```    Program f08bbfe

!     F08BBF Example Program Text

!     Mark 26 Release. NAG Copyright 2016.

!     .. Use Statements ..
Use nag_library, Only: dgemqrt, dgeqrt, dnrm2, dtpmqrt, dtpqrt, dtrtrs,  &
nag_wp, x04caf
!     .. Implicit None Statement ..
Implicit None
!     .. Parameters ..
Integer, Parameter               :: nbmax = 64, nin = 5, nout = 6
!     .. Local Scalars ..
Integer                          :: i, ifail, info, j, lda, ldb, ldt,    &
lwork, m, n, nb, nrhs
!     .. Local Arrays ..
Real (Kind=nag_wp), Allocatable  :: a(:,:), b(:,:), c(:,:), rnorm(:),    &
t(:,:), work(:)
!     .. Intrinsic Procedures ..
Intrinsic                        :: max, min
!     .. Executable Statements ..
Write (nout,*) 'F08BBF Example Program Results'
Write (nout,*)
Flush (nout)
!     Skip heading in data file
lda = m
ldb = m
nb = min(m,n,nbmax)
ldt = nb
lwork = nb*max(n,m)
Allocate (a(lda,n),b(ldb,nrhs),c(ldb,nrhs),rnorm(nrhs),t(ldt,min(m,      &
n)),work(lwork))

!     Read A and B from data file

c(1:m,1:nrhs) = b(1:m,1:nrhs)
!     Compute the QR factorization of first n rows of A
!     The NAG name equivalent of dgeqrt is f08abf
Call dgeqrt(n,n,nb,a,lda,t,ldt,work,info)

!     Compute C = (C1) = (Q**T)*B, storing the result in C
!                  (C2)
!     The NAG name equivalent of dgemqrt is f08acf
Call dgemqrt('Left','Transpose',n,nrhs,n,nb,a,lda,t,ldt,c,ldb,work,info)

b(1:n,1:nrhs) = c(1:n,1:nrhs)
!     Compute least squares solutions for first n rows by back-substitution in
!     R*X = C1
!     The NAG name equivalent of dtrtrs is f07tef
Call dtrtrs('Upper','No transpose','Non-Unit',n,nrhs,a,lda,c,ldb,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

!       Print solution using first n rows

!       ifail: behaviour on error exit
!              =0 for hard exit, =1 for quiet-soft, =-1 for noisy-soft
ifail = 0
Call x04caf('General',' ',n,nrhs,c,ldb,'solution(s) for n rows',ifail)

End If

!     Now add the remaining rows and perform QR update
!     The NAG name equivalent of dtpqrt is f08bbf
Call dtpqrt(m-n,n,0,nb,a,lda,a(n+1,1),lda,t,ldt,work,info)

!     Apply orthogonal transformations to C
!     The NAG name equivalent of dtpmqrt is f08bcf
Call dtpmqrt('Left','Transpose',m-n,nrhs,n,0,nb,a(n+1,1),lda,t,ldt,b,    &
ldb,b(5,1),ldb,work,info)
!     Compute least squares solutions for first n rows by bac-substitution in
!     R*X = C1
!     The NAG name equivalent of dtrtrs is f07tef
Call dtrtrs('Upper','No transpose','Non-Unit',n,nrhs,a,lda,b,ldb,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

!       Print least squares solutions
Write (nout,*)
ifail = 0
Call x04caf('G',' ',n,nrhs,b,ldb,                                      &
'Least squares solution(s) for all rows',ifail)

!       Compute and print estimates of the square roots of the residual
!       sums of squares

!       The NAG name equivalent of dnrm2 is f06ejf
Do j = 1, nrhs
rnorm(j) = dnrm2(m-n,b(n+1,j),1)
End Do

Write (nout,*)
Write (nout,*) 'Square root(s) of the residual sum(s) of squares'
Write (nout,99999) rnorm(1:nrhs)
End If

99999 Format (5X,1P,7E11.2)
End Program f08bbfe
```