c
c
c
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function zeroin(ax,bx,f,tol)
c =================================================
implicit double precision (a-h,o-z)
external f
c
c a zero of the function f(x) is computed in the interval ax,bx.
c (Standard routine from netlib)
c
c input..
c
c ax left endpoint of initial interval
c bx right endpoint of initial interval
c f function subprogram which evaluates f(x) for any x in
c the interval ax,bx
c tol desired length of the interval of uncertainty of the
c final result ( .ge. 0.0)
c
c output..
c
c zeroin abcissa approximating a zero of f in the interval ax,bx
c
c
c it is assumed that f(ax) and f(bx) have opposite signs
c without a check. zeroin returns a zero x in the given interval
c ax,bx to within a tolerance 4*macheps*dabs(x) + tol, where macheps
c is the relative machine precision.
c this function subprogram is a slightly modified translation of
c the algol 60 procedure zero given in richard brent, algorithms for
c minimization without derivatives, prentice - hall, inc. (1973).
c
c
c
c compute eps, the relative machine precision
c
eps = 1.0
10 eps = eps/2.0
tol1 = 1.0 + eps
if (tol1 .gt. 1.0) go to 10
c
c initialization
c
a = ax
b = bx
fa = f(a)
fb = f(b)
c
c begin step
c
20 c = a
fc = fa
d = b - a
e = d
30 if (dabs(fc) .ge. dabs(fb)) go to 40
a = b
b = c
c = a
fa = fb
fb = fc
fc = fa
c
c convergence test
c
40 tol1 = 2.0*eps*dabs(b) + 0.5*tol
xm = .5*(c - b)
if (dabs(xm) .le. tol1) go to 90
if (fb .eq. 0.0) go to 90
c
c is bisection necessary
c
if (dabs(e) .lt. tol1) go to 70
if (dabs(fa) .le. dabs(fb)) go to 70
c
c is quadratic interpolation possible
c
if (a .ne. c) go to 50
c
c linear interpolation
c
s = fb/fa
p = 2.0*xm*s
q = 1.0 - s
go to 60
c
c inverse quadratic interpolation
c
50 q = fa/fc
r = fb/fc
s = fb/fa
p = s*(2.0*xm*q*(q - r) - (b - a)*(r - 1.0))
q = (q - 1.0)*(r - 1.0)*(s - 1.0)
c
c adjust signs
c
60 if (p .gt. 0.0) q = -q
p = dabs(p)
c
c is interpolation acceptable
c
if ((2.0*p) .ge. (3.0*xm*q - dabs(tol1*q))) go to 70
if (p .ge. dabs(0.5*e*q)) go to 70
e = d
d = p/q
go to 80
c
c bisection
c
70 d = xm
e = d
c
c complete step
c
80 a = b
fa = fb
if (dabs(d) .gt. tol1) b = b + d
if (dabs(d) .le. tol1) b = b + dsign(tol1, xm)
fb = f(b)
if ((fb*(fc/dabs(fc))) .gt. 0.0) go to 20
go to 30
c
c done
c
90 zeroin = b
return
end