROS zeroes an R8LT matrix.
!
!  Discussion:
!
!    The R8LT storage format is used for an M by N lower triangular matrix,
!    and sets aside storage even for the entries that must be zero.
!
!  Licensing:
!
!    This code is distributed under the MIT license. 
!
!  Modified:
!
!    26 January 2013
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer M, N, the number of rows and columns of
!    the matrix.  M and N must be positive.
!
!    Output, real ( kind = rk ) A(M,N), the R8LT matrix.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer m
  integer n

  real ( kind = rk ) a(m,n)

  a(1:m,1:n) = 0.0D+00

  return
end
subroutine r8vec_indicator1 ( n, a )

!*****************************************************************************80
!
!! R8VEC_INDICATOR1 sets an R8VEC to the indicator1 vector.
!
!  Discussion:
!
!    A(1:N) = (/ 1 : N /)
!
!  Licensing:
!
!    This code is distributed under the MIT license. 
!
!  Modified:
!
!    06 September 2006
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer N, the number of elements of A.
!
!    Output, real ( kind = rk ) A(N), the array to be initialized.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer n

  real ( kind = rk ) a(n)
  integer i

  do i = 1, n
    a(i) = real ( i, kind = rk )
  end do

  return
end
subroutine r8vec_print ( n, a, title )

!*****************************************************************************80
!
!! R8VEC_PRINT prints an R8VEC.
!
!  Licensing:
!
!    This code is distributed under the MIT license. 
!
!  Modified:
!
!    16 December 1999
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    Input, integer N, the number of components of the vector.
!
!    Input, real ( kind = rk ) A(N), the vector to be printed.
!
!    Input, character ( len = * ) TITLE, a title.
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  integer n

  real ( kind = rk ) a(n)
  integer i
  character ( len = * ) title

  write ( *, '(a)' ) ' '
  write ( *, '(a)' ) trim ( title )
  write ( *, '(a)' ) ' '
  do i = 1, n
    write ( *, '(i8,g14.6)' ) i, a(i)
  end do

  return
end
subroutine timestamp ( )

!*****************************************************************************80
!
!! TIMESTAMP prints the current YMDHMS date as a time stamp.
!
!  Example:
!
!    31 May 2001   9:45:54.872 AM
!
!  Licensing:
!
!    This code is distributed under the MIT license.
!
!  Modified:
!
!    18 May 2013
!
!  Author:
!
!    John Burkardt
!
!  Parameters:
!
!    None
!
  implicit none

  integer, parameter :: rk = kind ( 1.0D+00 )

  character ( len = 8 ) ampm
  integer d
  integer h
  integer m
  integer mm
  character ( len = 9 ), parameter, dimension(12) :: month = (/ &
    'January  ', 'February ', 'March    ', 'April    ', &
    'May      ', 'June     ', 'July     ', 'August   ', &
    'September', 'October  ', 'November ', 'December ' /)
  integer n
  integer s
  integer values(8)
  integer y

  call date_and_time ( values = values )

  y = values(1)
  m = values(2)
  d = values(3)
  h = values(5)
  n = values(6)
  s = values(7)
  mm = values(8)

  if ( h < 12 ) then
    ampm = 'AM'
  else if ( h == 12 ) then
    if ( n == 0 .and. s == 0 ) then
      ampm = 'Noon'
    else
      ampm = 'PM'
    end if
  else
    h = h - 12
    if ( h < 12 ) then
      ampm = 'PM'
    else if ( h == 12 ) then
      if ( n == 0 .and. s == 0 ) then
        ampm = 'Midnight'
      else
        ampm = 'AM'
      end if
    end if
  end if

  write ( *, '(i2.2,1x,a,1x,i4,2x,i2,a1,i2.2,a1,i2.2,a1,i3.3,1x,a)' ) &
    d, trim ( month(m) ), y, h, ':', n, ':', s, '.', mm, trim ( ampm )

  return
end