NAG Library Manual, Mark 27.3
```    Program f08bffe

!     F08BFF Example Program Text

!     Mark 27.3 Release. NAG Copyright 2021.

!     .. Use Statements ..
Use nag_library, Only: dgeqp3, dnrm2, dormqr, dtrsm, nag_wp, x04caf
!     .. 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               :: inc1 = 1, nb = 64, nin = 5, nout = 6
!     .. Local Scalars ..
Real (Kind=nag_wp)               :: tol
Integer                          :: i, ifail, info, j, k, lda, ldb,      &
lwork, m, n, nrhs
!     .. Local Arrays ..
Real (Kind=nag_wp), Allocatable  :: a(:,:), b(:,:), rnorm(:), tau(:),    &
work(:)
Integer, Allocatable             :: jpvt(:)
!     .. Intrinsic Procedures ..
Intrinsic                        :: abs
!     .. Executable Statements ..
Write (nout,*) 'F08BFF Example Program Results'
Write (nout,*)
!     Skip heading in data file
lda = m
ldb = m
lwork = 2*n + (n+1)*nb
Allocate (a(lda,n),b(ldb,nrhs),rnorm(n),tau(n),work(lwork),jpvt(n))

!     Read A and B from data file

!     Initialize JPVT to be zero so that all columns are free

jpvt(1:n) = 0

!     Compute the QR factorization of A
!     The NAG name equivalent of dgeqp3 is f08bff
Call dgeqp3(m,n,a,lda,jpvt,tau,work,lwork,info)

!     Compute C = (C1) = (Q**T)*B, storing the result in B
!                  (C2)
!     The NAG name equivalent of dormqr is f08agf
Call dormqr('Left','Transpose',m,nrhs,n,a,lda,tau,b,ldb,work,lwork,info)

!     Choose TOL to reflect the relative accuracy of the input data

tol = 0.01_nag_wp

!     Determine and print the rank, K, of R relative to TOL

loop: Do k = 1, n
If (abs(a(k,k))<=tol*abs(a(1,1))) Then
Exit loop
End If
End Do loop
k = k - 1

Write (nout,*) 'Tolerance used to estimate the rank of A'
Write (nout,99999) tol
Write (nout,*) 'Estimated rank of A'
Write (nout,99998) k
Write (nout,*)
Flush (nout)

!     Compute least squares solutions by back-substitution in
!     R(1:K,1:K)*Y = C1, storing the result in B

Call dtrsm('Left','Upper','No transpose','Non-Unit',k,nrhs,one,a,lda,b,  &
ldb)

!     Compute estimates of the square roots of the residual sums of
!     squares (2-norm of each of the columns of C2)

!     The NAG name equivalent of dnrm2 is f06ejf
Do j = 1, nrhs
rnorm(j) = dnrm2(m-k,b(k+1,j),inc1)
End Do

!     Set the remaining elements of the solutions to zero (to give
!     the basic solutions)

b(k+1:n,1:nrhs) = zero

!     Permute the least squares solutions stored in B to give X = P*Y

Do j = 1, nrhs
work(jpvt(1:n)) = b(1:n,j)
b(1:n,j) = work(1:n)
End Do

!     Print least squares solutions

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

!     Print the square roots of the residual sums of squares

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

99999 Format (5X,1P,6E11.2)
99998 Format (1X,I8)
End Program f08bffe
```