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Truncation Error and the Taylor Series

Lecture NotesDr. Rakhmad Arief SiregarUniversiti Malaysia Perlis

Applied Numerical Method for Engineers

Chapter 4

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Background

Truncation error are those that results from using an approximation in place of an exact mathematical procedure.

A truncation error was introduce into numerical solution because difference equation only approximates the true value of the derivative

In order to gain insight into the properties of truncation error, the Taylor function is used

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Taylor’s Theorem

If the function f and its first n+1 derivatives are continuous on an interval containing a and x, then the value of the function at x is given by

where remainder Rn is defined as

2)(!2

)())(()()( ax

afaxafafxf

...)(!3

)( 3)3(

axaf

nn

n

Raxn

af )(

!

)()(

x

a

nn

n dttfn

txR )(

!

)( )1(

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Taylor’s Theorem

It is often convenient to simplify the Tailor series by defining a step size h = xi+1 - xi

where remainder Rn is defined as

is a value of x that lies somewhere between xi and xi+1

This value will be discussed later

nni

nii

iii Rhn

xfh

xfh

xfhxfxfxf

!

)(...

!3

)(

!2

)()()()(

)(3

)3(2

1

1)1(

)!1(

)(

n

n

n hn

fR

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Ex. 4.1 Taylor Approximation of a Polynomial

Use zero-through fourth-order Taylor series expansion to approximate the function

From xi = 0 with h = 1. That is, predict the function’s value at xi+1 = 1

2.125.05.015.01.0)( 234 xxxxxf

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Ex. 4.1 Taylor Approximation of a Polynomial

Solution For x = 0 then f(0) = 1.2 For x = 1 then f(1) = 0.2 this is the true that we are

trying to predict. Taylor series approximation with n = 0 Truncation error: Et = true value – approximation

Et = 0.2 -1.2 = -1.0 at x= 1 n = 1, the first derivative f’(0) = -0.4(0)3- 0.45(0)2 - 1.0(0) -0.25 = -0.25 Taylor series approximation with n = 1

2.125.05.015.01.0)( 234 xxxxxf

2.1)()( 1 ii xfxf

195.025.02.1)()( 1 hifhxfxf ii

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Ex. 4.1 Taylor Approximation of a Polynomial

Taylor series approximation with n = 1 Truncation error: Et = true value – approximation

Et = 0.2 -0.95 = -0.75 at x= 1 n = 2, the second derivative f’’(0) = -1.2(0)2- 0.9(0) - 1.0 = -1.0 Taylor series approximation with n = 2

Truncation error: Et = true value – approximation

Et = 0.2 -0.45 = -0.25 at x= 1

145.05.025.02.1)( 21 hifhhxf i

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Taylor series expansion

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Ex. 4.2

Use Taylor series expansion with n = 0 to 6 to approximate f(x) = cos x, at xi+1 = /3 on the bases of the value of f(x) and its derivatives at xi = /4.

Note this means that h = /3 - /4 = /12.

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Ex.4.1x

Find the truncation error in approximating the function

(a)

(b)

(c)

(d)

over the range 0 x 1

xxy )(12

2 2

1)( xxxy

4323 4

1

3

1

2

1)( xxxxxy

323 3

1

2

1)( xxxxy

)1ln()(2 xxy

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Ex.4.1x (Solution)

We consider a representative value of x, x = 0.5, and find the truncation error. The exact value of the function is given by

(a)

(b)

5.0)5.0(1 y

375.0)5.0(2

1)5.0()5.0( 2

2 y

405465108.0)5.1ln()1ln()(2 xxy

0094544892.05.0405465108.0 eT

030465108.0375.0405465108.0 eT

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Ex.4.1x (Solution)

We consider a representative value of x, x = 0.5, and find the truncation error. The exact value of the function is given by

(c)

(d)

416666667.0)5.0(3

1)5.0(

2

1)5.0()5.0( 32

3 y

405465108.0)5.1ln()1ln()(2 xxy

011201559.0416666667.0405465108.0 eT

004423441.0401041667.0405465108.0 eT

401041667.0)5.0(4

1)5.0(

3

1)5.0(

2

1)5.0()5.0( 432

4 y

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Truncation error

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1 1.2

y(x)

Te1

Te2

Te3

Te4

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Error Propagation

The purpose of this section is to study how error in numbers can propagate through mathematical functions

Assuming is an approximation of x, we would like to assess the effect of the discrepancy between x and on the value of the function.

We estimate

We can use Taylor can be employed, why?

Dropping the second and high order terms and rearranging yields

x~x~

)~()()~( xfxfxf

...)~(2

)~()~)(~()~()(

xx

xfxxxfxfxf

)~)(~()~()~()( xxxfxfxfxf

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Error Propagation

Dropping the second and high order terms and rearranging yields

Or can be rewrite as:

where: an estimate of the error of the function

an estimate of the error of x

)~)(~()~()~()( xxxfxfxfxf

)~()~()~( xxxfxf

)~()~()~( xxxfxf

xxx ~~

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Graphical depiction of the first-order error propagation

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Error Propagation in a function of a single variable

Given a value of = 2.5 with an error of = 0.01, estimate resulting error in the function f(x) = x3

Solution

We predict for = 2.5

x~

)~()~()~( xxxfxf

x~

1875.0)01.0()5.2(3)~( 2 xf

625.15)5.2( f

x~

1875.0625.15)5.2( f

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Function of more than one variable

For n independent variable having error following general relationship holds

nn

n xx

fx

x

fx

x

fxxxf ~...~~)~,...,~,~( 2

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121

nxxx ~,...,~,~ 21

nxxx ~,...,~,~ 21

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Ex. 4.6x

Modified the question

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Total Numerical Error

Total Numerical error is the summation of the truncation and round-off errors

To minimize round error is to increase the number of significant figure of the computer

The truncation error can be reduce by decreasing the step size

However the truncation errors are decrease as the round-off errors are increase

In actual cases, such situation relatively uncommon because most computer carry enough significant figures that round-off error do not redominate

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Blunders

Blunders or gross error could attributed to human imperfection

Occurs in computer programs, also can occurs in any stage of mathematical modeling.

Blunders are usually disregarded in discussion of numerical methods

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Formulation Errors

Formulation or model error relate to incomplete mathematical models.

Ex. A negligible formulation error is the fact that Newton’s second law does not account for relativistic problem.

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Data uncertainty

Errors sometimes enter into an analysis because of uncertainty in the physical data

Ex. In the case of falling parachutist Our sensor of velocity can overestimate

the velocity, etc