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MATH 174:MATH 174:
NUMERICAL ANALYSISNUMERICAL ANALYSISLECTURE 4
1st Semester AY 2010-2011
Lecturer: Jomar Fajardo Rabajante
IMSP, UPLB
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MEASURING QMEASURING Q--CONVERGENCECONVERGENCE
y Used for iterative methods (such asroot-finding)
y Deals with sequencesy Does not apply to discretization
methods (e.g. numerical quadrature,
solutions of ODEs)
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MEASURING QMEASURING Q--CONVERGENCECONVERGENCE
y Given an asymptotic error constant >0,the sequence {xk}
converges to x* withorder p>0 iff
p
kk
p
kk
p
k
k
k
xxxxor
xxxxor
xx
xx
*
1
*
*
1
*
*
1
*
lim
e
}
!
gp
P
P
P
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MEASURING QMEASURING Q--CONVERGENCECONVERGENCE
y Given an asymptotic error constant >0,the sequence {xk}
converges to x* withorder p>0 iff
p
kk
p
kk
p
k
k
k
xxxxor
xxxxor
xx
xx
*
1
*
*
1
*
*
1
*
lim
e
}
!
gp
P
P
P
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MEASURING CONVERGENCEMEASURING CONVERGENCE
When p=1
y If 0
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SOME a posteriori ERRORSOME a posteriori ERROR
ANALYSISANALYSIS
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ABSOLUTE & RELATIVE ERRORABSOLUTE & RELATIVE ERROR
TV
AV
TV
AVTVRE
AVTVAE
!
!
!
1
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FORWARD & BACKWARDFORWARD & BACKWARD
ANALYSISANALYSIS
%36.037.26.5
%69.16169.56.56169.5,6.5assume
digits,2use,37.2,. 2
}!
!!
!!
!!
!
!
backw
fwd
exactcalc
calcbackw
exactcalcfwd
E
Eyy
xxyExxxE
yyE
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LOSS OF SIGNIFICANT DIGITLOSS OF SIGNIFICANT DIGIT
ERRORERRORy The number x is said to approximate xT
to m significant digits ifm is the largest
nonnegative integer for which
44
34
10510004.4999.0
9986.0999.0
10510012.6998.0
9986.0998.0.
105
vv!
vv!
v
Ex
x
xx mT
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MAD, MAPD, MAPE, MSE,MAD, MAPD, MAPE, MSE,
Bias, Tracking SignalBias, Tracking Signaly Usually used for
Time Series forecasting
y Not applicable to interpolation
Len n be the number of periods
ni
TV
AVTVMAPD
nin
AVTVMAD
i
ii
ii
,...2,1,
,...2,1,
!
!
!
!
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Len n be the number of periods
ni
n
AVTVMSE
nin
TV
AVTV
MAPE
i
i
ii
,...2,1,
,...2,1,
2
!
!
!
!
MAD, MAPD, MAPE, MSE,MAD, MAPD, MAPE, MSE,
Bias, Tracking SignalBias, Tracking Signal
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Len n be the number of periods
MAD
AVTV
MAD
RSFETS
n
AVTV
nRSFEerrorAveBias
ii
ii
!!
!!).(
MAD, MAPD, MAPE, MSE,MAD, MAPD, MAPE, MSE,
Bias, Tracking SignalBias, Tracking Signal
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Standard Error of the Estimate,Standard Error of the
Estimate,
Correlation Coefficient & CoefficientCorrelation Coefficient
& Coefficient
of Determination (rof Determination (r22))
!
!
!
2222
2
,
2
,
2
2
)(
yynxxn
yxxynr
n
xybyayS
n
yyS
xy
c
xy
ForRegression
Lines
See CurveExpert for more information
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Stopping CriterionStopping Criterion
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Terminate when:Terminate when:
tolx
xx
tol
xx
fxf
k
kk
kk
tolk
e
e
e
1
1
1
)(
May meaniteration is
converging
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Terminate when:Terminate when:
tolx
xx
tol
xx
fxf
k
kk
kk
tolk
e
e
e
1
1
1
)(
May meaniteration is
converging
If tol=10-n, then xk+1 should approximate xT
to n decimal places/n significant digits
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Terminate when:Terminate when:
bigk
bigk
xxfxf
kk
u
u
u
)(
maxIterations have gone on
long enough
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CaveatsCaveats
gpgp
gpp
!
kxbut
kk
xx
k
kk
as,
,as,01
1
1
1
Notice that
tolxxuse
kxLet
kk
k
e
!
1
1...
3
1
2
11
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NORMNORM
y We also use the concept of norms as
stopping criteria (e.g. solution to systems
of linear equations)
y But we are going to delay the discussion
on norms (we will discuss them on
Chapter 2), and we will leave the
discussion on the use of norms asstopping criteria in Math
175.
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ASSIGNMENTASSIGNMENT
yGive instances when to use
the different stopping criteria
yState the WeierstrassApproximation Theorem
CHAPTER 2 NA TAYO NEXT
MEETING!!!
