MATH 175: Numerical Analysis II
Lecturer: Jomar Fajardo Rabajante
IMSP, UPLB
2nd Sem AY 2012-2013
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
rd
3
Method: Regula Falsi
y-axis
( x1 , f ( x1 ))
x-axis
( x3 ,0)
Approximate
root
( x2 , f ( x2 ))
rd
3
Method: Regula Falsi
We will use inverse linear interpolation (in slope-intercept
form):
x2  x1
x  x1 
( f ( x)  f ( x1 ))
f ( x2 )  f ( x1 )
To get x3, set f(x3)=0 (from the zero of the line):
x2  x1
x3  x1 
f ( x1 )
f ( x2 )  f ( x1 )
rd
3
Method: Regula Falsi
y-axis
( x1 , f ( x1 )) Set new x1 = old x3
( x3 ,0)
New Approximate
root
x-axis
( x3 ,0)
Old
Approximate
root
( x2 , f ( x2 ))
rd
3
y-axis
Method: Regula Falsi
( x1 , f ( x1 ))
( x3 ,0)
Old Approximate
root
x-axis
( x3 ,0)
New Approximate
root
( x2 , f ( x2 ))
Set new x2 = old x3
rd
3
Method: Regula Falsi
y-axis
3rd
1st
2nd
x-axis
rd
3
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
x2  x1
x1 f ( x2 )  x2 f ( x1 )
x3  x1 
f ( x1 ) 
f ( x2 )  f ( x1 )
f ( x2 )  f ( x1 )
rd
3
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.
rd
3
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: errork  errork 1 )

 1
where
x3,k  x3,k 1  tol

x3,k  x3,k 1
x3,k 1  x3,k 2
There’s an improved Regula Falsi method called
the Modified Regula Falsi.
y-axis
1/2
2nd
1st
3rd
x-axis
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.