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!1. Solving linear equations by QR decomposition/Gram-Schmidt and
        !                               QR decomposition/back-substitution.
        !2. Calculating the determinant of a matrix.
       
        program LIN_EQ
                implicit none

                         integer, parameter :: dbl=SELECTED_REAL_KIND(14), n=3, m=2
                         real(KIND=dbl) :: A(n,m), Q(n,m), R(m,m), QR(n,m), A_old(n,m), &
                                                         & b(n), x(m), DET
                         integer :: i

                A(1,1)=1 ; A(1,2)=1
                A(2,1)=0 ; A(2,2)=1
                A(3,1)=1 ; A(3,2)=1

                b(1)=6 ; b(2)=4 ; b(3)=6

                A_old = A

                CALL QR_GS(A, n, m, Q, R)
                CALL QR_BACK(Q, R, n, m, b, x)
                CALL QR_DETERMINANT(R, m, DET)

                call matout("A is", A, n, m)
                call matout("b is", b, n, 1)
                call matout("Q is", Q, n, m)
                call matout("R is", R, m, m)
                QR = MATMUL(Q,R)
                call matout("Q*R is", QR, n, m)
                call matout("x is", x, m, 1)

                if (n/=m) then
                   print '("The determinant of the matrix R is", f10.6)', DET
                   else
                          print '("The determinant of the matrix A is", f10.6)', DET
                endif

       

        end program LIN_EQ


        !Subroutine for QR decomposition by a modified Gram-Schmidt method
        subroutine QR_GS(A, n, m, Q, R)
        Implicit none

        integer, parameter :: dbl=SELECTED_REAL_KIND(14)
        integer :: n, m, i, j, k, l, i1
        real(KIND=dbl) :: A(n,m), Q(n,m), R(m,m)

        do i = 1, m
           R(i,i)=SQRT(DOT_PRODUCT(A(1:n,i),A(1:n,i)))
           do j = 1, n
                  Q(j,i) = A(j,i)/R(i,i)
           enddo
           do j = i+1, m
                  R(i,j)=DOT_PRODUCT(Q(1:n,i), A(1:n,j))
                  do k = 1, n
                         A(k,j) = A(k,j)-Q(k,i)*R(i,j)
                  enddo
           enddo
        enddo   

        end subroutine QR_GS

       
        !Subroutine for solving linear equations Ax=b => QRx=b by QR back-substitution.

        subroutine QR_BACK(Q, R, n, m, b, x)
        implicit none

        integer, parameter :: dbl=SELECTED_REAL_KIND(14)
        real(KIND=dbl) :: Q(n,m), R(m,m), x(m), b(n), Q_T(m,n), b_ny(m)
        !++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
       
        !++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
        integer :: n, m, i, k

        Q_T = TRANSPOSE(Q)
        b_ny = MATMUL(Q_T, b)

        x = 0.0

        do i = m, 1, -1
           do k = i+1, m
                  x(i) = x(i) - R(i,k)*x(k)
           enddo
           x(i) = (x(i) + b_ny(i))/R(i,i)
        enddo

        end subroutine QR_BACK

        !Subroutine for calculating the determinant of a matrix by QR decomposition.

        subroutine QR_DETERMINANT(R, m, DET)
        implicit none

         integer, parameter :: dbl=SELECTED_REAL_KIND(14)
         real(KIND=dbl) ::  R(m,m), DET
         integer :: m, i

        DET = 1

        do i = 1, m
        DET = DET*R(i,i)
        enddo

        end subroutine QR_DETERMINANT
È´±¨ÈçÏ´íÎó£º
:\Fortran_App\QR_Decomposition\LIN_EQ.f90(78) : Error: This name does not have a type, and must have an explicit type.   [N]
real(KIND=dbl) :: Q(n,m), R(m,m), x(m), b(n), Q_T(m,n), b_ny(m)
----------------------------^
D:\Fortran_App\QR_Decomposition\LIN_EQ.f90(78) : Error: A specification expression is invalid.   [N]
real(KIND=dbl) :: Q(n,m), R(m,m), x(m), b(n), Q_T(m,n), b_ny(m)
----------------------------^
D:\Fortran_App\QR_Decomposition\LIN_EQ.f90(78) : Error: This name does not have a type, and must have an explicit type.   [M]
real(KIND=dbl) :: Q(n,m), R(m,m), x(m), b(n), Q_T(m,n), b_ny(m)
------------------------------^
D:\Fortran_App\QR_Decomposition\LIN_EQ.f90(78) : Error: A specification expression is invalid.   [M]
real(KIND=dbl) :: Q(n,m), R(m,m), x(m), b(n), Q_T(m,n), b_ny(m)
------------------------------^
D:\Fortran_App\QR_Decomposition\LIN_EQ.f90(78) : Error: A specification expression is invalid.   [M]
real(KIND=dbl) :: Q(n,m), R(m,m), x(m), b(n), Q_T(m,n), b_ny(m)
------------------------------------^
D:\Fortran_App\QR_Decomposition\LIN_EQ.f90(78) : Error: A specification expression is invalid.   [M]
real(KIND=dbl) :: Q(n,m), R(m,m), x(m), b(n), Q_T(m,n), b_ny(m)
--------------------------------------^
D:\Fortran_App\QR_Decomposition\LIN_EQ.f90(78) : Error: A specification expression is invalid.   [M]
real(KIND=dbl) :: Q(n,m), R(m,m), x(m), b(n), Q_T(m,n), b_ny(m)
--------------------------------------------^
D:\Fortran_App\QR_Decomposition\LIN_EQ.f90(78) : Error: A specification expression is invalid.   [N]
real(KIND=dbl) :: Q(n,m), R(m,m), x(m), b(n), Q_T(m,n), b_ny(m)
--------------------------------------------------^
D:\Fortran_App\QR_Decomposition\LIN_EQ.f90(78) : Error: A specification expression is invalid.   [M]
real(KIND=dbl) :: Q(n,m), R(m,m), x(m), b(n), Q_T(m,n), b_ny(m)
----------------------------------------------------------^
D:\Fortran_App\QR_Decomposition\LIN_EQ.f90(78) : Error: A specification expression is invalid.   [N]
real(KIND=dbl) :: Q(n,m), R(m,m), x(m), b(n), Q_T(m,n), b_ny(m)
------------------------------------------------------------^
D:\Fortran_App\QR_Decomposition\LIN_EQ.f90(78) : Error: A specification expression is invalid.   [M]
real(KIND=dbl) :: Q(n,m), R(m,m), x(m), b(n), Q_T(m,n), b_ny(m)
---------------------------------------------------------------------^
D:\Fortran_App\QR_Decomposition\LIN_EQ.f90(104) : Error: This name does not have a type, and must have an explicit type.   [M]
  real(KIND=dbl) ::  R(m,m), DET
------------------------------^
D:\Fortran_App\QR_Decomposition\LIN_EQ.f90(104) : Error: A specification expression is invalid.   [M]
  real(KIND=dbl) ::  R(m,m), DET
------------------------------^
D:\Fortran_App\QR_Decomposition\LIN_EQ.f90(104) : Error: A specification expression is invalid.   [M]
  real(KIND=dbl) ::  R(m,m), DET
--------------------------------^
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