NACA Regula Falsi Method

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: First two guess values : a , b 2. 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:   Value of m in given function to obtain f(m) .

8. Numerical Analysis & Computer Applications Find points a and b such that a < b and f(a) * f(b) < 0 . Take the interval [a, b] and determine the next value of x1 . 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 . Repeat steps 2 & 3 until f(xi) = 0 Algorithm for the Regula–Falsi Method:

9. Numerical Analysis & Computer Applications Show that f(x) = x 3  + 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) = x 3  + 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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14. Numerical Analysis & Computer Applications 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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