# Physics 331 Introduction to Numerical Techniques in Physics

Save this PDF as:

Size: px
Start display at page:

Download "Physics 331 Introduction to Numerical Techniques in Physics"

## Transcription

1 Physics 331 Introduction to Numerical Techniques in Physics Instructor: Joaquín Drut Lecture 2

2 Any logistics questions?

3 Today: Number representation Sources of error Note: typo in HW! Two parts c. Call them c1 and c2.

4 Representing numbers on a computer

5 Numbers Our understanding of nature has led us to a description in terms of real and complex numbers... What makes real numbers so special? What are their defining properties?

6 Numbers Our understanding of nature has led us to a description in terms of real and complex numbers... What makes real numbers so special? What are their defining properties? Reals form a field (6 axioms about + and. )

7 Numbers Our understanding of nature has led us to a description in terms of real and complex numbers... What makes real numbers so special? What are their defining properties? Reals form a field (6 axioms about + and. ) Reals are ordered (3 axioms allowing to define > and < )

8 Numbers Our understanding of nature has led us to a description in terms of real and complex numbers... What makes real numbers so special? What are their defining properties? Reals form a field (6 axioms about + and. ) Reals are ordered (3 axioms allowing to define > and < ) Reals are complete (every convergent sequence has a real limit)

9 Numbers Our understanding of nature has led us to a description in terms of real and complex numbers... What makes real numbers so special? What are their defining properties? Reals form a field (6 axioms about + and. ) Reals are ordered (3 axioms allowing to define > and < ) Reals are complete (every convergent sequence has a real limit) Real numbers contain rational as well as irrational numbers. This is a problem! Why?

10 Numbers Our understanding of nature has led us to a description in terms of real and complex numbers... What makes real numbers so special? What are their defining properties? Reals form a field (6 axioms about + and. ) Reals are ordered (3 axioms allowing to define > and < ) Reals are complete (every convergent sequence has a real limit) Real numbers contain rational as well as irrational numbers. This is a problem! Why? How many rationals are there? How many irrationals?

11 Numbers Our understanding of nature has led us to a description in terms of real and complex numbers but computers use a discrete and finite set of numbers. You can already imagine that the range and precision of numbers we can represent will somehow be limited. An engineering decision had to be made regarding how to manage such limitations. But there is more to consider

12 Digital computers In digital computers, everything is ultimately reduced to on and off, or 0 and 1. Numbers on such a computer are encoded in base 2: Base 10 Base bits

13 Decimal and binary representations The decimal representation of numbers is so common that we often forget why we count the way we do. (Why?)

14 Decimal and binary representations The decimal representation of numbers is so common that we often forget why we count the way we do. Decimal Take the number 23, What does this mean in the decimal representation?

15 Decimal and binary representations The decimal representation of numbers is so common that we often forget why we count the way we do. Decimal Take the number 23, What does this mean in the decimal representation? 23, x x x x x x x x 10-3

16 Decimal and binary representations Binary We only have 1 and 0, and then we move on to the next digit! You may have something like What does this mean?

17 Decimal and binary representations Binary We only have 1 and 0, and then we move on to the next digit! You may have something like What does this mean? Exercise: What number is this in decimal representation? Take 2 minutes and try to figure it out!

18 Decimal and binary representations Binary We only have 1 and 0, and then we move on to the next digit! You may have something like What does this mean? Exercise: What number is this in decimal representation? Take 2 minutes and try to figure it out! Exercise: What number is 21 in binary representation? Take 2 minutes and try to figure it out!

19 On a computer, numbers are represented in terms of bits, but that s not all there is to it

20 Floating point representations Decimal floating point representation (a.k.a. scientific notation) d.ddddddd x 10 p d: 0,1,..,9 mantissa think precision exponent think order of magnitude Example f =

21 Floating point representations Decimal floating point representation (a.k.a. scientific notation) d.ddddddd x 10 p d: 0,1,..,9 mantissa think precision exponent think order of magnitude In general To find the first digit d (from the left), and thus the mantissa as well, divide by the highest power of 10 that is less than the number you want to represent. That power is the exponent p.

