Newton-Raphson Method Using Fortran 90 - fortran

Question

I developed the code below for finding the roots of the given polynomial. It works fine, but I need to adapt it to find all roots and not simply stop when it converges. How do I go about doing this? I thought about creating an outer do loop for values of x, but I'm uncertain whether this is the right approach. Any help is appreciated, thanks in advance.
PROGRAM nr
integer :: i
real :: x, f, df
write(*,*) "x=?"
read (*,*) x
write (*,*) '# Initial value: x=',x
do i=1,100
f= x**4 - 26*(x**3) + 131*(x**2) - 226*x + 120
df = 4*(x**3) - 3.0*26*(x**2) + 2.0*131*x - 226
write (*,*) i,x,f,df
x = x-f/df
end do
write (*,*) '#x = ',x
END PROGRAM

Answer 1

A possible algorithm to find all roots of the polynomial P consists in:
Start from some X0 and find a root R, using Newton's algorithm.
Divide P by (X-R): the division is exact (up to numerical error) since R is a root. (this step is called deflation)
Restart from the beginning if the quotient has degree > 1.
There are some subtleties:
If your polynomial has real coefficients and X0 is real, you will only find a real root, if there is any. To find complex roots, you then have to start from a complex X0, and of course use complex arithmetic.
Not all values of X0 will guarantee convergence, as Newton-Raphson is a local method. See Newton-Kantorovitch theorem and basins of attraction of the Newton method.
If there are multiple roots, convergence there is much slower. There are ways to adapt Newton's method to deal with this.
Numerical errors introduced in the deflation step add up and usually leads to poor accuracy of the subsequent roots. This is especially a problem for polynomials of high degree (how much high really depends). In extreme cases, the computed roots can be quite far from the exact roots. Furthermore, some polynomials are more "difficult" than others: see for instance Wilkinson's polynomial.
There are ways to find information on the roots (bound on the absolute values, circles enclosing the roots...), this can help to pick a good X0 in the algorithm: see Geometrical properties of polynomial roots. If you are only looking for real roots of a real polynomials, you first find bounds, then use Sturm's theorem to find intervals enclosing the roots. Another approach in the real case is to first find the roots of the derivative P' (and to find its roots, first find roots of P'', etc.), as the roots of P' separate the roots of P (except for multiple roots).
Another suggested reading: Roots of polynomials. Note that there are much better algorithms than Newton+deflation. There are also algorithms designed to find all the roots.
However, if you are only interested in this specific polynomial, I suggest to first have a look at the problem, as it's much easier than the general case: WolframAlpha.
Here, starting with integer values of X0 is going to work quite well...

Answer 2

Anyway, I give you below what I think is a better Newton-Raphson code, using Fortran-90 function declaration and a "while" rather than a blind "do". Best, Fabio M. S. Lima
PROGRAM Newton2
! Newton-Raphson method in Fortran-90
implicit none
integer i,imax
real x,xnew,tol
parameter (imax = 30)
parameter (tol = 0.00001)
print*, '*** Initial value: '
read*,x
xnew = 0.5 !Just to begin the loop below
i=0
do while ((abs(xnew-x)>tol) .and. (i<imax))
i=i+1
x = xnew
xnew = x -f(x)/f1(x)
print*, i,xnew,f(xnew)
end do
if (i>imax-2) then
print*,'# Error: Newton-Raphson has not converged after ',i,' iterations!'
else
print*,'* After ',i,' iterations, we found a root at ',xnew
endif
contains
function f(z)
real z,f
f = z**4 -26.0*(z**3) +131.0*(z**2) -226.0*z +120.0
end function f
function f1(z)
real z,f1
f1 = 4.0*(z**3) -3.0*26.0*(z**2) +2.0*131.0*z -226.0
end function f1
END PROGRAM Newton2