```
Program f08btfe
! F08BTF Example Program Text
! Mark 25 Release. NAG Copyright 2014.
! .. Use Statements ..
Use nag_library, Only: dznrm2, nag_wp, x04dbf, zgeqp3, ztrsm, zunmqr
! .. Implicit None Statement ..
Implicit None
! .. Parameters ..
Complex (Kind=nag_wp), Parameter :: one = (1.0E0_nag_wp,0.0E0_nag_wp)
Complex (Kind=nag_wp), Parameter :: zero = (0.0E0_nag_wp,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 ..
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,*) 'F08BTF Example Program Results'
Write (nout,*)
! Skip heading in data file
Read (nin,*)
Read (nin,*) m, n, nrhs
lda = m
ldb = m
lwork = (n+1)*nb
Allocate (a(lda,n),b(ldb,nrhs),tau(n),work(lwork),rnorm(nrhs), &
rwork(2*n),jpvt(n))
! Read A and B from data file
Read (nin,*)(a(i,1:n),i=1,m)
Read (nin,*)(b(i,1:nrhs),i=1,m)
! 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 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))) Exit loop
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 backsubstitution in
! R(1:K,1:K)*Y = 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 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 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,I8)
End Program f08btfe
```