22 Floating point representations Binary floating point representation (a.k.a. most of you have never seen this before) 1.bbbbbbb x 2 bbb b: 0,1 mantissa exponent

23 Floating point representations IEEE standard: f = s M B e E There is also quadruple precision! SP sign exponent mantissa

24 Floating point representations IEEE standard: f = s M B e E Overflow 0 Underflow Overflow Note: - There is a largest positive number! Overflow - There is a smallest positive number! Underflow - Only a finite number of exact values! - Interval between numbers depends on exponent!

25 Examples f = f = s M B e E

26 Examples f = f = s M B e E f =

27 Examples f = f = s M B e E f = f =

28 Floating point representations IEEE standard: f = s M B e E There is also quadruple precision! SP sign exponent mantissa What range of values can we represent for the exponent with 8 bits?

29 The exponent e : true exponent f = s M B e E E : bias (127 in SP; 1023 in DP)

30 Sources of error

31 Sources of error when programming... Syntax errors (aka compile errors: code does not even compile) Runtime errors (e.g. seg-fault; code compiles but fails when running) Numerical errors (more on this soon!) Physics errors (not a programming error: the code does what you want and it does it right... you just have the physics wrong!)

32 Sources of error when programming... Syntax errors (aka compile errors: code does not even compile) Runtime errors (e.g. seg-fault; code compiles but fails when running) Numerical errors (more on this soon!) Physics errors (not a programming error: the code does what you want and it does it right... you just have the physics wrong!) We will put aside discussions of the model/approach used to represent the physics. These could give... Systematic errors (controlled approximations; e.g. finite volume, finite mesh) Uncontrolled errors (the model may involve an approximation )

33 Sources of numerical error... Round-off errors (computers have finite precision) Round-off errors will propagate in iterative methods and potentially produce instabilities and give useless results.

34 Sources of numerical error... Round-off errors (computers have finite precision) Round-off errors will propagate in iterative methods and potentially produce instabilities and give useless results. Statistical error (e.g. if your method uses random numbers, like in MC) You just need more statistics - it s almost like a systematic error... but it can be much harder to deal with!

35 Statistical error / signal-to-noise problem. Example cosi loghxl x 0 x M x >

36 Statistical error / signal-to-noise problem. Example cosi loghxl x 0 x M x >

37 Sources of numerical error... Round-off errors (computers have finite precision) Round-off errors will propagate in iterative methods and potentially produce instabilities and give useless results. Statistical error (e.g. if your method uses random numbers, like in MC) You just need more statistics - it s almost like a systematic error... but it can be much harder to deal with!

38 Sources of numerical error... Round-off errors (computers have finite precision) Round-off errors will propagate in iterative methods and potentially produce instabilities and give useless results. Statistical error (e.g. if your method uses random numbers, like in MC) You just need more statistics - it s almost like a systematic error... but it can be much harder to deal with! Truncation errors (caused by the numerical method) How do you estimate the value of complicated functions, series expansions, integrals, derivatives...? Should be a systematic error, i.e. something we can treat. Why?

39 Total error Total error = True solution - Numerical solution Total relative error = True solution - Numerical solution True solution Usually we do not know the true solution; that is what we are looking for! The numerical solution can be systematically improved. Ideally the only error left is just the round-off error. We will always be limited by this!

### Floating Point Arithmetic

Floating Point Arithmetic CS 365 Floating-Point What can be represented in N bits? Unsigned 0 to 2 N 2s Complement -2 N-1 to 2 N-1-1 But, what about? very large numbers? 9,349,398,989,787,762,244,859,087,678

More information

### Scientific Computing. Error Analysis

ECE257 Numerical Methods and Scientific Computing Error Analysis Today s s class: Introduction to error analysis Approximations Round-Off Errors Introduction Error is the difference between the exact solution

More information

### IEEE-754 floating-point

IEEE-754 floating-point Real and floating-point numbers Real numbers R form a continuum - Rational numbers are a subset of the reals - Some numbers are irrational, e.g. π Floating-point numbers are an

