NAG Library Manual, Mark 28.7
```!   F12FBF Example Program Text
!   Mark 28.7 Release. NAG Copyright 2022.

Module f12fbfe_mod

!     F12FBF Example Program Module:
!            Parameters and User-defined Routines

!     .. Use Statements ..
Use nag_library, Only: nag_wp
!     .. Implicit None Statement ..
Implicit None
!     .. Accessibility Statements ..
Private
Public                           :: atv, av
!     .. Parameters ..
Real (Kind=nag_wp), Parameter, Public :: one = 1.0_nag_wp
Real (Kind=nag_wp), Parameter, Public :: two = 2.0_nag_wp
Real (Kind=nag_wp), Parameter, Public :: zero = 0.0_nag_wp
Integer, Parameter, Public       :: imon = 0, licomm = 140, nin = 5,     &
nout = 6
Contains
Subroutine av(m,n,x,w)

!       Computes  w <- A*x.

!       .. Scalar Arguments ..
Integer, Intent (In)           :: m, n
!       .. Array Arguments ..
Real (Kind=nag_wp), Intent (Out) :: w(m)
Real (Kind=nag_wp), Intent (In) :: x(n)
!       .. Local Scalars ..
Real (Kind=nag_wp)             :: h, k, s, t
Integer                        :: i, j
!       .. Intrinsic Procedures ..
Intrinsic                      :: real
!       .. Executable Statements ..
h = one/real(m+1,kind=nag_wp)
k = one/real(n+1,kind=nag_wp)
w(1:m) = zero
t = zero

Do j = 1, n
t = t + k
s = zero
Do i = 1, j
s = s + h
w(i) = w(i) + k*s*(t-one)*x(j)
End Do
Do i = j + 1, m
s = s + h
w(i) = w(i) + k*t*(s-one)*x(j)
End Do
End Do

Return
End Subroutine av

Subroutine atv(m,n,w,y)

!       Computes  y <- A'*w.

!       .. Scalar Arguments ..
Integer, Intent (In)           :: m, n
!       .. Array Arguments ..
Real (Kind=nag_wp), Intent (In) :: w(m)
Real (Kind=nag_wp), Intent (Out) :: y(n)
!       .. Local Scalars ..
Real (Kind=nag_wp)             :: h, k, s, t
Integer                        :: i, j
!       .. Intrinsic Procedures ..
Intrinsic                      :: real
!       .. Executable Statements ..
h = one/real(m+1,kind=nag_wp)
k = one/real(n+1,kind=nag_wp)
y(1:n) = zero
t = zero

Do j = 1, n
t = t + k
s = zero
Do i = 1, j
s = s + h
y(j) = y(j) + k*s*(t-one)*w(i)
End Do
Do i = j + 1, m
s = s + h
y(j) = y(j) + k*t*(s-one)*w(i)
End Do
End Do

Return
End Subroutine atv
End Module f12fbfe_mod
Program f12fbfe

!     F12FBF Example Main Program

!     .. Use Statements ..
Use f12fbfe_mod, Only: nout
Use nag_library, Only: nag_wp
!     .. Implicit None Statement ..
Implicit None
!     .. Executable Statements ..
Write (nout,*) 'F12FBF Example Program Results'

Call ex1

Call ex2

Contains
Subroutine ex1

!       .. Use Statements ..
Use f12fbfe_mod, Only: imon, licomm, nin, one, two, zero
Use nag_library, Only: dgttrf, dgttrs, dnrm2, f12faf, f12fbf, f12fcf,  &
f12fdf, f12fef
!       .. Local Scalars ..
Real (Kind=nag_wp)             :: h2, sigma
Integer                        :: ifail, info, irevcm, j, lcomm, ldv,  &
n, nconv, ncv, nev, niter, nshift
!       .. Local Arrays ..
comm(:), d(:,:), mx(:), resid(:),    &
v(:,:), x(:)
Integer                        :: icomm(licomm)
Integer, Allocatable           :: ipiv(:)
!       .. Intrinsic Procedures ..
Intrinsic                      :: real
!       .. Executable Statements ..
Write (nout,*)
Write (nout,*) 'F12FBF Example 1'
Write (nout,*)
Flush (nout)
!       Skip heading in data file

ldv = n
lcomm = 3*n + ncv*ncv + 8*ncv + 60
resid(n),v(ldv,ncv),x(n),ipiv(n))

ifail = 0
Call f12faf(n,nev,ncv,icomm,licomm,comm,lcomm,ifail)

!       Set the region of the spectrum that is required.
Call f12fdf('LARGEST MAGNITUDE',icomm,comm,ifail)
!       Use the Shifted Inverse mode.
