FORTRAN 90
Lecturer : Rafel Hekmat Hameed
University of Babylon
Subject : Fortran 90
College of Engineering
Year : Second B.Sc.
Mechanical Engineering Dep.
S
SO
OL
LU
UTTIIO
ON
N O
OF
F N
NO
ON
N-L
LIIN
NE
EA
AR
R E
EQ
QU
UA
ATTIIO
ON
N
Bisection Method
The bisection method in mathematics is a root-finding method which
#_>
#_>
repeatedly bisects an interval and then selects a subinterval in which a root
#_>
#_>
must lie for further processing. It is a very simple and robust method, but it is
also relatively slow. Because of this, it is often used to obtain a rough
approximation to a solution which is then used as a starting point for more
rapidly converging methods.
Consider a transcendental equation f (x) = 0 which has a zero in the
interval [a,b] and f (a)  f (b) < 0. We may refine our approximation to the
root by dividing the interval into two: find the midpoint c = (a + b)/2. In any
real world problem, it is very unlikely that f(c) = 0, however if we are that
lucky, then we have found a root. More likely, if f(a) and f(c) have opposite
signs, then a root must lie on the interval [a, c]. The only other possibility is
that f(c) and f(b) have opposite signs, and therefore the root must lie on the
interval [c, b].
We may repeat this process numerous times, each time halving the size
of the interval.
Numerical Example :
Find a root of f (x) = 3x + sin(x) - exp(x)=0.
The graph of this equation is given in the
figure.
ϭ
It's clear from the graph that there are two roots, one lies between
0 and 0.5 and the other lies between 1.5 and 2.0. Consider the function
f (x) in the interval [0, 0.5] since f (0)  f (0.5) is less than zero. Then the
bisection iterations are given by
Iteration
No.
1
2
3
4
5
6
a
b
c
f(a)  f(c)
0
0.25
0.25
0.34
0.34
0.354
0.5
0.5
0.393
0.393
0.367
0.367
0.25
0.393
0.34
0.367
0.354
0.3605
0.287 (+ve)
-0.015 (-ve)
9.69 E-3 (+ve)
-7.81 E-4 (-ve)
8.9 E-4 (+ve)
-3.1 E-6 (-ve)
So one of the roots of 3x + sin(x) - exp(x) = 0 is approximately 0.3605.
program bisection
implicit none
real,parameter::error=1e-4
real::a,b,f,c
10 read(*,*) a,b
15 if (f(a)*f(b).lt.0)then
c=(a+b)/2.0
else
write(*,*)"try with another value of a & b "
Ϯ
goto 10
endif
if(f(a)*f(c).lt.0)then
b=c
else
a=c
endif
if(abs(b-a).gt.error)goto 15
write(*,*)"the root is=",c
end
real function f(x)
implicit none
real::x
f=3*x + sin(x) - exp(x)
end
ϯ