More information

### Floating Point Representation. CS Summer 2008 Jonathan Kaldor

Floating Point Representation CS3220 - Summer 2008 Jonathan Kaldor Floating Point Numbers Infinite supply of real numbers Requires infinite space to represent certain numbers We need to be able to represent

More information

### Data Representation Floating Point

Data Representation Floating Point CSCI 2400 / ECE 3217: Computer Architecture Instructor: David Ferry Slides adapted from Bryant & O Hallaron s slides via Jason Fritts Today: Floating Point Background:

More information

### Floating Point Representation in Computers

Floating Point Representation in Computers Floating Point Numbers - What are they? Floating Point Representation Floating Point Operations Where Things can go wrong What are Floating Point Numbers? Any

More information

### 2 Computation with Floating-Point Numbers

2 Computation with Floating-Point Numbers 2.1 Floating-Point Representation The notion of real numbers in mathematics is convenient for hand computations and formula manipulations. However, real numbers

More information

### Number Systems. Binary Numbers. Appendix. Decimal notation represents numbers as powers of 10, for example

Appendix F Number Systems Binary Numbers Decimal notation represents numbers as powers of 10, for example 1729 1 103 7 102 2 101 9 100 decimal = + + + There is no particular reason for the choice of 10,

More information

### Chapter 03: Computer Arithmetic. Lesson 09: Arithmetic using floating point numbers

Chapter 03: Computer Arithmetic Lesson 09: Arithmetic using floating point numbers Objective To understand arithmetic operations in case of floating point numbers 2 Multiplication of Floating Point Numbers

More information

### CS321. Introduction to Numerical Methods

CS31 Introduction to Numerical Methods Lecture 1 Number Representations and Errors Professor Jun Zhang Department of Computer Science University of Kentucky Lexington, KY 40506 0633 August 5, 017 Number

More information

### 3.5 Floating Point: Overview

3.5 Floating Point: Overview Floating point (FP) numbers Scientific notation Decimal scientific notation Binary scientific notation IEEE 754 FP Standard Floating point representation inside a computer

More information

### 1.2 Round-off Errors and Computer Arithmetic

1.2 Round-off Errors and Computer Arithmetic 1 In a computer model, a memory storage unit word is used to store a number. A word has only a finite number of bits. These facts imply: 1. Only a small set

More information

### Floating Point Numbers

Floating Point Numbers Summer 8 Fractional numbers Fractional numbers fixed point Floating point numbers the IEEE 7 floating point standard Floating point operations Rounding modes CMPE Summer 8 Slides

More information

### Floating-Point Data Representation and Manipulation 198:231 Introduction to Computer Organization Lecture 3

Floating-Point Data Representation and Manipulation 198:231 Introduction to Computer Organization Instructor: Nicole Hynes nicole.hynes@rutgers.edu 1 Fixed Point Numbers Fixed point number: integer part

More information

### CS101 Lecture 04: Binary Arithmetic

CS101 Lecture 04: Binary Arithmetic Binary Number Addition Two s complement encoding Briefly: real number representation Aaron Stevens (azs@bu.edu) 25 January 2013 What You ll Learn Today Counting in binary

More information

### Floating Point. The World is Not Just Integers. Programming languages support numbers with fraction

1 Floating Point The World is Not Just Integers Programming languages support numbers with fraction Called floating-point numbers Examples: 3.14159265 (π) 2.71828 (e) 0.000000001 or 1.0 10 9 (seconds in

More information

### Classes of Real Numbers 1/2. The Real Line

Classes of Real Numbers All real numbers can be represented by a line: 1/2 π 1 0 1 2 3 4 real numbers The Real Line { integers rational numbers non-integral fractions irrational numbers Rational numbers

More information

### Floating-Point Numbers in Digital Computers

POLYTECHNIC UNIVERSITY Department of Computer and Information Science Floating-Point Numbers in Digital Computers K. Ming Leung Abstract: We explain how floating-point numbers are represented and stored

More information

### Floating point. Today! IEEE Floating Point Standard! Rounding! Floating Point Operations! Mathematical properties. Next time. !

