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!
! zeroin
!
! This procedure is adopted version of zeroin procedure
! by FMM library developed by Forsythe, Malcolm, and Moler
! (http://www.netlib.org/fmm/index.html) for Fortran 95.
! Some changes to code are introduced by FH:
! * epsilon by standard function
!
! This file is part of Munipack.
!
! Munipack is free software: you can redistribute it and/or modify
! it under the terms of the GNU General Public License as published by
! the Free Software Foundation, either version 3 of the License, or
! (at your option) any later version.
!
! Munipack is distributed in the hope that it will be useful,
! but WITHOUT ANY WARRANTY; without even the implied warranty of
! MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
! GNU General Public License for more details.
!
! You should have received a copy of the GNU General Public License
! along with Munipack. If not, see <http://www.gnu.org/licenses/>.
double precision function zeroin(ax,bx,f,tol)
implicit none
double precision, intent(in) :: ax,bx,tol
interface
function f(x)
double precision :: f
double precision, intent(in) :: x
end function f
end interface
!
! a zero of the function f(x) is computed in the interval ax,bx .
!
! input..
!
! ax left endpoint of initial interval
! bx right endpoint of initial interval
! f function subprogram which evaluates f(x) for any x in
! the interval ax,bx
! tol desired length of the interval of uncertainty of the
! final result ( .ge. 0.0d0)
!
!
! output..
!
! zeroin abcissa approximating a zero of f in the interval ax,bx
!
!
! it is assumed that f(ax) and f(bx) have opposite signs
! without a check. zeroin returns a zero x in the given interval
! ax,bx to within a tolerance 4*macheps*abs(x) + tol, where macheps
! is the relative machine precision.
! this function subprogram is a slightly modified translation of
! the algol 60 procedure zero given in richard brent, algorithms for
! minimization without derivatives, prentice - hall, inc. (1973).
!
!
double precision :: eps = epsilon(zeroin)
double precision a,b,c,d,e,fa,fb,fc,tol1,xm,p,q,r,s
double precision dabs,dsign
!
! compute eps, the relative machine precision
!
! FH: the loop is replaced by an internal function (to be faster)
! eps = 1.0d0
! 10 eps = eps/2.0d0
! tol1 = 1.0d0 + eps
! if (tol1 .gt. 1.0d0) go to 10
!
! initialization
!
a = ax
b = bx
fa = f(a)
fb = f(b)
!
! begin step
!
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
!
! convergence test
!
40 tol1 = 2.0d0*eps*dabs(b) + 0.5d0*tol
xm = .5*(c - b)
if (dabs(xm) .le. tol1) go to 90
if (fb .eq. 0.0d0) go to 90
!
! is bisection necessary
!
if (dabs(e) .lt. tol1) go to 70
if (dabs(fa) .le. dabs(fb)) go to 70
!
! is quadratic interpolation possible
!
if (a .ne. c) go to 50
!
! linear interpolation
!
s = fb/fa
p = 2.0d0*xm*s
q = 1.0d0 - s
go to 60
!
! inverse quadratic interpolation
!
50 q = fa/fc
r = fb/fc
s = fb/fa
p = s*(2.0d0*xm*q*(q - r) - (b - a)*(r - 1.0d0))
q = (q - 1.0d0)*(r - 1.0d0)*(s - 1.0d0)
!
! adjust signs
!
60 if (p .gt. 0.0d0) q = -q
p = dabs(p)
!
! is interpolation acceptable
!
if ((2.0d0*p) .ge. (3.0d0*xm*q - dabs(tol1*q))) go to 70
if (p .ge. dabs(0.5d0*e*q)) go to 70
e = d
d = p/q
go to 80
!
! bisection
!
70 d = xm
e = d
!
! complete step
!
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. 0.0d0) go to 20
go to 30
!
! done
!
90 zeroin = b
return
end function zeroin
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