college of engineering representing numbers in a computer section b, spring 2003 coe1361: computing...

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College of EngineeringCollege of EngineeringR

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COE1361: Computing for COE1361: Computing for EngineersEngineers

COE1361: Computing for Engineers

Learning Objectives

Understand how numbers are stored in a computer and what the implications are

Lecture 1c

Precision and accuracy

Numbers on a computer

Exercises

Summary

Not in our textbook … pay attention … create your own knowledge base

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Lecture References Online

Computer memory– http://www-und.ida.liu.se/~annsa582/tutorial/– http://www.crisinc.addr.com/memory.html– http://www.howstuffworks.com/computer-memory.htm

Numeric representation– http://www.scri.fsu.edu/~jac/MAD3401/Backgrnd/ieee.html– http://www.math.byu.edu/~schow/work/

floating_point_system.htm– http://www.cs.utah.edu/~zachary/isp/applets/FP/FP.html– http://www.research.microsoft.com/~hollasch/cgindex/coding/

ieeefloat.html– http://www.cs.unc.edu/~dm/UNC/COMP205/LECTURES/ERROR/lec23/

node4.html

Standards– http://standards.ieee.org/

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How Does a Computer Work?

In Lecture 1b we looked at:• The history of computers• The basic hardware structure of a microprocessor• How the hardware relates to the software

• Limitations in representation (precision/accuracy)• Binary Representation• IEEE Floating Point• Relation to Matlab

In Lecture 1c (this one), we will consider numeric representation:

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Numbers: Precision and Accuracy

Low precision: = 3.14

High precision: = 3.140101011

Low accuracy: = 3.10212

High accuracy: = 3.14159

High accuracy & precision: = 3.141592653

Good Accuracy Good Precision

Good PrecisionPoor Accuracy

Good AccuracyPoor Precision

Poor AccuracyPoor Precision

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Numbers

Numbers we use are DECIMAL (or base 10):– Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9– 123 = 1*102 + 2*101 + 3*100

But we can always use other bases:Octal (base 8):

– Digits: 0, 1, 2, 3, 4, 5, 6, 7– 123 = 1*82 + 2*81 + 3*80

– 1238 = 64+16+3 = 8310

Binary (base 2):– Digits: 0, 1– 1011 = 1*23 + 0*22 + 1*21 + 1*20

– 10112 = 8+0+2+1 = 1110

– 1238 = 001 010 011 = 10100112

Hexadecimal (base 16):– Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F– 123 = 1*162 + 2*161 + 3*160

– 12316 = 256 + 32 + 3 = 29110

– 12316 = 0001 0010 0011 = 1001000112

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Computer Memory

Results of a computation, graphics, word documents, everything that a computer does must be stored somewhere.

That “somewhere” is “memory”.

Memory comes in a variety of types and speeds:• Cache – in the CPU itself (fastest)• RAM - external to the CPU (fast)• Disk - physical media, external to the CPU, r/w• CDROM - physical media, (slow)• Tape - physical media, (slowest)

Memory is measured in “bytes” (and kilobytes, megabytes, gigabytes, and terabytes.)

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Inside the Bytes

A byte is the smallest memory allocation available.

A byte contains 8 bits so that:–Smallest: 0 0 0 0 0 0 0 0 = 010

–Largest: 1 1 1 1 1 1 1 1 = 1*27+1*26+1*25+1*24+1*23+1*22+1*21+1*20 = 25510 (or 28-1)

Result: a single byte can be used to store an integer number ranging from 0 to 255 (256 different numbers)

If negative numbers are included, one bit must be dedicated to the sign, leaving only 7 bits for the number–Smallest: 0

–Largest: +12710 or -12810

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Signed-Magnitude (OPTIONAL MATERIAL)

Example of signed magnitude:

000001012 => + 5

100001012 => - 5

Notice also that signed-magnitude gives plus and minus zero!

