NAG Library Manual, Mark 28.5
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```    Program f08bhfe

!     F08BHF Example Program Text

!     Mark 28.5 Release. NAG Copyright 2022.

!     .. Use Statements ..
Use nag_library, Only: dgeqp3, dnrm2, dormqr, dormrz, dtrsm, dtzrzf,     &
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,*) 'F08BHF Example Program Results'
Write (nout,*)
!     Skip heading in data file
Read (nin,*) m, n, nrhs
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 with column pivoting as
!     A = Q*(R11 R12)*(P**T)
!            ( 0  R22)

!     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 the RZ factorization of the K by K part of R as
!     (R11 R12) = (T 0)*Z
!     The NAG name equivalent of dtzrzf is f08bhf
Call dtzrzf(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 dtrsm is f06yjf
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 minimum-norm solutions), Y2 = 0

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

!     Form W = (Z**T)*Y

!     The NAG name equivalent of dormrz is f08bkf
Call dormrz('Left','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
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 f08bhfe
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