1
Solutions of Nonlinear Equations
and Systems of Equations
•Why?
•But
a
ac
b
b
x
c
bx
ax
2
4
0
2
2
!
!
=
"
=
+
+
m
?
0
sin
?
0
2
3
4
5
=
!
=
+
=
!
=
+
+
+
+
+
x
x
x
x
f
ex
dx
cx
bx
ax
Nonlinear Equation
Solvers
Bracketing
Graphical
Open Methods
Bisection
False Position
(RegulaFalsi
)
Newton Raphson
Secant
All Iterative
Bracketing Methods
(Or, two point methods for finding roots)
•
Two initial guesses for the root are required. These
guesses must “bracket” or be on either side of the root
Read Graphical Methods Sec. 5.2
If one root of a real and continuous function, f(x)=0, is
bounded by values x=x
l
, x
=x
u
then
f(x
l
) . f(x
u
) <0.
The function changes sign on opposite sides of the roo
Graphical Methods
Fig 5.4a
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2
Fig 5.4c
The Bisection Method
For the arbitrary equation of one variable, f(x)=0
1.
Pick x
l
and x
u
such that they bound the root of
interest, check if f(x
l
).f(x
u
) <0.
2.
Estimate the root by evaluating f[(x
l
+x
u
)/2].
3.
Find the pair
•
If f(x
l
). f[(x
l
+x
u
)/2]<0, root lies in the lower interval,
then x
u
=(x
l
+x
u
)/2 and go to step 2.
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 Spring '07
 seminario

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