Floating point Today! IEEE Floating Point Standard! Rounding! Floating Point Operations! Mathematical properties Next time! The machine model Chris Riesbeck, Fall 2011 Checkpoint IEEE Floating point Floating

More information

### Floating-point representation

Lecture 3-4: Floating-point representation and arithmetic Floating-point representation The notion of real numbers in mathematics is convenient for hand computations and formula manipulations. However,

More information

### Computational Methods. Sources of Errors

Computational Methods Sources of Errors Manfred Huber 2011 1 Numerical Analysis / Scientific Computing Many problems in Science and Engineering can not be solved analytically on a computer Numeric solutions

More information

### Homework 1 graded and returned in class today. Solutions posted online. Request regrades by next class period. Question 10 treated as extra credit

Announcements Homework 1 graded and returned in class today. Solutions posted online. Request regrades by next class period. Question 10 treated as extra credit Quiz 2 Monday on Number System Conversions

More information

### Signed umbers. Sign/Magnitude otation

Signed umbers So far we have discussed unsigned number representations. In particular, we have looked at the binary number system and shorthand methods in representing binary codes. With m binary digits,

More information

### 15213 Recitation 2: Floating Point

15213 Recitation 2: Floating Point 1 Introduction This handout will introduce and test your knowledge of the floating point representation of real numbers, as defined by the IEEE standard. This information

More information

### Module 2: Computer Arithmetic

Module 2: Computer Arithmetic 1 B O O K : C O M P U T E R O R G A N I Z A T I O N A N D D E S I G N, 3 E D, D A V I D L. P A T T E R S O N A N D J O H N L. H A N N E S S Y, M O R G A N K A U F M A N N

More information

### Floating Point Numbers. Lecture 9 CAP

Floating Point Numbers Lecture 9 CAP 3103 06-16-2014 Review of Numbers Computers are made to deal with numbers What can we represent in N bits? 2 N things, and no more! They could be Unsigned integers:

More information

### Declaration. Fundamental Data Types. Modifying the Basic Types. Basic Data Types. All variables must be declared before being used.

Declaration Fundamental Data Types All variables must be declared before being used. Tells compiler to set aside an appropriate amount of space in memory to hold a value. Enables the compiler to perform

More information

### 4.1 QUANTIZATION NOISE

DIGITAL SIGNAL PROCESSING UNIT IV FINITE WORD LENGTH EFFECTS Contents : 4.1 Quantization Noise 4.2 Fixed Point and Floating Point Number Representation 4.3 Truncation and Rounding 4.4 Quantization Noise

More information

### Inf2C - Computer Systems Lecture 2 Data Representation

Inf2C - Computer Systems Lecture 2 Data Representation Boris Grot School of Informatics University of Edinburgh Last lecture Moore s law Types of computer systems Computer components Computer system stack

More information

### Representing and Manipulating Floating Points

Representing and Manipulating Floating Points Jin-Soo Kim (jinsookim@skku.edu) Computer Systems Laboratory Sungkyunkwan University http://csl.skku.edu The Problem How to represent fractional values with

More information

### On a 64-bit CPU. Size/Range vary by CPU model and Word size.

On a 64-bit CPU. Size/Range vary by CPU model and Word size. unsigned short x; //range 0 to 65553 signed short x; //range ± 32767 short x; //assumed signed There are (usually) no unsigned floats or doubles.

More information

### Computers and programming languages introduction

Computers and programming languages introduction Eugeniy E. Mikhailov The College of William & Mary Lecture 01 Eugeniy Mikhailov (W&M) Practical Computing Lecture 01 1 / 19 Class goals and structure Primary

More information

### Lecture Notes: Floating-Point Numbers

Lecture Notes: Floating-Point Numbers CS227-Scientific Computing September 8, 2010 What this Lecture is About How computers represent numbers How this affects the accuracy of computation Positional Number

More information

### Computer Numbers and their Precision, I Number Storage

Computer Numbers and their Precision, I Number Storage Learning goal: To understand how the ways computers store numbers lead to limited precision and how that introduces errors into calculations. Learning

More information

### IT 1204 Section 2.0. Data Representation and Arithmetic. 2009, University of Colombo School of Computing 1

IT 1204 Section 2.0 Data Representation and Arithmetic 2009, University of Colombo School of Computing 1 What is Analog and Digital The interpretation of an analog signal would correspond to a signal whose

