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  • IQAir wants to estimate of the amount of air pollutants resulting from the combustion reaction at a coal-based PLTU around Jakarta which is one of the causes of air quality in Jakarta getting worse.
  • If the combustion reaction equation is known to be nonlinear with the function is $$\tag{1} f(x) = x^3 - 3x - 7, $$ help find the approximate value of the pollutant $(x)$ from the equation using the regula falsi OR bisection method as well as the desired number of iterations and pollutant limit values $[a,b]$.
  • Note: There is mission information, e.g.
    • What is the meaning of combustion reaction equation, when $f(x) = 0$?
    • Why if $f(x) = 0$, the solution $x$ is the approximate value of the pollutant?
  • Output
Method:
1. Bisection
2. Regula Falsi
Select Method: 1
Input pollutan lower limit value (mg/L): 1
Input pollutan upper limit value (mg/L): 8
Input the number of interations: 3

  Iteration no    Estimation Value of Pollutan Concentration (mg/L)    Error
  ------------    -------------------------------------------------    -----
       1                             4.5                                3.5
       2                             2.75                               1.75
       3                             1.875                              0.875

code Link to heading

  • Code
    def combustion(x):
  y = x**3 - 3*x - 7
  return y

print("Method:")
print("1. Bisection")
print("2. Regula Falsi")
print("Select Method: ", end='')
method = input()
print(method)

print("Input pollutan lower", end=' ')
print("limit value (mg/L): ", end='')
a = float(input())
print(a)

print("Input pollutan upper", end=' ')
print("limit value (mg/L): ", end='')
b = float(input())
print(b)

print("Input number of iteration: ", end='')
N = int(input())
print(N)

print()
print("i\tx\terr")

for i in range(1, N+1):
  fa = combustion(a)
  fb = combustion(b)

  if fa == fb:
    break;

  if method == '1':
    # apply bisection method
    c = (a + b) / 2

    # update a and b
    fc = combustion(c)
    if fa * fc < 0:
      b = c
    elif fc * fb < 0:
      a = c
    else:
      pass

  elif method == '2':
    # apply bisection regular falsi method
    c = (b * fa - a * fb) / (fa - fb)

    # update a and b
    a = b
    b = c
  else:
    # apply other methods
    pass

  # calculate error
  err = abs(a - b)

  # display results
  print(i, end='\t')
  print(c, end='\t')
  print(err)

print()
cs = f'{c:.20f}'
print("x =", cs)
print("error =", err)
print("interation =", i)
  • Method 1 (Bisection)
    Method:
1. Bisection
2. Regula Falsi
Select Method: 1
Input pollutan lower limit value (mg/L): 1.0
Input pollutan upper limit value (mg/L): 8.0
Input number of iteration: 100

