```    Program f08zefe

!     F08ZEF Example Program Text

!     Mark 26.1 Release. NAG Copyright 2017.

!     .. Use Statements ..
Use nag_library, Only: dgemv, dggqrf, dnrm2, dormqr, dormrq, dtrtrs,     &
nag_wp
!     .. Implicit None Statement ..
Implicit None
!     .. Parameters ..
Real (Kind=nag_wp), Parameter    :: one = 1.0E0_nag_wp
Real (Kind=nag_wp), Parameter    :: zero = 0.0E0_nag_wp
Integer, Parameter               :: nb = 64, nin = 5, nout = 6
!     .. Local Scalars ..
Real (Kind=nag_wp)               :: rnorm
Integer                          :: i, info, lda, ldb, lwork, m, n, p
!     .. Local Arrays ..
Real (Kind=nag_wp), Allocatable  :: a(:,:), b(:,:), d(:), taua(:),       &
taub(:), work(:), y(:)
!     .. Intrinsic Procedures ..
Intrinsic                        :: max, min
!     .. Executable Statements ..
Write (nout,*) 'F08ZEF Example Program Results'
Write (nout,*)
!     Skip heading in data file
lda = n
ldb = n
lwork = nb*(m+p)
Allocate (a(lda,m),b(ldb,p),d(n),taua(m),taub(m+p),work(lwork),y(p))

!     Read A, B and D from data file

!     Compute the generalized QR factorization of (A,B) as
!     A = Q*(R),   B = Q*(T11 T12)*Z
!            (0)          ( 0  T22)
!     The NAG name equivalent of dggqrf is f08zef
Call dggqrf(n,m,p,a,lda,taua,b,ldb,taub,work,lwork,info)

!     Compute c = (c1) = (Q**T)*d, storing the result in D
!                  (c2)
!     The NAG name equivalent of dormqr is f08agf
Call dormqr('Left','Transpose',n,1,m,a,lda,taua,d,n,work,lwork,info)

!     Putting Z*y = w = (w1), set w1 = 0, storing the result in Y1
!                        (w2)
y(1:m+p-n) = zero

If (n>m) Then

!       Copy c2 into Y2
y(m+p-n+1:p) = d(m+1:n)

!       Solve T22*w2 = c2 for w2, storing result in Y2
!       The NAG name equivalent of dtrtrs is f07tef
Call dtrtrs('Upper','No transpose','Non-unit',n-m,1,b(m+1,m+p-n+1),    &
ldb,y(m+p-n+1),n-m,info)

If (info>0) Then
Write (nout,*)                                                       &
'The upper triangular factor, T22, of B is singular, '
Write (nout,*) 'the least squares solution could not be computed'
Go To 100
End If

!       Compute estimate of the square root of the residual sum of squares
!       norm(y) = norm(w2)

!       The NAG name equivalent of dnrm2 is f06ejf
rnorm = dnrm2(n-m,y(m+p-n+1),1)

!       Form c1 - T12*w2 in D
!       The NAG name equivalent of dgemv is f06paf
Call dgemv('No transpose',m,n-m,-one,b(1,m+p-n+1),ldb,y(m+p-n+1),1,    &
one,d,1)
End If

!     Solve R*x = c1 - T12*w2 for x
!     The NAG name equivalent of dtrtrs is f07tef
Call dtrtrs('Upper','No transpose','Non-unit',m,1,a,lda,d,m,info)

If (info>0) Then
Write (nout,*) 'The upper triangular factor, R, of A is singular, '
Write (nout,*) 'the least squares solution could not be computed'
Else

!       Compute y = (Z**T)*w
!       The NAG name equivalent of dormrq is f08ckf
Call dormrq('Left','Transpose',p,1,min(n,p),b(max(1,                   &
n-p+1),1),ldb,taub,y,p,work,lwork,info)

!       Print least squares solution x

Write (nout,*) 'Generalized least squares solution'
Write (nout,99999) d(1:m)

!       Print residual vector y

Write (nout,*)
Write (nout,*) 'Residual vector'
Write (nout,99998) y(1:p)

!       Print estimate of the square root of the residual sum of squares

Write (nout,*)
Write (nout,*) 'Square root of the residual sum of squares'
Write (nout,99998) rnorm
End If
100   Continue

99999 Format (1X,7F11.4)
99998 Format (3X,1P,7E11.2)
End Program f08zefe
```