More information

### Independent Representation

Independent Representation 1 Integer magnitude + 1 2 3 4 5 6 7 8 9 0 Float sign sign of exponent mantissa On standard 32 bit machines: INT_MAX = 2147483647 which gives 10 digits of precision, (i.e. the

More information

### IEEE Standard 754 Floating Point Numbers

IEEE Standard 754 Floating Point Numbers Steve Hollasch / Last update 2005-Feb-24 IEEE Standard 754 floating point is the most common representation today for real numbers on computers, including Intel-based

More information

### CS429: Computer Organization and Architecture

CS429: Computer Organization and Architecture Dr. Bill Young Department of Computer Sciences University of Texas at Austin Last updated: September 18, 2017 at 12:48 CS429 Slideset 4: 1 Topics of this Slideset

More information

### CS 265. Computer Architecture. Wei Lu, Ph.D., P.Eng.

CS 265 Computer Architecture Wei Lu, Ph.D., P.Eng. 1 Part 1: Data Representation Our goal: revisit and re-establish fundamental of mathematics for the computer architecture course Overview: what are bits

More information

### IEEE Floating Point Numbers Overview

COMP 40: Machine Structure and Assembly Language Programming (Fall 2015) IEEE Floating Point Numbers Overview Noah Mendelsohn Tufts University Email: noah@cs.tufts.edu Web: http://www.cs.tufts.edu/~noah

More information

### Floating-point numbers. Phys 420/580 Lecture 6

Floating-point numbers Phys 420/580 Lecture 6 Random walk CA Activate a single cell at site i = 0 For all subsequent times steps, let the active site wander to i := i ± 1 with equal probability Random

More information

### Floating point. Today. IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Next time.

Floating point Today IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Next time The machine model Fabián E. Bustamante, Spring 2010 IEEE Floating point Floating point

More information

### Floating-point Arithmetic. where you sum up the integer to the left of the decimal point and the fraction to the right.

Floating-point Arithmetic Reading: pp. 312-328 Floating-Point Representation Non-scientific floating point numbers: A non-integer can be represented as: 2 4 2 3 2 2 2 1 2 0.2-1 2-2 2-3 2-4 where you sum

More information

### Floating Point Puzzles The course that gives CMU its Zip! Floating Point Jan 22, IEEE Floating Point. Fractional Binary Numbers.

class04.ppt 15-213 The course that gives CMU its Zip! Topics Floating Point Jan 22, 2004 IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Floating Point Puzzles For

More information

### System Programming CISC 360. Floating Point September 16, 2008

System Programming CISC 360 Floating Point September 16, 2008 Topics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Powerpoint Lecture Notes for Computer Systems:

More information

### The Design and Implementation of a Rigorous. A Rigorous High Precision Floating Point Arithmetic. for Taylor Models

The and of a Rigorous High Precision Floating Point Arithmetic for Taylor Models Department of Physics, Michigan State University East Lansing, MI, 48824 4th International Workshop on Taylor Methods Boca

More information

### Section 1.4 Mathematics on the Computer: Floating Point Arithmetic

Section 1.4 Mathematics on the Computer: Floating Point Arithmetic Key terms Floating point arithmetic IEE Standard Mantissa Exponent Roundoff error Pitfalls of floating point arithmetic Structuring computations

More information

### ECE 2020B Fundamentals of Digital Design Spring problems, 6 pages Exam Two 26 February 2014

Instructions: This is a closed book, closed note exam. Calculators are not permitted. If you have a question, raise your hand and I will come to you. Please work the exam in pencil and do not separate

More information

### Characters, Strings, and Floats

Characters, Strings, and Floats CS 350: Computer Organization & Assembler Language Programming 9/6: pp.8,9; 9/28: Activity Q.6 A. Why? We need to represent textual characters in addition to numbers. Floating-point

More information

### Table : IEEE Single Format ± a a 2 a 3 :::a 8 b b 2 b 3 :::b 23 If exponent bitstring a :::a 8 is Then numerical value represented is ( ) 2 = (

