bisection method
TRANSCRIPT
B.E. 4th Semester
Maths-IV: 140001
Numerical Analysis
Prof. K.K.PokarAssistant Professor in Mathematics
Government Engineering College,Bhuj
Numerical Methods
Bisection Method
Bisection Method
Bisection Method
Bisection Method
Algorithm for Bisection MethodStep-1:
Find the two points a and b such that f(a) f(b) < 0
Compute
Evaluate
Step-2:Decision Making for Replacing a / b with new
value
f(x0)
f(x0)=0 f(x0) > 0
f(x0) < 0
If f(x0)=0
Then x0 is the required root/solution
of the given equation
f(x) = 0
If f(x0) > 0
Then x0 replaces any one of a & b
f(x0) > 0
f(a) > 0
Root lies b/w b and
x0
f(b) > 0
Root lies b/w a and
x0
If f(x0) < 0
Then x0 replaces any one of a & b
f(x0) < 0
f(a) < 0
Root lies b/w b and
x0
f(b) < 0
Root lies b/w a and
x0
Step-3:
Repeating Step-2 for in place of
till the desired accuracy is obtained
Example: Find a real root of the equation Step-1:Find the two points a and b such that f(a) f(b)
< 0
Compute
Evaluate
Step-2:Decision Making for Replacing 2 / 3 with new
value =2.5
f(x0)=2.125
f(x0)=0 f(x0) > 0
f(x0) < 0
If f(x0)=2.125 > 0
Then x0 =2.5 replaces any one of 2 & 3
f(2.5) > 0
f(3) > 0
Root lies b/w 3 and
2.5
f(2) > 0
Root lies b/w a and
x0
Step-2:Decision Making for Replacing 2 / 2.5 with
new value 2.25
f(2.25)=-1.859375
f(x0)=0 f(x0) > 0
f(x0) < 0
If f(2.25) < 0
Then 2.25 replaces any one of 2 & 2.5
f(2.25) < 0
f(2) < 0
f(2.5) > 0
Root lies b/w
2.25 and 2.5
Table formn a f(a) b f(b) M=(a+b)
/2f(M)
0 2 -5 3 13 2.5 2.125
1 2 -5 2.5 2.125 2.25 -1.85938
2 2.25 -1.85938 2.5 2.125 2.375 0.021484
3 2.25 -1.85938 2.375 0.021484 2.3125 -0.94604
4 2.3125 -0.94604 2.375 0.021484 2.34375 -0.46915
5 2.34375 -0.46915 2.375 0.021484 2.359375 -0.22556
6 2.359375 -0.22556 2.375 0.021484 2.367188 -0.10247
7 2.367188 -0.10247 2.375 0.021484 2.371094 -0.0406
8 2.371094 -0.0406 2.375 0.021484 2.373047 -0.00959
9 2.373047 -0.00959 2.375 0.021484 2.374023 0.005942
10 2.373047 -0.00959 2.374023 0.005942 2.373535 -0.00182
11 2.373535 -0.00182 2.374023 0.005942 2.373779 0.002059
12 2.373535 -0.00182 2.373779 0.002059 2.373657 0.000118
Understanding the tablen a f(a) b f(b) M=(a+
b)/2f(M)
0 2 -5 3 3 2.5 2.125
1 2 -5 2.5 2.125 2.25 -1.85938
2 2.25 -1.85938 2.5 2.125 2.375 0.021484
3 2.25 -1.85938 2.375 0.021484 2.3125 -0.94604
4 2.3125 -0.94604 2.375 0.021484 2.34375 -0.46915
5 2.34375 -0.46915 2.375 0.021484 2.359375 -0.22556
6 2.359375 -0.22556 2.375 0.021484 2.367188 -0.10247
7 2.367188 -0.10247 2.375 0.021484 2.371094 -0.0406
8 2.371094 -0.0406 2.375 0.021484 2.373047 -0.00959
9 2.373047 -0.00959 2.375 0.021484 2.374023 0.005942
10 2.373047 -0.00959 2.374023 0.005942 2.373535 -0.00182
11 2.373535 -0.00182 2.374023 0.005942 2.373779 0.002059
12 2.373535 -0.00182 2.373779 0.002059 2.373657 0.000118
