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MEC
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MEC
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LectureLecture
44Numerical Methods for EngineeringNumerical Methods for Engineering
MECN 3500 MECN 3500
Professor: Dr. Omar E. Meza CastilloProfessor: Dr. Omar E. Meza [email protected]
http://www.bc.inter.edu/facultad/omeza
Department of Mechanical EngineeringDepartment of Mechanical Engineering
Inter American University of Puerto RicoInter American University of Puerto Rico
Bayamon CampusBayamon Campus
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Tentative Lectures ScheduleTentative Lectures Schedule
TopicTopic LectureLecture
Mathematical Modeling and Engineering Problem SolvingMathematical Modeling and Engineering Problem Solving 11
Introduction to MatlabIntroduction to Matlab 22
Numerical ErrorNumerical Error 33
Root FindingRoot Finding 33
System of Linear EquationsSystem of Linear Equations
Least Square Curve FittingLeast Square Curve Fitting
Polynomial Interpolation Polynomial Interpolation
Numerical IntegrationNumerical Integration
Ordinary Differential Equations Ordinary Differential Equations
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Taylor TheoremTaylor Theorem
Truncation Errors and the Truncation Errors and the Taylor SeriesTaylor Series
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To understand the use of Taylor Series in To understand the use of Taylor Series in the study of numerical methods.the study of numerical methods.
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Course ObjectivesCourse Objectives
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Truncation Errors: Truncation Errors: use approximation in use approximation in place of an exact mathematical procedure.place of an exact mathematical procedure.
Numerical Methods express functions in an Numerical Methods express functions in an approximate fashion: approximate fashion: The Taylor Series.The Taylor Series.
What is a Taylor Series?What is a Taylor Series?Some examples of Taylor series which you Some examples of Taylor series which you must have seenmust have seen
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IntroductionIntroduction
!6!4!2
1)cos(642 xxx
x
!7!5!3
)sin(753 xxx
xx
!3!2
132 xx
xe x
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The general form of the Taylor series is The general form of the Taylor series is given bygiven by
provided that all derivatives of f(x) are provided that all derivatives of f(x) are continuous and exist in the interval [x,x+h] continuous and exist in the interval [x,x+h]
What does this mean in plain English?What does this mean in plain English?
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General Taylor SeriesGeneral Taylor Series
32
!3!2h
xfh
xfhxfxfhxf
As Archimedes would have said, “Give me the value of the function at a single point, and the value of all (first, second, and so on) its derivatives at that single point, and I can give you the value of the function at any other point” (fine print
excluded)
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Example: Example: Find the value of f(6) given that Find the value of f(6) given that f(4)=125, f’(4)=74, f’’(4)=30, f’’’(4)=6 and f(4)=125, f’(4)=74, f’’(4)=30, f’’’(4)=6 and all other higher order derivatives of f(x) at all other higher order derivatives of f(x) at x=4 are zero.x=4 are zero.
Solution: Solution: x=4, x+h=6 x=4, x+h=6 h=6-x=2 h=6-x=2
Since the higher order derivatives are zero,Since the higher order derivatives are zero,
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General Taylor SeriesGeneral Taylor Series
!3
24
!2
2424424
32
fffff
!3
26
!2
2302741256
32
f
860148125 341
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(x(xi+1i+1-x-xii)= h)= h step sizestep size (define first) (define first)
Reminder term, RReminder term, Rnn, accounts for all terms , accounts for all terms from (n+1) to infinity.from (n+1) to infinity.
99
nn
ii
n
iiiiiii
Rxxn
f
xxf
xxxfxfxf
)(!
