newton.f

c      PROGRAM TO FIND A ROOT OF  f(x) = 0  USING VANILLA NEWTON's METHOD
         program newt
       implicit none
       real xold,tol
       real xnew
       real ff,ffprime,fold,fpold
       integer iter,itstart,itend
       external ff, ffprime

c      THE INITIAL GUESS FOR THE ROOT IS xold
c      FIND ROOT TO WITHIN A TOLERANCE  tol
c      THE FUNCTIONS f(x),f'(x) ARE DEFINED IN A SUBROUTINES GIVING ff,ffprime

c      DO A MAXIMUM OF itend ITERATIONS
       itstart=1
       itend=50

c      ENTER INITIAL GUESS, called xold, and TOL

c      IF |ff(xold)| < tol  THEN xold IS A GOOD APPROXIMATION OF THE ROOT

C      MAIN ITERATION LOOP
       do iter=itstart,itend

c        DEFINE f(xold) and f'(xold)

c        AVOID DIVIDING BY 0

c        GET THE NEW ITERATE xnew BY THE NEWTON FORMULA

c        CHECK IF ff(xnew) IS A ROOT

c        PUT xold INTO xnew

       enddo


999    stop
       end
c****6****************************************************************72
       real function ff(x)
c      THIS FUNCTION SUBROUTINE COMPUTES THE VALUE OF f AT A POINT x

c      DEFINE ff

       return
       end

c****6****************************************************************72
       real function ffprime(x)
c      THIS FUNCTION SUBROUTINE COMPUTES THE VALUE OF f' AT A POINT x

c      DEFINE ffprime

       return
       end

/subsectionFinding f' Newton's method requires a formula for both f(x) and f'(x). It is sometimes true that we can figure out how to evaluate the function f whose root(s) we wish to find, but may be unable to determine f'. To use Newton's method, we need to figure out how to evaluate f', and here we present two possible approaches.

• Approximate the derivative by using the previous iterate. That is, if is the current iterate, use the approximation

  

  Substituting this into Newton's method gives the Secant method.

• Approximate the derivative by a finite difference

  

  For good accuracy, should be about .

In either case, one must be careful - the approximations both involve taking the ratio of two numbers which are small in magnitude, and such a division is prone inaccuracies. You should try using finite differences to approximate f' with various values of , in both single and double precision.