Floating Point Numbers in Java by Michael L. Overton Virtually all modern computers follow the IEEE 2 floating point standard in their representation of floating point numbers. The Java programming language

More information

### Computer Systems C S Cynthia Lee

Computer Systems C S 1 0 7 Cynthia Lee 2 Today s Topics LECTURE: Floating point! Real Numbers and Approximation MATH TIME! Some preliminary observations on approximation We know that some non-integer numbers

More information

### Systems I. Floating Point. Topics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties

Systems I Floating Point Topics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties IEEE Floating Point IEEE Standard 754 Established in 1985 as uniform standard for

More information

### Giving credit where credit is due

CSCE 230J Computer Organization Floating Point Dr. Steve Goddard goddard@cse.unl.edu http://cse.unl.edu/~goddard/courses/csce230j Giving credit where credit is due Most of slides for this lecture are based

More information

### CS101 Introduction to computing Floating Point Numbers

CS101 Introduction to computing Floating Point Numbers A. Sahu and S. V.Rao Dept of Comp. Sc. & Engg. Indian Institute of Technology Guwahati 1 Outline Need to floating point number Number representation

More information

### Numeric Variable Storage Pattern

Numeric Variable Storage Pattern Sreekanth Middela Srinivas Vanam Rahul Baddula Percept Pharma Services, Bridgewater, NJ ABSTRACT This paper presents the Storage pattern of Numeric Variables within the

More information

### Integers and Floating Point

CMPE12 More about Numbers Integers and Floating Point (Rest of Textbook Chapter 2 plus more)" Review: Unsigned Integer A string of 0s and 1s that represent a positive integer." String is X n-1, X n-2,

More information

### Number Systems and Computer Arithmetic

Number Systems and Computer Arithmetic Counting to four billion two fingers at a time What do all those bits mean now? bits (011011011100010...01) instruction R-format I-format... integer data number text

More information

### CS61C L10 MIPS Instruction Representation II, Floating Point I (6)

CS61C L1 MIPS Instruction Representation II, Floating Point I (1) inst.eecs.berkeley.edu/~cs61c CS61C : Machine Structures Lecture #1 Instruction Representation II, Floating Point I 25-1-3 There is one

More information

### The type of all data used in a C (or C++) program must be specified

The type of all data used in a C (or C++) program must be specified A data type is a description of the data being represented That is, a set of possible values and a set of operations on those values

More information

### Computer Arithmetic Ch 8

Computer Arithmetic Ch 8 ALU Integer Representation Integer Arithmetic Floating-Point Representation Floating-Point Arithmetic 1 Arithmetic Logical Unit (ALU) (2) Does all work in CPU (aritmeettis-looginen

More information

### Floating Point Considerations

Chapter 6 Floating Point Considerations In the early days of computing, floating point arithmetic capability was found only in mainframes and supercomputers. Although many microprocessors designed in the

More information

### Chapter 2 Float Point Arithmetic. Real Numbers in Decimal Notation. Real Numbers in Decimal Notation

Chapter 2 Float Point Arithmetic Topics IEEE Floating Point Standard Fractional Binary Numbers Rounding Floating Point Operations Mathematical properties Real Numbers in Decimal Notation Representation

More information

### Computer Architecture Review. Jo, Heeseung

Computer Architecture Review Jo, Heeseung Computer Abstractions and Technology Jo, Heeseung Below Your Program Application software Written in high-level language System software Compiler: translates HLL

More information

### Floating Point Numbers

Floating Point Floating Point Numbers Mathematical background: tional binary numbers Representation on computers: IEEE floating point standard Rounding, addition, multiplication Kai Shen 1 2 Fractional

More information

### What Every Programmer Should Know About Floating-Point Arithmetic

What Every Programmer Should Know About Floating-Point Arithmetic Last updated: October 15, 2015 Contents 1 Why don t my numbers add up? 3 2 Basic Answers 3 2.1 Why don t my numbers, like 0.1 + 0.2 add

More information

### Floating Point. EE 109 Unit 20. Floating Point Representation. Fixed Point

2.1 Floating Point 2.2 EE 19 Unit 2 IEEE 754 Floating Point Representation Floating Point Arithmetic Used to represent very numbers (fractions) and very numbers Avogadro s Number: +6.247 * 1 23 Planck

More information

### Truncation Errors. Applied Numerical Methods with MATLAB for Engineers and Scientists, 2nd ed., Steven C. Chapra, McGraw Hill, 2008, Ch. 4.

