1. MATH 175: Numerical Analysis II
Lecturer: Jomar Fajardo Rabajante
IMSP, UPLB
2nd Sem AY

2. 3rd Method: Regula Falsi/False Position/Inverse Linear Interpolation (also a bracketing method)
We will still use the Intermediate Zero Theorem
The function should be continuous
The function values of the two endpoints of the line should be of opposite signs

3. 3rd Method: Regula Falsi
y-axis
x-axis
Approximate root

4. 3rd Method: Regula Falsi
We will use inverse linear interpolation (in slope-intercept form):
To get x3, set f(x3)=0 (from the zero of the line):

5. 3rd Method: Regula Falsi
y-axis
Set new x1 = old x3
New Approximate root
x-axis
Old Approximate root

6. 3rd Method: Regula Falsi
y-axis
Old Approximate root
x-axis
Set new x2 = old x3
New Approximate root

7. 3rd Method: Regula Falsi
y-axis
3rd
2nd
x-axis
1st

8. 3rd Method: Regula Falsi
Try this at home: create a flowchart of the Regula Falsi method
Hint: use bisection algorithm, however change the formula for x3 to

10. 3rd Method: Regula Falsi
Like bisection method, Regula Falsi is guaranteed to converge to the root (assuming IZT is met).
The rate at which this method converges will depend on how nearly linear f(x) is near its zero. If f(x) is sufficiently differentiable then it is well approximated by a straight line over small intervals.
Regula Falsi is often (not always) faster than bisection method, but still the order of convergence is linear.
Bullet 1: global convergence

11. 3rd Method: Regula Falsi
We can use abs(x3,k-x3,k-1)<tol as our stopping criterion but not (b-a)/2k<tol (where [a,b] is the initial bracket) since the width of the bracket may not converge to zero like in bisection.
But a better stopping criterion is
(Let x3,k be the approximate root at iteration k, and λ be the asymptotic error constant. Invoking the error evolution equation: )
Tol=10^(-m): accurate at least up to m decimal places
where

12. There’s an improved Regula Falsi method called the Modified Regula Falsi.
y-axis
1/2
3rd
2nd
x-axis
1st

13. Bisection, Regula Falsi and Modified Regula Falsi methods are called Interval/Bracketing Methods. Next topics: Iterative methods, such as secant method, Newton’s method, fixed point iteration, Mϋller’s method, Bairstow’s method Usually, faster methods require more assumptions and offer fewer guarantees.