olortegui 2.pdf

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Numerical Methods Applied to Mechatronics

Lecture No 3Escuela de Ingeniería Mecatrónica

Universidad Nacional de Trujillo

ROUNDOFF AND TRUNCATIONERRORS

Dr. Jorge A. Olortegui Yume Ph.D.



Roudoff and Truncation Errors Dr. Jorge A. Olortegui Yume Ph.D.2

COURSE PROJECTSThe dimensions of the planar mechanisms, shown in the Figures are given in theTables which go along with them. The angle of the driver link 1 with the horizontal

axis is φ . The constant angular speed of the driver link 1 isn

and is given in thetables.




COURSE PROJECTS




PROJECT No 1




PROJECT No 2




PROJECT No 3




ACCURACY AND PRECISION

Accuracy

How closely computed /measured value agreeswith true value

Precision (Reproducibility)

How closely computed /measured values agreewith each other.

inaccurate

and imprecise

accurate andimprecise

inaccurate andprecise

accurate andprecise




ERROR DEFINITIONS

• True error ( E t): Difference between true

value and approximation.

• Absolute error (| E t |): Absolutedifference between true value andapproximation.

• True fractional relative error: True errordivided by the true value.

• Relative error ( t): True fractionalrelative error in percentage.

approxtruet V V E

approxtruet V V E

true

approxtrue

true

t fract t V

V V

V E

100100 true

approxtrue

true

t t V

V V

V E




ERROR DEFINITIONSActual Error Determination

•

Previous definitions rely on knowing a true value• Having a true value is illogical/rare

approximations to true valuee.g. : the relative error ( t) is approximated as:

• Challenge: Find adequate “ E approx .”

at

100

approx

approxa V

E

100100 approx

approxtrue

approx

approxa

V

V V

V

E

Don´t know “V true ”




ERROR DEFINITIONSActual Error Determination (cont´d)

• Challenge: Find adequate “ E approx .”

Example: Iterative calculations

100

present

previous present a V

V V




ERROR DEFINITIONSStoping Criterion

Repeat calculations until:

Where: s : Prespecified error tolerance

sa




ERROR DEFINITIONSRoundoff Error• Roundoff errors arise because digital computers

cannot represent some quantities exactly. Thereare two major facets of roundoff errors involved

in numerical calculations: – Digital computers have size and precision limits on

their ability to represent numbers. – Certain numerical manipulations are highly sensitive

to roundoff errors.

TRABAJO DOMICILIARIO




ERROR DEFINITIONSTruncation Errors• Result from using an approximation in place of an exact mathematical procedure.

Example 1 : Approximation to a derivative using a finite-difference equation:

dv

dt

v

t

v(t i 1) v(t i )

t i 1 t iGain insight in error calculation use Taylor s Series.

Taylor´s theorem: Any smooth function can be approximated as apolynomial.

f xi 1 f xi f ' xi h f '' xi

2!h 2 f (3) xi

3!h 3

f (n ) xi n!

h n Rn

Rn : Remainder




ERROR DEFINITIONSTruncation ErrorsExample 1 (cont´d):

Taylor´s theorem:

f xi 1 f xi f ' xi h f '' xi

2!h 2 f (3) xi

3!h 3

f (n ) xi n!

h n Rn

• Usually, the nth order Taylorseries expansion will be exactfor an nth order polynomial.

• In other cases, the remainderterm Rn is of the order of hn+1 ,meaning:

– More terms used less error – Smaller spacing, smaller

error for a given number ofterms.




ERROR DEFINITIONSTruncation ErrorsExample 1 (cont´d): Applying to the case of velocity

Taylor´s theorem:

nn

ni

nii

iii Rt

ndt

t vd

t dt t vd

t dt t vd

t dt

t dvt vt v

!!3!2

33

3

22

2

1

t R

t n

dt t vd

t dt t vd

t dt t vd

dt t dv

t t vt v nn

ninii

iii 123

3

2

2

1

!!3!2

ii t t t Donde 1:

t t vt v

t R

t ndt

t vd

t dt

t vd

t dt

t vd

dt t dv

iinnn

in

ii

i 112

3

3

2

2

!!3!2

t

t vt vt O

dt t dv iii 12n For




ERROR DEFINITIONSTruncation ErrorsExample 1 (cont´d): Applying to the case of velocity

Taylor´s theorem: ii

iii

t t t vt v

t Odt

t dv

1

1

TRUNCATION ERROR with order of magnitude “ t ”

Note that TRUNCATION ERROR depends on:

• “n”• “ t ”• Whether we have approxs. for 2nd, 3rd, … derivatives of function




ERROR DEFINITIONSTruncation ErrorsApplication: Numerical Differentiation

• The first order Taylor series can be used to calculateapproximations to derivatives:

– Given:

– Then:

This is termed a “forward” difference because it utilizes data ati and i +1 to estimate the derivative.

f ( xi 1) f ( xi ) f '( xi )h O(h 2 )

f '( xi ) f ( xi 1) f ( xi)

hO(h)




ERROR DEFINITIONS

Truncation ErrorsApplication: Numerical Differentiation

• There are also backward difference and centereddifference approximations, depending on thepoints used:

• Forward:

• Backward:

•

Centered:

f '( xi ) f ( xi 1) f ( xi)h O(h)

f '( xi) f ( xi ) f ( xi 1)

hO(h)

f '( xi ) f ( xi 1) f ( xi 1)

2hO(h 2 )




ERROR DEFINITIONS

Other Error types

• Blunders - errors caused by malfunctions of the computer orhuman imperfection.

• Model errors - errors resulting from incomplete mathematicalmodels.

• Data uncertainty - errors resulting from the accuracy and/or

precision of the data.

olortegui 2.pdf

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