Question

Write a program in python to find a single root of a function f(x) on an interval [a,b] using the bisectional
method. Test your program on an equation that you may easily solve analytically.

Answer 1

#*******Python code***************

def findFuncValue(c,f):
#Lets say f(x)= a*x^n+b*x^n-1+c*x^n-3+..........+1 . Here f =[a,b,c,....,1]
#value to store the function value
value = 0.0
for i in range(0,len(f)):
#find value at each coefficient
value+=(f[i]*(c**(len(f)-i-1)))
return value
  
def getSingleRootOfFuncf(f,a,b):
#variable to count the iterations
iteration =1
#mid point of a and b
c = (a+b)/2.0
#run while loop till func value at c is 0 or
#diff between b-a is very small
while findFuncValue(c,f)!=0 or (b-a)/2 > 0.001:
#print the value at each iteration
print "iteration=%s\ta=%s\tb=%s\tc=%s\tf(c)=%s"%(iteration,a,b,c,findFuncValue(c,f))
#if sign of func value at c and
#function value at a is of same sigh
if findFuncValue(c,f)*findFuncValue(a,f) > 0:
#then update value of a to c
a=c
else:
#if funcation value at a and c are of different sign
b=c
#update the mid point c,based on updated a and b for next iteration
c = (a+b)/2.0
iteration+=1
return c

if __name__ == '__main__':

#***Test case 1**
f =[1,0,-1,-2]
a =1
b=2
#***Test case 2****
f=[1,-2]
a = 1
b =4
#call function to get a root using bisection method
root = getSingleRootOfFuncf(f,a,b)
print 'A Root of f is: ',root

  

#***********Python code screenshot*******

  

#*********test for function f = x^2-2x where a =1 and b=4

#***Value of iteration number , a, b and c and f(c) till it converges******

iteration=1 a=1 b=4 c=2.5 f(c)=0.5

iteration=2 a=1 b=2.5 c=1.75 f(c)=-0.25

iteration=3 a=1.75 b=2.5 c=2.125 f(c)=0.125

iteration=4 a=1.75 b=2.125 c=1.9375 f(c)=-0.0625

iteration=5 a=1.9375 b=2.125 c=2.03125 f(c)=0.03125

iteration=6 a=1.9375 b=2.03125 c=1.984375 f(c)=-0.015625

iteration=7 a=1.984375 b=2.03125 c=2.0078125 f(c)=0.0078125

iteration=8 a=1.984375 b=2.0078125 c=1.99609375 f(c)=-0.00390625

iteration=9 a=1.99609375 b=2.0078125 c=2.001953125 f(c)=0.001953125

iteration=10 a=1.99609375 b=2.001953125 c=1.9990234375 f(c)=-0.0009765625

iteration=11 a=1.9990234375 b=2.001953125 c=2.00048828125 f(c)=0.00048828125

iteration=12 a=1.9990234375 b=2.00048828125 c=1.99975585938 f(c)=-0.000244140625

iteration=13 a=1.99975585938 b=2.00048828125 c=2.00012207031 f(c)=0.0001220703125

iteration=14 a=1.99975585938 b=2.00012207031 c=1.99993896484 f(c)=-6.103515625e-05

iteration=15 a=1.99993896484 b=2.00012207031 c=2.00003051758 f(c)=3.0517578125e-05

iteration=16 a=1.99993896484 b=2.00003051758 c=1.99998474121 f(c)=-1.52587890625e-05

iteration=17 a=1.99998474121 b=2.00003051758 c=2.00000762939 f(c)=7.62939453125e-06

iteration=18 a=1.99998474121 b=2.00000762939 c=1.9999961853 f(c)=-3.81469726562e-06

iteration=19 a=1.9999961853 b=2.00000762939 c=2.00000190735 f(c)=1.90734863281e-06

iteration=20 a=1.9999961853 b=2.00000190735 c=1.99999904633 f(c)=-9.53674316406e-07

iteration=21 a=1.99999904633 b=2.00000190735 c=2.00000047684 f(c)=4.76837158203e-07

iteration=22 a=1.99999904633 b=2.00000047684 c=1.99999976158 f(c)=-2.38418579102e-07

iteration=23 a=1.99999976158 b=2.00000047684 c=2.00000011921 f(c)=1.19209289551e-07

iteration=24 a=1.99999976158 b=2.00000011921 c=1.9999999404 f(c)=-5.96046447754e-08

iteration=25 a=1.9999999404 b=2.00000011921 c=2.0000000298 f(c)=2.98023223877e-08

iteration=26 a=1.9999999404 b=2.0000000298 c=1.9999999851 f(c)=-1.49011611938e-08

iteration=27 a=1.9999999851 b=2.0000000298 c=2.00000000745 f(c)=7.45058059692e-09

iteration=28 a=1.9999999851 b=2.00000000745 c=1.99999999627 f(c)=-3.72529029846e-09

iteration=29 a=1.99999999627 b=2.00000000745 c=2.00000000186 f(c)=1.86264514923e-09

iteration=30 a=1.99999999627 b=2.00000000186 c=1.99999999907 f(c)=-9.31322574615e-10

iteration=31 a=1.99999999907 b=2.00000000186 c=2.00000000047 f(c)=4.65661287308e-10

iteration=32 a=1.99999999907 b=2.00000000047 c=1.99999999977 f(c)=-2.32830643654e-10

iteration=33 a=1.99999999977 b=2.00000000047 c=2.00000000012 f(c)=1.16415321827e-10

iteration=34 a=1.99999999977 b=2.00000000012 c=1.99999999994 f(c)=-5.82076609135e-11

iteration=35 a=1.99999999994 b=2.00000000012 c=2.00000000003 f(c)=2.91038304567e-11

iteration=36 a=1.99999999994 b=2.00000000003 c=1.99999999999 f(c)=-1.45519152284e-11

iteration=37 a=1.99999999999 b=2.00000000003 c=2.00000000001 f(c)=7.27595761418e-12

iteration=38 a=1.99999999999 b=2.00000000001 c=2.0 f(c)=-3.63797880709e-12

iteration=39 a=2.0 b=2.00000000001 c=2.0 f(c)=1.81898940355e-12

iteration=40 a=2.0 b=2.0 c=2.0 f(c)=-9.09494701773e-13

iteration=41 a=2.0 b=2.0 c=2.0 f(c)=4.54747350886e-13

iteration=42 a=2.0 b=2.0 c=2.0 f(c)=-2.27373675443e-13

iteration=43 a=2.0 b=2.0 c=2.0 f(c)=1.13686837722e-13

iteration=44 a=2.0 b=2.0 c=2.0 f(c)=-5.68434188608e-14

iteration=45 a=2.0 b=2.0 c=2.0 f(c)=2.84217094304e-14

iteration=46 a=2.0 b=2.0 c=2.0 f(c)=-1.42108547152e-14

iteration=47 a=2.0 b=2.0 c=2.0 f(c)=7.1054273576e-15

iteration=48 a=2.0 b=2.0 c=2.0 f(c)=-3.5527136788e-15

iteration=49 a=2.0 b=2.0 c=2.0 f(c)=1.7763568394e-15

iteration=50 a=2.0 b=2.0 c=2.0 f(c)=-8.881784197e-16

iteration=51 a=2.0 b=2.0 c=2.0 f(c)=4.4408920985e-16

iteration=52 a=2.0 b=2.0 c=2.0 f(c)=-2.22044604925e-16

A Root of f is: 2.0

#*****************Output Screenshot**********

#***Please do let me know if you have any doubts or want me to modify the code*************