College Call Girls Nashik Nehal 7001305949 Independent Escort Service Nashik
Β
PO_groupVI_Term assignment.pptx
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
ο§ 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.
ο§ Such methods are always convergent.
ο΅ Examples of bracketing methods:
ο΅ Bisection method
ο΅ False position method
4. ABOUT BISECTION METHOD
ο§ Assumptions:
Given an interval [a, b]
f(x) is continuous on [a, b]
f(a) and f(b) have opposite signs.
ο§ 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.
ο§ 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..
ο§ If f(c) > 0, let anew = a and bnew = c and repeat
process.
ο§ If f(c) < 0, let anew = c and bnew = b and repeat
process.
ο§ This reassignment ensures the root is always
bracketed!! initial point βaβ
root βdβ
initial point βbβ
a
c b
d
ο§ Bisection is an iterative process, where the initial interval is halved
until the size of the interval decreases below some predefined
tolerance ο₯:|a - b| ο³ ο₯ or f(x) falls below a tolerance ο€:|f(c ) β f(c-1)| ο£
ο€.
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,
π β ππ
ππ β ππ
=
π β ππ
ππ β ππ
βΉ
π(π) β π(π)
π(π) β π(π)
=
π β π
π β π
βΉ
βπ(π)
π(π) β π(π)
=
ππ β π
π β π
πππ π = ππ, π π = π βΉ ππ = π β
(π β π)π(π)
π(π) β π(π)
βΉ ππ =
βππ(π) + ππ(π)
π(π) β π(π)
βΉ ππ =
ππ π β ππ(π)
π(π) β π(π)
Iterative formula for
Method of False Position
X
f(x)
Root
a
b
ππ ππ
12. CONTD..
Assumptions:
Given an interval [a, b]
f(x) is continuous on [a, b]
f(a) and f(b) have opposite signs.
ο§ 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.
ο§ For that we perform the following steps:
1. Compute the in between point on x-axis,
2. Evaluate f(ππ)
3. If f(a) f(ππ) < 0 then new interval [a, ππ]
If f(a) f(ππ) > 0 then new interval [ππ, b]
4. Repeat the procedure until π(ππ) < tolerance value.
ππ =
ππ π β ππ(π)
π(π) β π(π)