1. Process Optimization
Term assignment
Submitted to : Prof . Amit Kumar
By : (Group VI)
19bch049 - Punit patel
19bch050 - Raj patel
19bch051 - Riya patel
19bch052 - Sahil patel
19bch053 - Yash patel
2. Bracketing Method
In bracketing methods, the method starts with an interval that
contains the root and a procedure is used to obtain a smaller interval
containing the root.
Such methods are always convergent.
Examples of bracketing methods:
Bisection method
False position method
4. ABOUT BISECTION METHOD
Assumptions:
Given an interval [a, b]
f(x) is continuous on [a, b]
f(a) and f(b) have opposite signs.
These assumptions ensure the existence of at least one zero in the interval [a, b] and the
bisection method can be used to obtain a smaller interval that contains the zero.
For that we perform the following steps:
1. Compute the mid point c = (a + b) / 2
2. Evaluate f(c)
3. If f(a) f(c) < 0 then new interval [a, c]
If f(a) f(c) > 0 then new interval [c, b]
4. Repeat the procedure until we get convergence.
a
b
f(a)
f(b)
c
a C1 C2
5. CONTD..
If f(c) > 0, let anew = a and bnew = c and repeat
process.
If f(c) < 0, let anew = c and bnew = b and repeat
process.
This reassignment ensures the root is always
bracketed!! initial point ‘a’
root ‘d’
initial point ‘b’
a
c b
d
Bisection is an iterative process, where the initial interval is halved
until the size of the interval decreases below some predefined
tolerance :|a - b| or f(x) falls below a tolerance :|f(c ) – f(c-1)|
.
11. ABOUT REGULA-FALSI METHOD
This technique is similar to the bisection
method except that the next iteration is
taken as the line of interception between
the pair of x-values and the x-axis rather
than at the midpoint.
(a, f(a))
(b, f(b))
By two point line
formula,
𝒚 − 𝒚𝟏
𝒚𝟏 − 𝒚𝟐
=
𝒙 − 𝒙𝟏
𝒙𝟏 − 𝒙𝟐
⟹
𝒇(𝒙) − 𝒇(𝒂)
𝒇(𝒂) − 𝒇(𝒃)
=
𝒙 − 𝒂
𝒂 − 𝒃
⟹
−𝒇(𝒂)
𝒇(𝒂) − 𝒇(𝒃)
=
𝒙𝟏 − 𝒂
𝒂 − 𝒃
𝒇𝒐𝒓 𝒙 = 𝒙𝟏, 𝒇 𝒙 = 𝟎 ⟹ 𝒙𝟏 = 𝒂 −
(𝒂 − 𝒃)𝒇(𝒂)
𝒇(𝒂) − 𝒇(𝒃)
⟹ 𝒙𝟏 =
−𝒂𝒇(𝒃) + 𝒃𝒇(𝒂)
𝒇(𝒂) − 𝒇(𝒃)
⟹ 𝒙𝟏 =
𝒂𝒇 𝒃 − 𝒃𝒇(𝒂)
𝒇(𝒃) − 𝒇(𝒂)
Iterative formula for
Method of False Position
X
f(x)
Root
a
b
𝒙𝟏 𝒙𝟐
12. CONTD..
Assumptions:
Given an interval [a, b]
f(x) is continuous on [a, b]
f(a) and f(b) have opposite signs.
These assumptions ensure the existence of at least one zero in the interval [a, b] and
the false position method can be used to obtain a smaller interval that contains the
zero.
For that we perform the following steps:
1. Compute the in between point on x-axis,
2. Evaluate f(𝒙𝟏)
3. If f(a) f(𝒙𝟏) < 0 then new interval [a, 𝒙𝟏]
If f(a) f(𝒙𝟏) > 0 then new interval [𝒙𝟏, b]
4. Repeat the procedure until 𝒇(𝒙𝒊) < tolerance value.
𝒙𝟏 =
𝒂𝒇 𝒃 − 𝒃𝒇(𝒂)
𝒇(𝒃) − 𝒇(𝒂)