CAAM 353 -Errors and Floating Point Notation 17pp

7/28/2019 CAAM 353 -Errors and Floating Point Notation 17pp

1/17

CAAM 353

Errors and Floating Point Notation


2/17

Types of Errors

Data Error vs. Computational Error

,

, Error =

= + ()

Computational Data

(Sine Example)


3/17

Types of Computational Errors

Rounding

Truncation

Accumulation Cancelation


4/17

Floating Point System

Real numbers can have infinite decimal

representations, but computer memory is finite

+

+

+ +

+

+

+

+ +

base of the number system

precision (# of digits) Lower Bound on Exponent e

Upper Bound on Exponent e

0 . . 1, 0


5/17

Chopping and Rounding

How do you cut off the infinite decimal

representation to write fl(x)?

Chop: Drop the (t, t+1, t+2, .) last Digits

Round: If < /2 then chop

else if > /2, last digit gets

modified to be: +1else /2, round to nearest even


6/17

Rounding Unit and Error

When we round, the WORST weve done is to

create a relative error of:

||

Quantity:

known as Machine Precision MATLAB: eps = 2.2204e-016


7/17

IEEE System

Many computer systems (including MATLAB)

run on this system

Base is 2

Single Precision = 32 bit word

Double Precision = 64 bit word

1 bit for sign (0 = +, 1 = -) 11 bits for exponent, e = 2^11 possibilities

52 bits for the number (d0 to d51)

Eps = * 2^(1-52) = 2^(-52)


8/17

IEEE System

Exponent possibilities 2^11 = 2048

exponent = b 1023

b = 11 bit number

Infinity = 11111111111 = 2047-1023 = 1024

0 = 00000000000 = 0-1023 = -1023

Normal Numbers are in the exponent range

1023 -1022

1024 reserved for infinity, -1023 reserved for 0


9/17

How many numbers?

How many numbers can you represent with a, , , system?

p 1 + 1 2 + 1

How are these numbers spaced?

Ex. 2, 2, 1, 1

+

2 2

(floatgui.m)


10/17

Floating Point Arithmetic

Add 4.567 10 and 4.567 10

Real answer:

0.4567 + 0.004567 0.46127

Rounded real answer: 0.4613

Answer in system with t = 4, with chop

0.4567 + 0.0045 0.4612

Most computer systems have guard digits to

avoid this (allow leading zeros that dont

count in the precision)


11/17

Floating Point Arithmetic

Multiplication:

4.567 10times 4.567 10

Real Answer: 2.0857489 10

Chopped Answer for t=4:

2.085 10

If you use this number for anothercomputation in your code, the error propagates

called Error Accumulation


12/17

Another Interesting Example

From calculus, you should remember that the

following series is the Taylyer series of exp(x)

!

=

Lets look at this sum in MATLAB for:

x = 10 and x = -10

Can you foresee a problem?

(exptest.m)


13/17

Cancelation Error

Previous slides error for x = -10 can also be

called cancelation error (loss of digits due to

canceling of the beginning digits)

Sometimes you can avoid cancellation error,

by being smart with your arithmetic

Example: 1 +


14/17

Subtracting Close Numbers

Back to our

derivative example

Solution should

improve as h 0,

right? WRONG

Total Error:

(sinapprox.m)

+ ()

+


15/17

Patriot Missile Failure

On February 25, 1991, during the Gulf War, an

American Patriot Missile battery in Dharan,

Saudi Arabia, failed to track and intercept an

incoming Iraqi Scud missile.

28 people died, 100 injured

The cause was an inaccurate calculation of the

time since boot due to computer arithmetic

errors


16/17

Patriot Missile Disaster

The time in tenths of second as measured by

the system's internal clock was multiplied by

1/10 to produce the time in seconds

Calculations done with 24-bit (base 2 numbers)

1/10 has a non-terminating rep. in base 2

Chop error, multiplied by the large time (in

tenths of seconds) produced a time in seconds

that was inaccurate


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Patriot Missile Disaster

Detector had been on 100 hours = 3600000

tenths of seconds

Multiply by base 2 (24-bit, roughly 6 digits of

accuracy) version of 1/10 and you are off by

about 0.34 sec

A Scud travels at about 1,676 meters per

second, and so travels more than half a

kilometer in 0.34 sec!

http://www.ima.umn.edu/~arnold/disasters/patriot.html

CAAM 353 -Errors and Floating Point Notation 17pp

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