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EE103 Lecture Notes, Winter 2009, Prof S. Jacobsen
Section 3
SECTION 3: ROOTS OF AN EQUATION OF A SINGLE VARIABLE
.............................................
31
Fixed Point Approach (Method of Successive Approximations):
.......................................................
31
Newton's Method:
...................................................................................................................................
35
The Secant Method:
...............................................................................................................................
40
Newton's Method and the Roots of Polynomials:
................................................................................
43
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View Full Document EE103 Lecture Notes, Winter 2009, Prof S. Jacobsen
Section 3
31
SECTION 3: ROOTS OF AN EQUATION OF A SINGLE VARIABLE
In this section we concern ourselves with an introduction to finding a root of the equation
() 0
fx
=
where we assume that f is a continuous function and
x
is a scalar.
We've already seen
the bisection algorithm that, generally, is excellent for finding an interval of reasonably
small length in which we are assured that a root is present.
Of course, the word "root" is
nothing more than an expression for a "solution" of the equation.
That is,
x
is said to be
a solution
or root
of the above equation if
=
.
Fixed Point Approach (Method of Successive Approximations):
The fixed point or successive approximation method is one that's important to present
because it provides a method of analysis for other methods, including Newton's method,
one of the best.
The fixed point method assumes that the equation to be solved is
()
x
gx
=
where g is a continuous function.
Note that if we take ( )
( )
f
xx
g
x
=
−
, the problem is
one of finding a root of the equation ( )
0
=
.
Ex:
Assume
5
2 1
f
x
=−+
.
We can think of finding a root of this polynomial as a
fixed point problem by writing it as
5
1
2
x
x
+
=
where, of course,
5
() (
1
)
/2
x
=+
.
The idea of the fixed point method, or method of successive approximations, is nothing
more than to successively apply the following operation
x
←
That is, we select, say,
0
x
and we compute
EE103 Lecture Notes, Winter 2009, Prof S. Jacobsen
Section 3
32
10
()
x
gx
=
,
21
x
=
32
x
=
………….
1
kk
x
+
=
Therefore, if the sequence { }
k
x
converges to say
*
x
, i.e.,
*
k
x
x
→
, we have, by continuity
of the function g
*1
*
lim
lim (
)
(lim
)
( )
k
k
x
x
g
xg
x
+
→∞
→∞
→∞
==
=
=
That is, if the sequence
*
k
x
x
→
it must be the case that
*
x
is a solution of the equation
x
=
and hence
*
x
is a fixed point.
Figure 1: Example of Fixed Point Method
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Section 3
33
Figure 2: Typical sequence of iterates of FP method
Figure 3: Example of nonconvergenc of FP method
EE103 Lecture Notes, Winter 2009, Prof S. Jacobsen
Section 3
34
Therefore, we have examples that demonstrate that the fixed point method may produce
solutions and may not, even when there are fixed points.
Moreover, as seen above, the
fixed point method, when it does find a fixed point, may not find a fixed point that is the
nearest to the starting point.
The reasons will become clear when as we address the rate
of convergence issue.
That is, when convergence does occur we can develop a result for
the rate of convergence.
Assume
k
x
x
→
and assume ''
g
, the second derivative, exists and is continuous.
Let
kk
exx
=−
denote the error at the
th
k
iteration.
Then Taylor's theorem states that we can
represent, exactly,
(
)
k
gx
by the first three terms of the Taylor expansion about the
point
x
; the last term is, of course, the socalled remainder term.
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This note was uploaded on 03/30/2009 for the course EE 103 taught by Professor Vandenberghe,lieven during the Winter '08 term at UCLA.
 Winter '08
 VANDENBERGHE,LIEVEN

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