– 000000002 => + 0

– 100000002 => - 0

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Two’s Complement Used (OPTIONAL MATERIAL)

If first bit is zero, the number is positive and there is no further interpretation needed:

– 011111112 => + 127

If first bit is one, the number is negative. Complement all bits and add one to get to the positive magnitude of the negative number.

111111112

00000000

+ 1 000000012

Therefore original number = -1

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Two’s Complement (OPTIONAL MATERIAL)

• Two’s Complement has no negative zero.

• 11111111 = - 1,

• 10000000 = -128

• Range: [ -128, 127 ]

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Inside the Bytes

28-1 (255) is not a very big number, so computers generally use multiple bytes to represent numbers:

Here is some new vocabulary:–byte = 8 bits–short (precision) = 2 bytes–single (precision) = 4 bytes–double (precision) = 8 bytes–quad (precision) = 16 bytes

–char = 2 bytes (used to be 1 byte)

Note:–A word is the basic size of the CPU registers and for Pentium

chips it is 4 bytes or 32 bits; it is 8 bytes for the new Itanium and some unix chipsets. It was 2 bytes for early Intel chips; some new game consoles use 16 byte words.

Java and other languages use

different names!

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Inside the Bytes

How are fractional numbers (called real or floating point numbers) stored in a computer?

The IEEE 754 double format consists of three fields:– a 52-bit fraction, f (also called the mantissa)– an 11-bit biased exponent, e– and a 1-bit sign, s

These fields are stored contiguously in 8 bytes (or 2 successively addressed 4-byte words):

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Inside the Bytes

Because only a finite number of bits are used for each part of the number, not all possible real numbers can be represented in a computer using IEEE 754

0

Positive numbers smaller than 2-1022 (positive underflow, Matlab “realmin”)

Positive numbers greater than (2-2-52) x 21023 (positive overflow, Matlab “realmax”)

Negative numbers less than -(2-2-52) x 21023 (negative overflow)

Negative numbers greater than -2-1022

(negative underflow)

RESULT: about 16 digits of precision in the range ±10±308

Zero (actually is a special combination of bits)

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Inside the Bytes

Others sources of error in computation:

•Errors in the input data - measurement errors, errors introduced by the conversion of decimal data to binary, roundoff errors.

•Roundoff errors during computation (as discussed)

•Truncation errors - using approximate calculation is inevitable when computing quantities involving limits and other infinite processes on a computer– Never try to compare two floating point numbers for equality

because all 16+ digits would have to match perfectly…

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Describing Error in Matlab

AccuracyHow close your answer is to the “actual” or “real” answer.

PrecisionThe smallest difference that can be represented on the computer (help eps)

Recognize:MATLAB (and other programs that use IEEE doubles) give you 15-16 “good” digitsMATLAB will also store exact integer values using only the mantissa (for about 15-16 digits total)

Matlab commands:realmin, realmax, eps, bitmax (try with help)

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Memory in MATLAB

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What it Means…

In the previous slide, we see:

What is the size of the variable “i”

What does “class” represent?

How many bytes are used to store the value?

Name Size Bytes Class

i 1x1 8 double array

s 1x43 86 char array

t 1x200 1600 double array

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Back to MATLAB - what does all this mean?

• We must pay attention to the math! (on any computer!)

• Given a finite (limited) number of bits, almost all computations will result in numbers that can’t be represented exactly.

• Remember that the result of a floating-point computation must be truncated to fit back into it’s finite representation.

Always check your results - design programs to allow for independent verification of computations!

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SUMMARY

Describe memory. List different kinds of memory.

What is IEEE 754? Describe how MATLAB represents

numbers.

Draw a number line and identify ranges where

computers will generate errors.

Describe three potential sources of errors in

computation.

Describe precision. Describe accuracy.

Describe how we can protect ourselves from

computation error.

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Summary

Action Items

• Review the lecture

• How will you use this information in your current work? … future work at GT?

• Start Exploring Matlab!

• Increment your record of commands learned

• Come prepared to ask questions about MATLAB

Lecture 1c

Numbers on a computer

Precision and accuracy

college of engineering representing numbers in a computer section b, spring 2003 coe1361: computing...

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