Lecturer Jomar Fajardo Rabajante IMSP UPLB 2 nd Sem AY 20122013 httpwwwyoutubecomwatchvhhT25CO6wDI Theorem Assume that r is a zero of the differentiable function ID: 377417
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Slide1
MATH 175: Numerical Analysis II
Lecturer: Jomar Fajardo Rabajante
IMSP, UPLB
2
nd
Sem
AY
2012-2013Slide2
http://www.youtube.com/watch?v=hhT25CO6wDI
____________________________________________
Theorem: Assume that r is a zero of the differentiable function f. Then if but then f has a zero of multiplicity q at r. The root is called simple if the multiplicity is one.
5th Method: Newton’s Method/Newton-Raphson IterationSlide3
5
th
Method: Newton’s Method/Newton-Raphson Iteration
Newton’s Method converges, but is slowing down near x=0.
y=x^2, has root of multiplicity 2.Slide4
Assume that Newton’s method converges to the root r
. Usually, Newton’s method is
quadratically (fast) convergent if the root r is simple. If the root is NOT simple Newton’s method converges linearly.Newton’s Method Convergence Theorem: Assume that the (q+1)-times continuously differentiable function f on [a,b]
has a multiplicity q root at rє
[a,b]. Then Newton’s method is locally convergent to r, and the error error
k at iteration k satisfies5th
Method: Newton’s Method/Newton-Raphson IterationSlide5
Example: If q=3
Hence,
Near convergence, this means that the error will decrease by 2/3 on each iteration (O((2/3)^k). 5th Method: Newton’s Method/Newton-Raphson Iteration
For the previous methods, you can interpret lambda same
as this.
This interpretation is not applicable when lambda=0Slide6
If f is (
q
+1)-times continuously differentiable on [a,b], which contains a root r of multiplicity q>1, then Modified Newton’s method converges locally and quadratically to r.
5th Method: Newton’s Method/Newton-Raphson IterationSlide7
Assuming Newton’s method is applied to a function with a zero of multiplicity
q>1
. The multiplicity of the zero can be estimated as the integer nearest to5th
Method: Newton’s Method/Newton-Raphson
Iteration
Apply this only if we do not know
q
.Slide8
Example: Newton’s Convergence Theorem guarantees convergence to a simple zero if
f
is twice differentiable and if we start the iteration sufficiently close to the zero.An iterative method is called locally convergent to r if the method converges to r for initial guesses sufficiently close
to r.Slide9
Another Failure of Newton’s MethodSlide10
Stopping Criterion:
If Newton’s method is linearly (i.e. not
quadratically) convergent use (same as Regula Falsi)5
th Method: Newton’s Method/Newton-
Raphson IterationSlide11
Stopping Criterion:
If Newton’s method is
superlinearly (e.g. quadratically) convergent or if we do not know the order of convergence use5
th Method: Newton’s Method/Newton-
Raphson IterationSlide12
Definition:An iteration of the form:
is called a
fixed point (or functional) iteration.And any x* such that is called a fixed point of g.A GENERAL METHOD: FIXED POINT ITERATIONSlide13
Questions:
Is secant method a fixed point iteration?
Is Newton’s method a fixed point iteration?We are going to study the generalization of Secant and Newton’s methods and look at the origin of the sufficient conditions for the methods to converge to a zero of a function (e.g. Newton’s Method Convergence Theorem was proved using the Fixed Point Theorem)… Actually, everything that is true for Fixed Point Iteration is applicable to Secant and Newton’s Method.A GENERAL METHOD: FIXED POINT ITERATIONSlide14
How to generate a fixed point iteration for solving roots?
Example:
1st Method: Use Secant or Newton’s Transform Newton’s Transform:A GENERAL METHOD: FIXED POINT ITERATIONSlide15
2nd Method:
Separate
x in the equation. Examples: A GENERAL METHOD: FIXED POINT ITERATIONSlide16
2nd Method:
What is the idea of this method?
A GENERAL METHOD: FIXED POINT ITERATION
Solving the zero of
Solving the intersection of Slide17Slide18
Fixed point:
y-value is equal to x-valueSlide19
x-value of the fixed point is the root of the original functionSlide20
Exercise:Let us go back to our original equation:
From
Derive this fixed point formula: A GENERAL METHOD: FIXED POINT ITERATIONSlide21
Question: Can every equation f(x)=0 be turned into a fixed point problem?
