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ECIV 301
Programming & Graphics
Numerical Methods for Engineers
Lecture 7
Roots of Equations
Bracketing Methods
Last Time The Problem
)(1)( tvec
gmcf
tm
c
Define Function
0)( cfc must satisfy
c is the ROOT of the equation
Last Time ClassificationMethods
Bracketing Open
• Graphical• Bisection Method• False Position
• Fixed Point Iteration• Newton-Raphson• Secand
Last Time Graphical Methods)(1)( tve
c
gmcf
tm
c
f(c)
-10
-5
0
5
10
15
20
25
0 50 100 150 200
f(c)
c
f(c)
v=10 m/st=3 secm=65 kgg=9.81
Last Time Graphical Methods
No Roots Even Number of Roots
Lower and Upper Bounds of interval yield values of same sign
Last Time Graphical MethodsLower and Upper Bounds of interval yield values of opposite sign
Odd number of Roots
Last Time Bisection Method
Choose Lower, xl and Upper xu guesses that bracket the root
f(c)
-10
-5
0
5
10
15
20
25
0 50 100 150 200
f(c)
xl xu
Last Time Bisection Method
Calculate New Estimate xr and f(xr)
f(c)
-10
-5
0
5
10
15
20
25
0 50 100 150 200
f(c)
xl xu
xr=0.5(xl+xu)
Last Time Bisection MethodDefine New Interval that Brackets the RootCheck sign of
f(xl)*f(xr) and f(xu)*f(xr)
f(c)
-10
-5
0
5
10
15
20
25
0 50 100 150 200
f(c)
xl xu
Previous Guess
xu
Last Time Bisection MethodRepeat until convergence
f(c)
-10
-5
0
5
10
15
20
25
0 50 100 150 200
f(c)
xl xu
Previous Guess
xr=0.5(xl+xu)
Last Time Bisection MethodCheck Convergence
ErrorAcceptablex
xxnewr
oldr
newr
%100
Root = newrx
If Error
Objectives
• Master methods to compute roots of equations
• Assess reliability of each method
• Choose best method for a specific problem
• REGULA FALSI Method (False Position)
False Position Method
f(c)
-10
-5
0
5
10
15
20
25
0 50 100 150 200
f(c)
xl xu
xr=0.5(xl+xu)
Recall Bisection Method
No consideration on function values
False Position Methodf(c)
-10
-5
0
5
10
15
20
25
0 50 100 150 200
f(c)
f(xl)
f(xu)xl
xuxr
NEW ESTIMATE
False Position Methodf(xl)
f(xu)xl
xuxr
From Similar Triangles
ur
lr
u
l
xx
xx
xf
xf
lruurl xxxfxxxf
False Position Method
ul
ulu
ul
luur xfxf
xfxx
xfxf
xfxxx
Add and subtract
ul
uluur xfxf
xxxfxx
New Estimate
Loop
xold=xr
Error=100*abs(x-xold)/xr
Sign=f(xl)*f(xr)
Sign
xu=xr
fu=f(xu)xl=xr
fl=f(xl)Error=0
Error<Eall ROOT=xr
FALSE
<0 >0
ululuur ffxxfxx
fu=f(xu), fl=f(xl)