```    Program f08kbfe

!     F08KBF Example Program Text

!     Mark 27.0 Release. NAG Copyright 2019.

!     .. Use Statements ..
Use nag_library, Only: dgesvd, nag_wp
!     .. Implicit None Statement ..
Implicit None
!     .. Parameters ..
Integer, Parameter               :: nb = 64, nin = 5, nout = 6,          &
prerr = 0
!     .. Local Scalars ..
Integer                          :: i, info, lda, ldu, ldvt, lwork, m, n
!     .. Local Arrays ..
Real (Kind=nag_wp), Allocatable  :: a(:,:), a_copy(:,:), b(:), s(:),     &
u(:,:), vt(:,:), work(:)
Real (Kind=nag_wp)               :: dummy(1,1)
!     .. Intrinsic Procedures ..
Intrinsic                        :: max, min, nint
!     .. Executable Statements ..
Write (nout,*) 'F08KBF Example Program Results'
Write (nout,*)
!     Skip heading in data file
lda = m
ldu = m
ldvt = n
Allocate (a(lda,n),a_copy(m,n),s(n),vt(ldvt,n),u(ldu,m),b(m))

!     Read the m by n matrix A from data file

!     Read the right hand side of the linear system

a_copy(1:m,1:n) = a(1:m,1:n)

!     Use routine workspace query to get optimal workspace.
lwork = -1
!     The NAG name equivalent of dgesvd is f08kbf
Call dgesvd('A','S',m,n,a,lda,s,u,ldu,vt,ldvt,dummy,lwork,info)

!     Make sure that there is enough workspace for block size nb.
lwork = max(m+4*n+nb*(m+n),nint(dummy(1,1)))
Allocate (work(lwork))

!     Compute the singular values and left and right singular vectors
!     of A.

!     The NAG name equivalent of dgesvd is f08kbf
Call dgesvd('A','S',m,n,a,lda,s,u,ldu,vt,ldvt,work,lwork,info)

If (info/=0) Then
Write (nout,99999) 'Failure in DGESVD. INFO =', info
99999   Format (1X,A,I4)
Go To 100
End If

!     Print the significant singular values of A

Write (nout,*) 'Singular values of A:'
Write (nout,99998) s(1:min(m,n))
99998 Format (1X,4(3X,F11.4))

If (prerr>0) Then
Call compute_error_bounds(m,n,s)
End If

If (m>n) Then
!       Compute V*Inv(S)*U^T * b to get least squares solution.
Call compute_least_squares(m,n,a_copy,m,u,ldu,vt,ldvt,s,b)
End If

100   Continue

Contains
Subroutine compute_least_squares(m,n,a,lda,u,ldu,vt,ldvt,s,b)

!       .. Use Statements ..
Use nag_library, Only: dgemv, dnrm2
!       .. Implicit None Statement ..
Implicit None
!       .. Scalar Arguments ..
Integer, Intent (In)           :: lda, ldu, ldvt, m, n
!       .. Array Arguments ..
Real (Kind=nag_wp), Intent (In) :: a(lda,n), s(n), u(ldu,m),           &
vt(ldvt,n)
Real (Kind=nag_wp), Intent (Inout) :: b(m)
!       .. Local Scalars ..
Real (Kind=nag_wp)             :: alpha, beta, norm
!       .. Local Arrays ..
Real (Kind=nag_wp), Allocatable :: x(:), y(:)
!       .. Intrinsic Procedures ..
Intrinsic                      :: allocated
!       .. Executable Statements ..
Allocate (x(n),y(n))

!       Compute V*Inv(S)*U^T * b to get least squares solution.

!       y = U^T b
!       The NAG name equivalent of dgemv is f06paf
alpha = 1._nag_wp
beta = 0._nag_wp
Call dgemv('T',m,n,alpha,u,ldu,b,1,beta,y,1)

y(1:n) = y(1:n)/s(1:n)

!       x = V y
Call dgemv('T',n,n,alpha,vt,ldvt,y,1,beta,x,1)

Write (nout,*)
Write (nout,*) 'Least squares solution:'
Write (nout,99999) x(1:n)

!       Find norm of residual ||b-Ax||.
alpha = -1._nag_wp
beta = 1._nag_wp
Call dgemv('N',m,n,alpha,a,lda,x,1,beta,b,1)

norm = dnrm2(m,b,1)

Write (nout,*)
Write (nout,*) 'Norm of Residual:'
Write (nout,99999) norm

If (allocated(x)) Then
Deallocate (x)
End If
If (allocated(y)) Then
Deallocate (y)
End If

99999   Format (1X,4(3X,F11.4))

End Subroutine compute_least_squares

Subroutine compute_error_bounds(m,n,s)

!       Error estimates for singular values and vectors is computed
!       and printed here.

!       .. Use Statements ..
Use nag_library, Only: ddisna, nag_wp, x02ajf
!       .. Implicit None Statement ..
Implicit None
!       .. Scalar Arguments ..
Integer, Intent (In)           :: m, n
!       .. Array Arguments ..
Real (Kind=nag_wp), Intent (In) :: s(n)
!       .. Local Scalars ..
Real (Kind=nag_wp)             :: eps, serrbd
Integer                        :: i, info
!       .. Local Arrays ..
Real (Kind=nag_wp), Allocatable :: rcondu(:), rcondv(:), uerrbd(:),    &
verrbd(:)
!       .. Executable Statements ..
Allocate (rcondu(n),rcondv(n),uerrbd(n),verrbd(n))

!       Get the machine precision, EPS and compute the approximate
!       error bound for the computed singular values.  Note that for
!       the 2-norm, S(1) = norm(A)

eps = x02ajf()
serrbd = eps*s(1)

!       Call DDISNA (F08FLF) to estimate reciprocal condition
!       numbers for the singular vectors

Call ddisna('Left',m,n,s,rcondu,info)
Call ddisna('Right',m,n,s,rcondv,info)

!       Compute the error estimates for the singular vectors

Do i = 1, n
uerrbd(i) = serrbd/rcondu(i)
verrbd(i) = serrbd/rcondv(i)
End Do

!       Print the approximate error bounds for the singular values
!       and vectors

Write (nout,*)
Write (nout,*) 'Error estimate for the singular values'
Write (nout,99999) serrbd
Write (nout,*)
Write (nout,*) 'Error estimates for the left singular vectors'
Write (nout,99999) uerrbd(1:n)
Write (nout,*)
Write (nout,*) 'Error estimates for the right singular vectors'
Write (nout,99999) verrbd(1:n)

99999   Format (4X,1P,6E11.1)

End Subroutine compute_error_bounds

End Program f08kbfe
```