YES, and in many different ways…
However, not all are converging!!! And actually, not all has fixed points (no intersection with y=x)!!!A GENERAL METHOD: FIXED POINT ITERATIONSlide22Slide23Slide24Slide25Slide26Slide27Slide28
How to do the iterations work? THE COBWEB DIAGRAM (Geometric representation of FPI)
draw a line segment vertically to the function
then draw a horizontal line segment to the diagonal y=x.repeat.A GENERAL METHOD: FIXED POINT ITERATIONSlide29
DIVERGING!!!
FPI fails!Slide30
What if we start here?Slide31
Intuitively, what do you think is the condition for convergence? (Hint: look at the slopes)
A GENERAL METHOD: FIXED POINT ITERATION
CONVERGING!Slide32
Slope=1
|Slopes|>1
|Slopes|<1
|Slopes|<1|Slopes|>1Slide33
FIXED POINT THEOREM: Let the iteration function
g
be continuous on the closed interval [a,b] with g:[a,b][a,b]
. Furthermore, suppose that g is differentiable on the open interval (a,b
) and there exists a positive constant K<1 (capital letter K)
such that |g’(x)|<K<1 for all xє
(a,b). ThenThe fixed point x* in
[
a,b
]
exists and is unique.
The sequence
{
x
k
} generated by x
k
=g(x
k-1
)
converges to
x*
for any
x
0
є
[
a,b
]
.
A GENERAL METHOD: FIXED POINT ITERATION
Assignment: PROVE!!!Slide34
Example:
x
k=sqrt(xk-1) on [0.3,2]. We should be sure that there is a fixed point in [0.3,2]1. g(x)=sqrt(x), continuous on [0.3,2]?
YES!2. g:[0.3,2][0.3,2]? YES!
Why? This square root function is monotonically increasing. sqrt(0.3)=0.5477… and
sqrt(2)=1.4142... So [sqrt(0.3),sqrt(2)] is a subset of [0.3,2].
3. |g’(x)|=0.5/sqrt(x)<1 for all x in [0.3,2]?
YES!
Why?
|g’(x)| is monotonically decreasing so |0.5/
sqrt
(0.3)|=0.91287…<1. Actually, K=0.5/
sqrt
(0.3)
Hence, the fixed point in [0.3,2] exists and is unique. And if you use any starting value in [0.3,2], the iteration will converge to the fixed point.
Try [0.1,2] would the theorem hold?Slide35
0.547723
0.740083
0.8602810.9275130.963075
0.981364
0.9906380.995308
0.9976510.998825
0.999412
0.999706
0.999853
0.999927
0.999963
0.999982
0.999991
0.999995
0.999998
0.999999
0.999999
1
1Slide36
Note that the hypotheses of the fixed point theorem are sufficient conditions for convergence of the iteration scheme, but not necessary.
However, we may add
for all xє(a,b): if |g’(x)|>1, then the iteration diverges. If |g’(x)|=1, no conclusion can be made.If
|g’(x)|<1, then the fixed point is said to be attracting.If |g’(x)|>1, then the fixed point is said to be repelling.
A GENERAL METHOD: FIXED POINT ITERATIONSlide37
Assume that the hypotheses of the fixed point theorem are met, also assume
g’
is continuous on (a,b). If g’(x*)≠0, then for any starting value in [a,b], the iteration will converge only linearly to the fixed point. Example: try to get g’(x) consider [a,b]=[-1,1]
A GENERAL METHOD: FIXED POINT ITERATIONSlide38
To obtain a higher-order convergence, the iteration function must have a zero derivative at the fixed point. The more derivatives of the iteration function which are zero at the fixed point, the higher will be the order of convergence.
A GENERAL METHOD: FIXED POINT ITERATIONSlide39
Stopping Criterion:
If fixed point iteration is linearly convergent use
A GENERAL METHOD: FIXED POINT ITERATIONSlide40
Stopping Criterion:
If fixed point iteration is
superlinearly convergent or if we do not know the order of convergence use
A GENERAL METHOD: FIXED POINT ITERATIONSlide41
ANOTHER FIXED-POINT TRANSFORM:
HALLEY’S METHOD
(has cubic order of convergence for simple zeros)Halley’s Method is under the group of Householder’s Methods for root-finding.A GENERAL METHOD: FIXED POINT ITERATIONSlide42
Fixed Point Iteration has many other applications other than root-finding.
It is also used in the analysis of Discrete Dynamical Systems leading to the study of Chaos.
A GENERAL METHOD: FIXED POINT ITERATION