```    Program f08zffe

!     F08ZFF Example Program Text

!     Mark 27.1 Release. NAG Copyright 2020.

!     .. Use Statements ..
Use nag_library, Only: dgemv, dggrqf, dnrm2, dormqr, dormrq, dtrmv,      &
dtrtrs, nag_wp
!     .. Implicit None Statement ..
Implicit None
!     .. Parameters ..
Real (Kind=nag_wp), Parameter    :: one = 1.0E0_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(:,:), c(:), d(:), taua(:), &
taub(:), work(:), x(:)
!     .. Intrinsic Procedures ..
Intrinsic                        :: min
!     .. Executable Statements ..
Write (nout,*) 'F08ZFF Example Program Results'
Write (nout,*)
!     Skip heading in data file
Read (nin,*)
Read (nin,*) m, n, p
lda = m
ldb = p
lwork = nb*(p+n)
Allocate (a(lda,n),b(ldb,n),c(m),d(p),taua(n),taub(n),work(lwork),x(n))

!     Read B, A, C and D from data file
Read (nin,*)(a(i,1:n),i=1,m)
Read (nin,*)(b(i,1:n),i=1,p)
Read (nin,*) c(1:m)
Read (nin,*) d(1:p)

!     Compute the generalized RQ factorization of (B,A) as
!     B = (0 T12)*Q,   A = Z*(R11 R12)*Q, where T12, R11
!                            ( 0  R22)
!     are upper triangular
!     The NAG name equivalent of dggrqf is f08zff
Call dggrqf(p,m,n,b,ldb,taub,a,lda,taua,work,lwork,info)

!     Set Qx = y. The problem then reduces to:
!                 minimize (Ry - Z^Tc) subject to Ty = d
!     Update c = Z^T*c -> minimize (Ry-c)
!     The NAG name equivalent of dormqr is f08agf
Call dormqr('Left','Transpose',m,1,min(m,n),a,lda,taua,c,m,work,lwork,   &
info)

!     Putting y = (y1), solve T12*w = d for w, storing result in d
!                 (w )
!     The NAG name equivalent of dtrtrs is f07tef
Call dtrtrs('Upper','No transpose','Non-unit',p,1,b(1,n-p+1),ldb,d,p,    &
info)

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

!     From first n-p rows of (Ry-c) we have: R11*y1 + R12*w = c(1:n-p) = c1
!     Form c1 = c1 - R12*w = R11*y1
!     The NAG name equivalent of dgemv is f06raf
Call dgemv('No transpose',n-p,p,-one,a(1,n-p+1),lda,d,1,one,c,1)

!     Solve R11*y1 = c1 for y1, storing result in c(1:n-p)
!     The NAG name equivalent of dtrtrs is f07tef
Call dtrtrs('Upper','No transpose','Non-unit',n-p,1,a,lda,c,n-p,info)

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

!     Copy y into X (first y1, then w)
x(1:n-p) = c(1:n-p)
x(n-p+1:n) = d(1:p)

!     Compute x = (Q**T)*y
!     The NAG name equivalent of dormrq is f08ckf
Call dormrq('Left','Transpose',n,1,p,b,ldb,taub,x,n,work,lwork,info)

!     The least squares solution is in x, the remainder here is to compute
!     the residual, which equals c2 - R22*w.

!     Upper triangular part of R22 first
!     The NAG name equivalent of dtrmv is f06pff
Call dtrmv('Upper','No transpose','Non-unit',min(m,n)-n+p,               &
a(n-p+1,n-p+1),lda,d,1)
Do i = 1, min(m,n) - n + p
c(n-p+i) = c(n-p+i) - d(i)
End Do

If (m<n) Then

!       Additional rectangular part of R22
!       The NAG name equivalent of dgemv is f06paf
Call dgemv('No transpose',m-n+p,n-m,-one,a(n-p+1,m+1),lda,d(m-n+p+1),  &
1,one,c(n-p+1),1)
End If

!     Compute norm of residual sum of squares.
rnorm = dnrm2(m-(n-p),c(n-p+1),1)

!     Print least squares solution x
Write (nout,*) 'Constrained least squares solution'
Write (nout,99999) x(1:n)

!     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

100   Continue

99999 Format (1X,7F11.4)
99998 Format (3X,1P,E11.2)
End Program f08zffe
```