i	x	err
1	4.5	3.5
2	2.75	1.75
3	1.875	0.875
4	2.3125	0.4375
5	2.53125	0.21875
6	2.421875	0.109375
7	2.4765625	0.0546875
8	2.44921875	0.02734375
9	2.435546875	0.013671875
10	2.4287109375	0.0068359375
11	2.42529296875	0.00341796875
12	2.427001953125	0.001708984375
13	2.4261474609375	0.0008544921875
14	2.42572021484375	0.00042724609375
15	2.425933837890625	0.000213623046875
16	2.4260406494140625	0.0001068115234375
17	2.4259872436523438	5.340576171875e-05
18	2.426013946533203	2.6702880859375e-05
19	2.4260005950927734	1.33514404296875e-05
20	2.4259939193725586	6.67572021484375e-06
21	2.425990581512451	3.337860107421875e-06
22	2.4259889125823975	1.6689300537109375e-06
23	2.4259880781173706	8.344650268554688e-07
24	2.425988495349884	4.172325134277344e-07
25	2.4259887039661407	2.086162567138672e-07
26	2.425988808274269	1.043081283569336e-07
27	2.425988756120205	5.21540641784668e-08
28	2.425988782197237	2.60770320892334e-08
29	2.425988769158721	1.30385160446167e-08
30	2.425988762639463	6.51925802230835e-09
31	2.425988759379834	3.259629011154175e-09
32	2.4259887577500194	1.6298145055770874e-09
33	2.425988756935112	8.149072527885437e-10
34	2.425988757342566	4.0745362639427185e-10
35	2.4259887575462926	2.0372681319713593e-10
36	2.425988757444429	1.0186340659856796e-10
37	2.4259887573934975	5.093170329928398e-11
38	2.4259887573680317	2.546585164964199e-11
39	2.4259887573552987	1.2732925824820995e-11
40	2.425988757361665	6.366462912410498e-12
41	2.425988757358482	3.183231456205249e-12
42	2.4259887573600736	1.5916157281026244e-12
43	2.4259887573608694	7.958078640513122e-13
44	2.4259887573612673	3.979039320256561e-13
45	2.4259887573614662	1.9895196601282805e-13
46	2.4259887573615657	9.947598300641403e-14
47	2.4259887573616155	4.973799150320701e-14
48	2.4259887573616403	2.4868995751603507e-14
49	2.425988757361628	1.2434497875801753e-14
50	2.4259887573616217	6.217248937900877e-15
51	2.425988757361625	3.1086244689504383e-15
52	2.4259887573616234	1.7763568394002505e-15
53	2.4259887573616226	8.881784197001252e-16
54	2.425988757361622	8.881784197001252e-16
55	2.425988757361622	8.881784197001252e-16
56	2.425988757361622	8.881784197001252e-16
57	2.425988757361622	8.881784197001252e-16
58	2.425988757361622	8.881784197001252e-16
59	2.425988757361622	8.881784197001252e-16
60	2.425988757361622	8.881784197001252e-16
61	2.425988757361622	8.881784197001252e-16
62	2.425988757361622	8.881784197001252e-16
63	2.425988757361622	8.881784197001252e-16
64	2.425988757361622	8.881784197001252e-16
65	2.425988757361622	8.881784197001252e-16
66	2.425988757361622	8.881784197001252e-16
67	2.425988757361622	8.881784197001252e-16
68	2.425988757361622	8.881784197001252e-16
69	2.425988757361622	8.881784197001252e-16
70	2.425988757361622	8.881784197001252e-16
71	2.425988757361622	8.881784197001252e-16
72	2.425988757361622	8.881784197001252e-16
73	2.425988757361622	8.881784197001252e-16
74	2.425988757361622	8.881784197001252e-16
75	2.425988757361622	8.881784197001252e-16
76	2.425988757361622	8.881784197001252e-16
77	2.425988757361622	8.881784197001252e-16
78	2.425988757361622	8.881784197001252e-16
79	2.425988757361622	8.881784197001252e-16
80	2.425988757361622	8.881784197001252e-16
81	2.425988757361622	8.881784197001252e-16
82	2.425988757361622	8.881784197001252e-16
83	2.425988757361622	8.881784197001252e-16
84	2.425988757361622	8.881784197001252e-16
85	2.425988757361622	8.881784197001252e-16
86	2.425988757361622	8.881784197001252e-16
87	2.425988757361622	8.881784197001252e-16
88	2.425988757361622	8.881784197001252e-16
89	2.425988757361622	8.881784197001252e-16
90	2.425988757361622	8.881784197001252e-16
91	2.425988757361622	8.881784197001252e-16
92	2.425988757361622	8.881784197001252e-16
93	2.425988757361622	8.881784197001252e-16
94	2.425988757361622	8.881784197001252e-16
95	2.425988757361622	8.881784197001252e-16
96	2.425988757361622	8.881784197001252e-16
97	2.425988757361622	8.881784197001252e-16
98	2.425988757361622	8.881784197001252e-16
99	2.425988757361622	8.881784197001252e-16
100	2.425988757361622	8.881784197001252e-16

x = 2.42598875736162211680380096368025988340377807617187500
error = 8.881784197001252e-16
interation = 100
    Note: It has not yet stopped in 100 iterations since there is always new c from previous a and b.
  • Method 2 (Regula Falsi)
    Method:
1. Bisection
2. Regula Falsi
Select Method: 2
Input pollutan lower limit value (mg/L): 1.0
Input pollutan upper limit value (mg/L): 8.0
Input number of iteration: 100

i	x	err
1	1.1285714285714286	6.871428571428572
2	1.2540693397752025	0.12549791120377396
3	8.22098176347065	6.966912423695447
4	1.3690203109391845	6.851961452531466
5	1.4789269031521688	0.10990659221298427
6	4.136642366324688	2.6577154631725195
7	1.844814336803726	2.2918280295209623
8	2.093593129020052	0.24877879221632582
9	2.5681447002446642	0.47455157122461245
10	2.4009115872524283	0.16723311299223598
11	2.4242907675446332	0.023379180292204982
12	2.4260101070971984	0.0017193395525652022
13	2.4259887393490422	2.136774815619802e-05
14	2.4259887573614316	1.8012389357835445e-08
15	2.4259887573616217	1.900701818158268e-13
16	2.425988757361622	4.440892098500626e-16
17	2.425988757361622	0.0

x = 2.42598875736162211680380096368025988340377807617187500  
error = 0.0
interation = 18
    Note: It stops in 18 iterations after fa == fb.