# Try out NAG Library functions

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Function ID
C02AGF
Name
nagf_zeros_poly_real
Description
Zeros of a polynomial with real coefficients
Keywords
Laguerre's method | root-finding
For this routine two examples are presented. There is a single example program for c02agf, with a main program and the code to solve the two example problems given in the subroutines EX1 and EX2.
!   C02AGF Example Program Text
!   Mark 26.1 Release. NAG Copyright 2016.

Module c02agfe_mod

!     C02AGF Example Program Module:
!            Parameters

!     .. Implicit None Statement ..
Implicit None
!     .. Accessibility Statements ..
Private
!     .. Parameters ..
Integer, Parameter, Public       :: nin = 5, nout = 6
Logical, Parameter, Public       :: scal = .True.
End Module c02agfe_mod
Program c02agfe

!     C02AGF Example Main Program

!     .. Use Statements ..
Use c02agfe_mod, Only: nout
!     .. Implicit None Statement ..
Implicit None
!     .. Executable Statements ..
Write (nout,*) 'C02AGF Example Program Results'

Call ex1

Call ex2

Contains
Subroutine ex1

!       .. Use Statements ..
Use c02agfe_mod, Only: nin, scal
Use nag_library, Only: c02agf, nag_wp
!       .. Local Scalars ..
Real (Kind=nag_wp)             :: zi, zr
Integer                        :: i, ifail, n, nroot
!       .. Local Arrays ..
Real (Kind=nag_wp), Allocatable :: a(:), w(:), z(:,:)
!       .. Intrinsic Procedures ..
Intrinsic                      :: abs
!       .. Executable Statements ..
Write (nout,*)
Write (nout,*)
Write (nout,*) 'Example 1'

!       Skip heading in data file

Allocate (a(0:n),w(2*(n+1)),z(2,n))

Write (nout,*)
Write (nout,99999) 'Degree of polynomial = ', n

ifail = 0
Call c02agf(a,n,scal,z,w,ifail)

Write (nout,99998) 'Computed roots of polynomial'

nroot = 1

Do While (nroot<=n)

zr = z(1,nroot)
zi = z(2,nroot)
If (zi==0.0E0_nag_wp) Then
Write (nout,99997) 'z = ', zr
nroot = nroot + 1
Else
Write (nout,99997) 'z = ', zr, ' +/- ', abs(zi), '*i'
nroot = nroot + 2
End If

End Do

99999   Format (/,1X,A,I4)
99998   Format (/,1X,A,/)
99997   Format (1X,A,1P,E12.4,A,E12.4,A)
End Subroutine ex1
Subroutine ex2

!       .. Use Statements ..
Use c02agfe_mod, Only: nin, scal
Use nag_library, Only: a02abf, c02agf, nag_wp, x02ajf, x02alf
!       .. Local Scalars ..
Real (Kind=nag_wp)             :: deltac, deltai, di, eps, epsbar, f,  &
r1, r2, r3, rmax
Integer                        :: i, ifail, j, jmin, n
!       .. Local Arrays ..
Real (Kind=nag_wp), Allocatable :: a(:), abar(:), r(:), w(:), z(:,:),  &
zbar(:,:)
Integer, Allocatable           :: m(:)
!       .. Intrinsic Procedures ..
Intrinsic                      :: abs, max, min
!       .. Executable Statements ..
Write (nout,*)
Write (nout,*)
Write (nout,*) 'Example 2'

!       Skip heading in data file

Allocate (a(0:n),abar(0:n),r(n),w(2*(n+1)),z(2,n),zbar(2,n),m(n))

!       Read in the coefficients of the original polynomial.

!       Compute the roots of the original polynomial.

ifail = 0
Call c02agf(a,n,scal,z,w,ifail)

!       Form the coefficients of the perturbed polynomial.

eps = x02ajf()
epsbar = 3.0_nag_wp*eps

Do i = 0, n

If (a(i)/=0.0_nag_wp) Then
f = 1.0_nag_wp + epsbar
epsbar = -epsbar
abar(i) = f*a(i)
Else
abar(i) = 0.0E0_nag_wp
End If

End Do

!       Compute the roots of the perturbed polynomial.

ifail = 0
Call c02agf(abar,n,scal,zbar,w,ifail)

!       Perform error analysis.

!       Initialize markers to 0 (unmarked).

m(1:n) = 0

rmax = x02alf()

!       Loop over all unperturbed roots (stored in Z).

Do i = 1, n
deltai = rmax
r1 = a02abf(z(1,i),z(2,i))

!         Loop over all perturbed roots (stored in ZBAR).

Do j = 1, n

!           Compare the current unperturbed root to all unmarked
!           perturbed roots.

If (m(j)==0) Then
r2 = a02abf(zbar(1,j),zbar(2,j))
deltac = abs(r1-r2)

If (deltac<deltai) Then
deltai = deltac
jmin = j
End If

End If

End Do

!         Mark the selected perturbed root.

m(jmin) = 1

!         Compute the relative error.

If (r1/=0.0E0_nag_wp) Then
r3 = a02abf(zbar(1,jmin),zbar(2,jmin))
di = min(r1,r3)
r(i) = max(deltai/max(di,deltai/rmax),eps)
Else
r(i) = 0.0_nag_wp
End If

End Do

Write (nout,*)
Write (nout,99999) 'Degree of polynomial = ', n
Write (nout,*)
Write (nout,*) 'Computed roots of polynomial   ', '  Error estimates'
Write (nout,*) '                            ',                         &
'   (machine-dependent)'
Write (nout,*)

Do i = 1, n
Write (nout,99998) 'z = ', z(1,i), z(2,i), '*i', r(i)
End Do

99999   Format (1X,A,I4)
99998   Format (1X,A,1P,E12.4,Sp,E12.4,A,5X,Ss,E9.1)
End Subroutine ex2
End Program c02agfe

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