Regula Falsi (False position) Method

Regula Falsi or False Position Method is one of the iterative (bracketing) Method for solving root(s) of nonlinear equation under Numerical Methods or Analysis.

1. Regula Falsi (False Position) Method
Numerical Analysis
Isaac Amornortey Yowetu
NIMS-Ghana
September 12, 2020

2. Background of Regula Falsi MethodDerivation of the Regula Falsi MethodRegula Falsi Algorithm to nd approximate SolutionApplication of Regula Falsi Method
Application of Regula Falsi Method
Question
Find a root ofxe
x
=1 onI= [0;1]using Regula Falsi method.

3. Background of Regula Falsi MethodDerivation of the Regula Falsi MethodRegula Falsi Algorithm to nd approximate SolutionApplication of Regula Falsi Method
Outline
Background of Regula Falsi Method
Derivation of the Regula Falsi Method
Graphical Example
Regula Falsi Algorithm to nd approximate Solution
Application of Regula Falsi Method

4. Background of Regula Falsi MethodDerivation of the Regula Falsi MethodRegula Falsi Algorithm to nd approximate SolutionApplication of Regula Falsi Method
Background of Regula Falsi Method
Introduction
It is one of the bracketing iterative methods in nding
roots of a nonlinear equations.
The approximated root is found by the use of straight
lines or slopes.
It is also noted to be based on Bolzano's theorem for
continuous functions.
Theorem (Bolzano)
If a functionf(x)is continuous on an interval[a;b]and
f(a)f(b)<0, then a valuec2(a;b)exists for which
f(c) =0.

5. Background of Regula Falsi MethodDerivation of the Regula Falsi MethodRegula Falsi Algorithm to nd approximate SolutionApplication of Regula Falsi Method
Graphical Example
Figure:

6. Background of Regula Falsi MethodDerivation of the Regula Falsi MethodRegula Falsi Algorithm to nd approximate SolutionApplication of Regula Falsi Method
Derivation of the Regula Falsi Method
y
x
=
f(a)0
ac
=
f(b)0
bc
(1)
OR
y
x
=
f(a)f(b)
ab
=
f(a)0
ac
(2)
OR
y
x
=
f(a)f(b)
ab
=
f(b)0
bc
(3)

7. Background of Regula Falsi MethodDerivation of the Regula Falsi MethodRegula Falsi Algorithm to nd approximate SolutionApplication of Regula Falsi Method
Derivation Continues...
Using any of the 3 approaches, can help us derived Regula
Falsi Method. Using eqn(1):
f(a)0
ac
=
f(b)0
bc
(4)
f(a)
ac
=
f(b)
bc
(5)
f(a)(bc) =f(b)(ac) (6)
bf(a)cf(a) =af(b)cf(b) (7)
cf(b)cf(a) =af(b)bf(a) (8)
c=
af(b)bf(a)
f(b)f(a)
(9)

8. Background of Regula Falsi MethodDerivation of the Regula Falsi MethodRegula Falsi Algorithm to nd approximate SolutionApplication of Regula Falsi Method
Regula Falsi Algorithm
iff(a)f(b)<0:
root exists
else:
root doesn't exist
Iteration Processes when root exists
1. c=
af(b)bf(a)
f(b)f(a)
2. f(c) =0,
3. f(a)f(c)<0:
setb c
4.
seta c
5.

9. Background of Regula Falsi MethodDerivation of the Regula Falsi MethodRegula Falsi Algorithm to nd approximate SolutionApplication of Regula Falsi Method
Application of Regula Falsi Method
Example 1
Find a root ofxe
x
=1 onI= [0;1]using Regula Falsi method.
Solution
Considering ourf(x) =xe
x
1=0 anda1=0;b1=1
f(0) =0e
0
1=1
f(1) =1e
1
1=1:7183
f(0)f(1)<0, hencec2[0;1].
c1=
af(b)bf(a)
f(b)f(a)
=0:3679 (10)
f(c1) =0:4685 (11)
choose[a2;b2] = [0:3679;1]

10. Background of Regula Falsi MethodDerivation of the Regula Falsi MethodRegula Falsi Algorithm to nd approximate SolutionApplication of Regula Falsi Method
Solution Continue...
c2=
a2f(b2)b2f(a2)
f(b2)f(a2)
=0:5033 (12)
f(c2) =0:1674 (13)
choose[a3;b3] = [0:5033;1]
c3=
a3f(b3)b3f(a3)
f(b3)f(a3)
=0:5474 (14)
f(c3) =0:0536 (15)
choose[a4;b4] = [0:5474;1]

11. Background of Regula Falsi MethodDerivation of the Regula Falsi MethodRegula Falsi Algorithm to nd approximate SolutionApplication of Regula Falsi Method
Solution summary
iterationa b c f(c)
1 0 10.3679-0.4685
2 0.367910.5033-0.1674
3 0.503310.5474-0.0536
4 0.547410.5611-0.0166
5 0.561110.5666-0.0051
6 0.566610.5670-0.0015
7 0.567010.5671-0.0005
8 0.567110.5671-0.0001
9 0.567110.5671-0.00004
10 0.567110.5671-0.00001
Conclusion: The approximate solution cn=0:5671

12. Background of Regula Falsi MethodDerivation of the Regula Falsi MethodRegula Falsi Algorithm to nd approximate SolutionApplication of Regula Falsi Method
End
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