ES 240: Scientific and Engineering Computation. Chapter 4 Chapter 4: Errors Uchechukwu Ofoegbu Temple University.

ES 240: Scientific and Engineering Computation. Chapter 4

Chapter 4: Errors

Uchechukwu Ofoegbu

Temple University


Murphy's Laws of Computing

1. When computing, whatever happens, behave as though you meant it to happen.

2. For every action, there is an equal and opposite malfunction.

3. To err is human... to blame your computer for your mistakes is even more than human--it's downright natural.

4. He who laughs last probably made a backup.

5. If at first you don't succeed, blame your computer.

6. The number one cause of computer problems is computer solutions.

7. A computer program will always do what you tell it to do, but rarely what you want to do.


Accuracy and PrecisionAccuracy and Precision

Accuracy– how closely a computed or measured value agrees with

the true value

Precision – how closely individual computed or measured values

agree with each other.


Error DefinitionsError Definitions

True error:– Et = True Value - Approximation

Absolute error:– |Et|

True fractional relative error:–

Relative error–

ValueTrue

Et

_

%100_ValueTrue

Ett


Error ApproximationError Approximation

The approximate percent relative error

–

For iterative processes, the error can be approximated as the difference in values between successive iterations.

%100_

_

valuetrue

errorionapproximata

%100_

__

ionapproximatpresent

ionapproximatpreviousionapproximatpresenta


Using Error EstimatesUsing Error Estimates

Stopping Criterion– The sign of the error is not the problem– We just want the error to be less than a specific value

– The computation is repeated until | a |< s

Correct to at least n significant figures– set )%105.0( 2 n

s


Error Estimates - ExampleError Estimates - Example

Write an m-file that – approximates cos(x) using 1-10 terms– plots the relative error versus the number of terms

Determine the stopping criterion that will ensure that the cosine approximation is correct to at least 5 significant figures

Write an m-file that will– Approximate cos(x) correctly up to 10 significant figures. – Display the approximation until it stops– Be creative – Display the number of terms it took


Roundoff ErrorsRoundoff Errors

Roundoff errors arise because digital computers cannot represent some quantities exactly.

There are two major facets of roundoff errors involved in numerical calculations:– Digital computers can only represent numbers up to a

certain size and precision, so they round-off.– Some numerical manipulations are easily affected by the

roundoff errors.


Roundoff Errors ExampleRoundoff Errors Example

Let’s try 4.8


The Taylor Theorem and SeriesThe Taylor Theorem and Series

The Taylor theorem states that any smooth function can be approximated as a polynomial.

The Taylor series provides a means to express this idea mathematically.


The Taylor SeriesThe Taylor Series

f x i1 f x i f ' x i h f '' x i 2!

h2 f (3) x i 3!

h3 f (n ) x i

n!hn Rn

Step SizeStep Size


Truncation ErrorTruncation Error

In general, the nth order Taylor series expansion will be exact for an nth order polynomial.

In other cases, the remainder term Rn is of the order of hn+1, meaning:– The more terms are used, the smaller the error, and– The smaller the spacing, the smaller the error for a given

number of terms.


Taylor Series - ExampleTaylor Series - Example

In Matlab: – Use Taylor Series for n = 0-5 to Evaluate f(x) = sin x, at xi+1 = π

on the basis of f(x) and its derivatives at π/2. – Display the truncation error as n increases– Vary h and observe the difference


Numerical DifferentiationNumerical Differentiation

The first order Taylor series can be used to calculate approximations to derivatives:– Given:

– Then:

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

f (x i1) f (x i) f '(x i)h O(h2)

f '(x i)f (x i1) f (x i)

h O(h)


Differentiation (cont)Differentiation (cont)

There are also backward difference and centered difference approximations, depending on the points used:

Forward:

Backward:

Centered:

f '(x i)f (x i1) f (x i)

h O(h)

f '(x i)f (x i) f (x i 1)

h O(h)

f '(x i)f (x i1) f (x i 1)

2h O(h2)


ExampleExample

234 2.03.0)( xxxxf

xxxxf 26.02.1)(' 23

x = 0.5; h = 0.5


Total Numerical ErrorTotal Numerical Error

The total numerical error is the summation of the truncation and roundoff errors.

The truncation error generally increases as the step size increases, while the roundoff error decreases as the step size increases - this leads to a point of diminishing returns for step size.


Other ErrorsOther Errors

Blunders - errors caused by malfunctions of the computer or human imperfection.

Model errors - errors resulting from incomplete mathematical models.

Data uncertainty - errors resulting from the accuracy and/or precision of the data.


Taylor Series - LabTaylor Series - Lab

4.12


HWHW

4.11, 15, 17,

ES 240: Scientific and Engineering Computation. Chapter 4 Chapter 4: Errors Uchechukwu Ofoegbu Temple University.

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