Numerical Analysis and Computer Applications Regula Falsi Method. (False position method).

1. Chemical Engineering Department
Presented by:
Mujeeb UR Rahman 17CH106
Under the Supervision: Sher Khan Awan
BSRS Mehran UET Jamshoro, Pk
Date: 9Nov2020
Mehran University of Engineering & Technology
Jamshoro, Pakistan
Numerical Analysis & Computer Applications
Regula Falsi Method

2. Regula Falsi Method:
Numerical Analysis & Computer Applications
The Regula–Falsi Method is a numerical method for
estimating the roots of a polynomial f(x).
The objective is to make convergence faster.
Assume that f(x) is continuous.
A value x replaces the midpoint in the Bisection
Method and serves as the new approximation of a root
of f(x).

3. Numerical Analysis & Computer Applications
It was developed because the bisection method converges
at a slow speed. In simple terms, the method is the trial-
and-error technique of using test ("false") values for the
variable and then adjusting the test value according to the
outcome.
Regula Falsi Method: Cont…

4. Numerical Analysis & Computer Applications
Note that the line segment drawn from f(a) to f(b) is called
the interpolation line.
Regula Falsi Method: Cont…

5. Numerical Analysis & Computer Applications
Theorem (Bolzano): If the function f(x) is continuous in [a,
b] and f(a)f(b) < 0 (i.e. f(x) has opposite signs at a and b)
then a value c ∈ (a, b) exists such that f(c) = 0.
Regula Falsi Method: Cont…

6. Numerical Analysis & Computer Applications
Regula Falsi Method:
1. First two guess values:
a , b2. Putting values of a and b
in given f(x).
3. Obtain f(a) and f(b).
4. Finding m.

7. Numerical Analysis & Computer Applications
Regula Falsi Method:
Formula:
𝑚 =
𝑎𝑓 𝑏 − 𝑏𝑓(𝑎)
𝑓 𝑏 − 𝑓(𝑎)
Value of m in given function to
obtain f(m).

8. Numerical Analysis & Computer Applications
1. Find points a and b such that a < b and f(a) * f(b) < 0.
2. Take the interval [a, b] and determine the next value of x1.
3. If f(x1) = 0 then x1 is an exact root, else if f(x1) * f(b) < 0 then
let a = x1, else if f(a) * f(x1) > 0 then let b = x1.
4. Repeat steps 2 & 3 until f(xi) = 0
Algorithm for the Regula–Falsi Method:

9. Numerical Analysis & Computer Applications
1. Show that f(x) = x3 + 3x - 5 has a root in [1,2], and use the
Regula Falsi Method to determine an approximation to the root
that is accurate to at least within 3 decimal places.
Now, the information required to perform the Regula Falsi Method is
as follow:
•f(x) = x3 + 3x - 5,
•Lower Guess a = 1,
•Upper Guess b = 2,
•And error = 0.001
Problem:

10. Numerical Analysis & Computer Applications
We know that f(a) = f(1) = -1 (negative) and f(b) = f(2) = 9 (positive)
so the Intermediate Value Theorem ensures that the root of the
function f(x) lies in the interval [1,2].
Problem: Cont…
C++ Programming for Regula Falsi Method

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Result:
C++ Programming for Regula Falsi Method

15. Numerical Analysis & Computer Applications
Limitations:
While Regula Falsi Method like Bisection Method is always convergent, meaning
that it is always leading towards a definite limit and relatively simple to
understand but there are also some drawbacks when this algorithm is used. As
both regula falsi and bisection method are similar there are some common
limitations both the algorithms have.
•Rate of convergence
The convergence of the regula falsi method can be very slow in some cases(May
converge slowly for functions with big curvatures).
•Relies on sign changes
If a function f (x) is such that it just touches the x -axis for example say f(x) = x2
then it will not be able to find lower guess (a) such that f(a)*f(b) < 0
•Cannot detect Multiple Roots
Like Bisection method, Regula Falsi Method fails to identify multiple different
roots, which makes it less desirable to use compared to other methods that can
identify multiple roots.

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