NAG Library Manual, Mark 28.7
```    Program f08bvfe

!     F08BVF Example Program Text

!     Mark 28.7 Release. NAG Copyright 2022.

!     .. Use Statements ..
Use nag_library, Only: dznrm2, nag_wp, x04dbf, zgeqp3, ztrsm, ztzrzf,    &
zunmqr, zunmrz
!     .. Implicit None Statement ..
Implicit None
!     .. Parameters ..
Complex (Kind=nag_wp), Parameter :: one = (1.0_nag_wp,0.0_nag_wp)
Complex (Kind=nag_wp), Parameter :: zero = (0.0_nag_wp,0.0_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 ..
Complex (Kind=nag_wp), Allocatable :: a(:,:), b(:,:), tau(:), work(:)
Real (Kind=nag_wp), Allocatable  :: rnorm(:), rwork(:)
Integer, Allocatable             :: jpvt(:)
Character (1)                    :: clabs(1), rlabs(1)
!     .. Intrinsic Procedures ..
Intrinsic                        :: abs
!     .. Executable Statements ..
Write (nout,*) 'F08BVF Example Program Results'
Write (nout,*)
!     Skip heading in data file
lda = m
ldb = m
lwork = (n+1)*nb
Allocate (a(lda,n),b(ldb,nrhs),tau(n),work(lwork),rnorm(n),rwork(2*n),   &
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 with column pivoting as
!     A = Q*(R11 R12)*(P**T)
!            ( 0  R22)
!     The NAG name equivalent of zgeqp3 is f08btf
Call zgeqp3(m,n,a,lda,jpvt,tau,work,lwork,rwork,info)

!     Compute C = (C1) = (Q**H)*B, storing the result in B
!                  (C2)
!     The NAG name equivalent of zunmqr is f08auf
Call zunmqr('Left','Conjugate 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 the RZ factorization of the K by K part of R as
!     (R1 R2) = (T 0)*Z
!     The NAG name equivalent of ztzrzf is f08bvf
Call ztzrzf(k,n,a,lda,tau,work,lwork,info)

!     Compute least squares solutions of triangular problems by
!     back substitution in T*Y1 = C1, storing the result in B
!     The NAG name equivalent of ztrsm is f06zjf
Call ztrsm('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 dznrm2 is f06jjf
Do j = 1, nrhs
rnorm(j) = dznrm2(m-k,b(k+1,j),inc1)
End Do

!     Set the remaining elements of the solutions to zero (to give
!     the minimum-norm solutions), Y2 = 0

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

!     Form W = (Z**H)*Y
!     The NAG name equivalent of zunmrz is f08bxf
Call zunmrz('Left','Conjugate transpose',n,nrhs,k,n-k,a,lda,tau,b,ldb,   &
work,lwork,info)

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

Do j = 1, nrhs
Do i = 1, n
work(jpvt(i)) = b(i,j)
End Do
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 x04dbf('General',' ',n,nrhs,b,ldb,'Bracketed','F7.4',               &
'Least squares solution(s)','Integer',rlabs,'Integer',clabs,80,0,      &
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 (3X,1P,7E11.2)
99998 Format (1X,I6)
End Program f08bvfe
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