Lecture 4 - Numerical Errors CVEN 302 June 10, 2002

Lecture 4 - Numerical ErrorsLecture 4 - Numerical Errors

CVEN 302

June 10, 2002

Lecture’s GoalsLecture’s Goals

• Understanding Computer Errors

• Uncertainty in Data and Input

• Well-defined problem

• Numerical Errors– Round-off Error– Truncation Error

Unavoidable Errors in ComputingUnavoidable Errors in Computing

• Hardware problems:Example: PentiumTM Chip

Documentation is critical for any code that is not going to be used and immediately discarded. Documentation takes the form of comment statements that describe the input and output parameters of a function as well as the steps performed in the analysis.


• Some software bugs are caused by deterministic errors in the execution of the problem.

Example:

Problems in the built-in functions such as sine or

cosine and a series of operational commands.

Matlab Numerical ProblemsMatlab Numerical Problems

example>format long e>2.6 + 0.2

ans = 2.800000000000e+000>ans + 0.2

ans = 3.000000000000e+000>ans + 0.2

ans = 3.200000000001e+000 ^Note The program has changed the value of ‘ans’

Matlab Numerical ProblemsMatlab Numerical Problems

ExampleSame method but different results.

>format long e>2.6 + 0.6

ans = 3.200000000000e+000>ans + 0.6

ans = 3.800000000000e+000>ans + 0.6

ans = 4.400000000000e+000>ans + 0.6

ans = 5


• Numerical Errors are based on the mathematics of the problem.

– Round-off Errors

– Truncation Errors

Numerical ErrorsNumerical Errors

Round-off errors occur in computer calculation whenever digits to the right of the decimal are discarded.


Example:> b = 1/3> b = 0.333333> b*3 - 1 = 0

or> b =4/3 - 1> b = 0.33333> b*3 - 1 = -2.2204e-16 ????????


Truncation error is introduced whenever a number computational uses a formula involving discrete values.


Example:

> x = tan(pi/6)> y = sin(pi/6)/cos(pi/6)

where,

> x - y = 1.1102e-16 ????????

Computer ErrorsComputer Errors

Why do you want to know about errors in

computer programs?

To recognize what is a good algorithm!

Digital Representation of NumbersDigital Representation of Numbers

• Bits, Bytes and Words - the binary language of the computer programmer, and electrical engineer.– Bit is a single unit of information (0 or 1)– Byte is a combination of 8 bits– Word is 32, 64, or 128 bit pieces of information

Types of VariablesTypes of Variables

• Integer - two types a regular (16 bits) or a long (32 bits)

• Float - two types a single precision (32 bits) and double precision(64 bits)

• Complex - two single precision real numbers (64 bits)


• The type of variable determines the size of the under and overflow limits.– Underflow is the lowest value the computer can

reach without major problems.– Overflow is the highest value the computer can

reach without major problems.– Matlab commands to see the limits are:

realmax or realmin

Example: Float PointExample: Float Point

The program: halfDif (x,y)

Shows how the numbers converge by halving

the difference and at what point does the delta

term becomes insignificant.

Examples: Finite Precision ArithmeticExamples: Finite Precision Arithmetic

• An earlier example was the difference between a using 1/3 and 4/3 to show round-off errors.

• An example program (epprox) shows the convergence on the exponential term of 1.

exp(1) = [ 1 + (1/n) ]n

Example: epproxExample: epprox

The program has 2 round-off errors. The first error is a relatively minor one and second is catastrophic.

• The minor error is due to the inability to exactly represent 1/n with powers of 2.

• The major round-off error occurs due to round-off of (1 + 1/n) term at a high power.

Measuring ErrorsMeasuring Errors

Absolute Error

Eabs = | x - xtrue |

Relative Error

Erel = | x - xtrue | / | xref |

Convergence of Iterative SequencesConvergence of Iterative Sequences

Using the example problem we will look at four test

cases for convergence for newtsqrt (program) Change:

– NOT_CONVERGED change to r~=rold (comparison)– NOT_CONVERGED change to (r - rold)> (error)– NOT_CONVERGED change to

abs(r-rold)/rold >abs (absolute error)

– NOT_CONVERGED change to

abs((r-rold)/rold)>rel (relative error)

Testing example: newtsqrt.mTesting example: newtsqrt.m

The program newtsqrt has a driver ‘testSqrt’

program. The program will input data into the

function and compare the results with the

actual results.

Truncation Error in AlgorithmsTruncation Error in Algorithms

Truncation error results from approximating

continuous mathematical expressions with

discrete algebraic formulas. Unlike round-off,

which is controlled by the hardware and

computer language being used, truncation

error is under the control of the programmer.

Set of examples of truncation errorsSet of examples of truncation errors

• Sinser.m is a program for calculating sin(x) using a series expansion.

• expSeriesPlot shows the convergence of the absolute error for an exp() as a series expansion.

• demoTaylor shows a Taylor Series expansion for a simple equation with a variable delta step and different number of derivatives.

• fidiff is a finite difference program to test both round-off and truncation errors.

Sinser ExampleSinser Example

The program examines how the addition of

more terms helps the sinusoid to converge .

sin(x) = (-1)k-1 (x2k-1 /(2k-1)!)

expSeriesPlotexpSeriesPlot

The program shows how the solution

converges on the true value of exp(x) as a

series expansion.

exp(x) = 1 + ( xn / n!)

Taylor SeriesTaylor Series

A series of Taylor series are used to represent

the function at a location with each series

containing an addition derivative. The

function is:

x1

1xf

demoTaylordemoTaylor

The program looks at successive derivatives.

0

3

0

2

003

0

2

002

0001

00

*!3

*!2

*

*!2

*

where, *

1

1

xfh

xfh

xfhxfxP

xfh

xfhxfxP

x -xΔhxfhxfxP

xxf

Round-off and Truncation with Round-off and Truncation with Finite DifferenceFinite Difference

The program fidiff(x) is a simple finite difference program to calculate the first order derivative of an exponential function with variable stepsize.

dh

xfdhxf

dx

xexpd

Accuracy and PrecisionAccuracy and Precision• Accuracy - How closely a measured or computed value

agrees with the true value• Precision - How closely individual measured or

computed values agree with each other

More More AccurateAccurate

More More PrecisePrecise

Precision is getting them close together.

Accuracy is getting all your shots near the target.

HomeworkHomework

• Modify demoTaylor program

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Lecture 4 - Numerical Errors CVEN 302 June 10, 2002