# About the First Curve (ref. to my note).
#
# With (c1,c2,...,c9) := (w,z,y,x,1-x,1-y,1-z,1-w,1),
# we ask when
#
#       sum_{i=2,...,9} |ci-c_{i-1}|        (*)
#
# achieves its minimum on the first curve and what is it.
#
# Looking at the curve, one finds that the quantity (*) is
# equals to
#
#       2(w+z-w+z-y+x-y)+1-2x=4(z-y)+1.
#
# Now think y,z,w are functions of x, so we are looking for
# the point for which
#
#              z'-y'=0.
#
# Let g(x,y) be defined by Fxy, and let f(x,y,z) be defined
# by f13 which in linear in z. It can be shown that the following
# systems of equations has to be solved.
#
#          g(x,y)=0,                  (*)
#          f(x,y,z)=0,                (**)
#          f g -f g + f g =0.         (***)
#           y x  x y   z x
#
# We will solve it as follows: think x and y and functions of z defined
# by (*) and (**); then LHS of (***) is a function of z also. We wolve
# (***) by secant method.
#
# The following short MAPLE code compute the point.
#
#                                        Ren-Cang Li, June 1, 1996
#                                        na.rcli@na-net.ornl.gov

x0:=.10071776138031890797:
y0:= .18840211732259462202e-1:
zL:=.72506749291982080566-1.0e-10:
zU:=.72506749291982080566+1.0e-10:

read('zfixed'):  # define x(z) and y(z).

read('zitern'):
zItern(x0,y0,zL,zU,'x1','y1','z1'):