Call f12fdf('SHIFTED INVERSE',icomm,comm,ifail)

h2 = one/real((n+1)*(n+1),kind=nag_wp)
sigma = zero
!       The NAG name equivalent of dgttrf is f07cdf

irevcm = 0
ifail = -1
revcm:  Do
Call f12fbf(irevcm,resid,v,ldv,x,mx,nshift,comm,icomm,ifail)
If (irevcm==5) Then
Exit revcm
Else If (irevcm==-1 .Or. irevcm==1) Then
!           Perform matrix vector multiplication
!           y <--- inv[A-sigma*I]*x
!           The NAG name equivalent of dgttrs is f07cef
Else If (irevcm==4 .And. imon/=0) Then
!           Output monitoring information
Call f12fef(niter,nconv,d,d(1,2),icomm,comm)
!           The NAG name equivalent of dnrm2 is f06ejf
Write (6,99999) niter, nconv, dnrm2(nev,d(1,2),1)
Flush (nout)
End If
End Do revcm

If (ifail==0) Then
!         Post-Process using F12FCF to compute eigenvalues/vectors.
Call f12fcf(nconv,d,v,ldv,sigma,resid,v,ldv,comm,icomm,ifail)
Write (nout,99998) nconv, sigma
Write (nout,99997)(j,d(j,1),j=1,nconv)
End If

Return

99999   Format (1X,'Iteration',1X,I3,', No. converged =',1X,I3,', norm o',     &
'f estimates =',E16.8)
99998   Format (1X,/,' The ',I4,' Ritz values of closest to ',F8.4,' are:',/)
99997   Format (1X,I8,5X,F12.4)
End Subroutine ex1
Subroutine ex2

!       .. Use Statements ..
Use f12fbfe_mod, Only: atv, av, licomm, nin, one
Use nag_library, Only: daxpy, dnrm2, dscal, f12faf, f12fbf, f12fcf,    &
nag_wp, x04caf
!       .. Local Scalars ..
Real (Kind=nag_wp)             :: sigma, temp
Integer                        :: ifail, ifail1, irevcm, j, lcomm,     &
ldu, ldv, m, n, nconv, ncv, nev,     &
nshift
!       .. Local Arrays ..
Real (Kind=nag_wp), Allocatable :: ax(:), comm(:), d(:,:), mx(:),      &
resid(:), u(:,:), v(:,:), x(:)
Integer                        :: icomm(licomm)
!       .. Intrinsic Procedures ..
Intrinsic                      :: sqrt
!       .. Executable Statements ..
Write (nout,*)
Write (nout,*) 'F12FBF Example 2'
Write (nout,*)
Flush (nout)
!       Skip heading in data file
Read (nin,*) m, n, nev, ncv

ldu = m
ldv = n
lcomm = 3*n + ncv*ncv + 8*ncv + 60
Allocate (ax(m),comm(lcomm),d(ncv,2),mx(m),resid(n),u(ldu,nev),        &
v(ldv,ncv),x(m))

!       Initialize for problem.
ifail = 0
Call f12faf(n,nev,ncv,icomm,licomm,comm,lcomm,ifail)

!       Main reverse communication loop.
irevcm = 0
ifail = -1
revcm:  Do
Call f12fbf(irevcm,resid,v,ldv,x,mx,nshift,comm,icomm,ifail)
If (irevcm==5) Then
Exit revcm
Else If (irevcm==-1 .Or. irevcm==1) Then
!           Perform the operation X <- A'AX
Call av(m,n,x,ax)
Call atv(m,n,ax,x)
End If
End Do revcm

If (ifail==0) Then
!         Post-Process using F12FCF.
!         Computed singular values may be extracted.
!         Singular vectors may also be computed now if desired.
ifail1 = 0
Call f12fcf(nconv,d,v,ldv,sigma,resid,v,ldv,comm,icomm,ifail1)

!         Singular values (squared) are returned in the first column
!         of D and the corresponding right singular vectors are
!         returned in the first NEV columns of V.

Do j = 1, nconv
d(j,1) = sqrt(d(j,1))

!           Compute the left singular vectors from the formula
!           u = Av/sigma/norm(Av).

Call av(m,n,v(1,j),ax)
u(1:m,j) = ax(1:m)
!           The NAG name equivalent of dnrm2 is f06ejf
temp = one/dnrm2(m,u(1,j),1)
!           The NAG name equivalent of dscal is f06edf
Call dscal(m,temp,u(1,j),1)

!           Compute the residual norm ||A*v - sigma*u|| for the nconv
!           accurately computed singular values and vectors.
!           Store the result in 2nd column of array D.

!           The NAG name equivalent of daxpy is f06ecf
Call daxpy(m,-d(j,1),u(1,j),1,ax,1)
d(j,2) = dnrm2(m,ax,1)

End Do

!         Print computed residuals

Call x04caf('G','N',nconv,2,d,ncv,                                   &
' Singular values and direct residuals',ifail1)
End If

Deallocate (ax,comm,d,mx,resid,u,v,x)

Return

End Subroutine ex2
End Program f12fbfe
```