)(!2
))(()()(
1
)(
2111
)1()1(
)!1(
)(
n
n
n hn
fR
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Zero-order approximationZero-order approximation
First-order approximationFirst-order approximation
Second-order approximationSecond-order approximation
1010
)x(f)x(f i1i
)xx)(x(f)x(f)x(f i1iii1i
2i1ii1iii1i )xx(
!2
f)xx)(x(f)x(f)x(f
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Example 4.1: Taylor Series Approximation Example 4.1: Taylor Series Approximation of a polynomial of a polynomial Use zero- through fourth-Use zero- through fourth-order Taylor Series approximation to order Taylor Series approximation to approximate the function:approximate the function:
From xFrom xii=0 with h=1. That is, predict the =0 with h=1. That is, predict the function’s value at xfunction’s value at xi+1i+1=1=1
f(0)=1.2f(0)=1.2
f(1)=0.2 -f(1)=0.2 - True value True value
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The Taylor SeriesThe Taylor Series
2.1x25.0x5.0x15.0x1.0xf 234
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Zero-order approximationZero-order approximation
First-order approximationFirst-order approximation
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The Taylor SeriesThe Taylor Series
0.12.12.0E
ionapproximatvalueTrueE
2.1)1(f
2.1)0(fxfxf
t
t
i1i
75.095.02.0E
95.0)1(25.02.1h)0('f0f1f
25.0)0('f
25.0x1x45.0x4.0)x('f
h)x('fxfxf
t
23i
ii1i
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Second-order approximationSecond-order approximation
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The Taylor SeriesThe Taylor Series
25.045.02.0E
45.012
1)1(25.02.1
!2
h)0(''fh)0('f0f1f
1)0(''f
1x9.0x2.1)x(''f
!2
h)x(''fh)x('fxfxf
t
22
2i
2i
ii1i
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Third-order approximationThird-order approximation
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The Taylor SeriesThe Taylor Series
1.03.02.0E
3.06
19.01
2
1)1(25.02.1
!3
h)0('''f
!2
h)0(''fh)0('f0f1f
9.0)0(''f
9.0x4.2)x('''f!3
h)x('''f
!2
h)x(''fh)x('fxfxf
t
32
32
i
3i
2i
ii1i
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Fourth-order approximationFourth-order approximation
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The Taylor SeriesThe Taylor Series
02.02.0E
2.024
14.2
6
19.01
2
1)1(25.02.1
!4
h)0(f
!3
h)0('''f
!2
h)0(''fh)0('f0f1f
4.2)0(f
4.2)x(f
!4
h)x(f
!3
h)x('''f
!2
h)x(''fh)x('fxfxf
t
432
4iv32
iv
iiv
4i
iv3i
2i
ii1i
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The Taylor SeriesThe Taylor Series
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If we truncate the series after the first If we truncate the series after the first derivative termderivative term
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Taylor Series to Estimate Truncation ErrorsTaylor Series to Estimate Truncation Errors
nn
i1i
)n(
2i1ii1iii1i
R)tt(!n
v
)tt(!2
''v)tt)(t('v)t(v)t(v
1i1iii1i R)tt)(t('v)t(v)t(v
)tt(
R
)tt(
)t(v)t(v)t('v
i1i
1
i1i
i1ii
First-order approximation Truncation Error
)tt(O i1i
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Forward Difference ApproximationForward Difference Approximation
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Numerical DifferentiationNumerical Differentiation
)xx(O)xx(
)x(f)x(f)x('f i1i
i1i
i1ii
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Backward Difference ApproximationBackward Difference Approximation
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Numerical DifferentiationNumerical Differentiation
)h(Oh
)x(f)x(f)x('f 1iii
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Centered Difference ApproximationCentered Difference Approximation
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Numerical DifferentiationNumerical Differentiation
)h(Oh2
)x(f)x(f)x('f 21i1ii
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Example 4.4: Example 4.4: To find the forward, backward To find the forward, backward and centered difference approximation for and centered difference approximation for f(x) at x=0.5 using step size of h=0.5, f(x) at x=0.5 using step size of h=0.5, repeat using h=0.25. The true value is -repeat using h=0.25. The true value is -0.91250.9125
h=0.5h=0.5
xxi-1i-1=0 -=0 - f(x f(xi-1i-1)=1.2)=1.2
xxii=0.5 -=0.5 - f(x f(xii)=0.925)=0.925
XXi+1i+1=1 -=1 - f(x f(xi+1i+1)=0.2)=0.2
2323
The Taylor SeriesThe Taylor Series
25.0x1x45.0x4.0x'f
2.1x25.0x5.0x15.0x1.0xf23
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Forward Difference ApproximationForward Difference Approximation
Backward Difference ApproximationBackward Difference Approximation
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The Taylor SeriesThe Taylor Series
%9.58
45.15.0
925.02.0
h
)x(f)x(f)5.0('f
t
i1i
%7.39
55.05.0
2.1925.0
h
)x(f)x(f)5.0('f
t
1ii
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Centered Difference ApproximationCentered Difference Approximation
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The Taylor SeriesThe Taylor Series
%6.9
15.0
2.12.0
h
)x(f)x(f)5.0('f
t
1i1i
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h=0.25h=0.25
xxi-1i-1=0.25 -=0.25 - f(x f(xi-1i-1)=1.10351563)=1.10351563
xxii=0.5 -=0.5 - f(x f(xii)=0.925)=0.925
XXi+1i+1=0.75 -=0.75 - f(x f(xi+1i+1)=0.63632813)=0.63632813
Forward Difference ApproximationForward Difference Approximation
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The Taylor SeriesThe Taylor Series
%5.26
155.125.0
925.063632813.0
h
)x(f)x(f)5.0('f
t
i1i
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Backward Difference ApproximationBackward Difference Approximation
Centered Difference ApproximationCentered Difference Approximation
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The Taylor SeriesThe Taylor Series
%4.2
934.025.0
10351563.163632813.0
h
)x(f)x(f)5.0('f
t
1i1i
%7.21
714.025.0
10351563.1925.0
h
)x(f)x(f)5.0('f
t
1ii
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Homework3 Homework3 www.bc.inter.edu/facultad/omeza
Omar E. Meza Castillo Ph.D.Omar E. Meza Castillo Ph.D.
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