Chapter 4: Roundoff and Truncation Errors Applied Numerical Methods with MATLAB for Engineers and Scientists, 2nd ed., Steven C. Chapra, McGraw Hill, 2008, Ch. 4. 1 Outline Errors Accuracy and Precision

More information

### Midterm Exam Answers Instructor: Randy Shepherd CSCI-UA.0201 Spring 2017

Section 1: Multiple choice (select any that apply) - 20 points 01. Representing 10 using the 4 byte unsigned integer encoding and using 4 byte two s complements encoding yields the same bit pattern. (a)

More information

### Most nonzero floating-point numbers are normalized. This means they can be expressed as. x = ±(1 + f) 2 e. 0 f < 1

Floating-Point Arithmetic Numerical Analysis uses floating-point arithmetic, but it is just one tool in numerical computation. There is an impression that floating point arithmetic is unpredictable and

More information

### AMTH142 Lecture 10. Scilab Graphs Floating Point Arithmetic

AMTH142 Lecture 1 Scilab Graphs Floating Point Arithmetic April 2, 27 Contents 1.1 Graphs in Scilab......................... 2 1.1.1 Simple Graphs...................... 2 1.1.2 Line Styles........................

More information

### IEEE Standard for Floating-Point Arithmetic: 754

IEEE Standard for Floating-Point Arithmetic: 754 G.E. Antoniou G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: 754 1 / 34 Floating Point Standard: IEEE 754 1985/2008 Established in 1985 (2008)

More information

### CS61C : Machine Structures

inst.eecs.berkeley.edu/~cs61c CS61C : Machine Structures Lecture #10 Instruction Representation II, Floating Point I 2005-10-03 Lecturer PSOE, new dad Dan Garcia www.cs.berkeley.edu/~ddgarcia #9 bears

More information

### ecture 25 Floating Point Friedland and Weaver Computer Science 61C Spring 2017 March 17th, 2017

ecture 25 Computer Science 61C Spring 2017 March 17th, 2017 Floating Point 1 New-School Machine Structures (It s a bit more complicated!) Software Hardware Parallel Requests Assigned to computer e.g.,

More information

### Digital Computers and Machine Representation of Data

Digital Computers and Machine Representation of Data K. Cooper 1 1 Department of Mathematics Washington State University 2013 Computers Machine computation requires a few ingredients: 1 A means of representing

More information

### Numerical Precision. Or, why my numbers aren t numbering right. 1 of 15

Numerical Precision Or, why my numbers aren t numbering right 1 of 15 What s the deal? Maybe you ve seen this #include int main() { float val = 3.6f; printf( %.20f \n, val); Print a float with

More information

### Chapter 2 Bits, Data Types, and Operations

Chapter 2 Bits, Data Types, and Operations How do we represent data in a computer? At the lowest level, a computer is an electronic machine. works by controlling the flow of electrons Easy to recognize

More information

### Binary Addition & Subtraction. Unsigned and Sign & Magnitude numbers

Binary Addition & Subtraction Unsigned and Sign & Magnitude numbers Addition and subtraction of unsigned or sign & magnitude binary numbers by hand proceeds exactly as with decimal numbers. (In fact this

More information

### Data Representation and Introduction to Visualization

Data Representation and Introduction to Visualization Massimo Ricotti ricotti@astro.umd.edu University of Maryland Data Representation and Introduction to Visualization p.1/18 VISUALIZATION Visualization

More information

### 4 Operations On Data 4.1. Foundations of Computer Science Cengage Learning

4 Operations On Data 4.1 Foundations of Computer Science Cengage Learning Objectives After studying this chapter, the student should be able to: List the three categories of operations performed on data.

More information

### Floating Point. CSC207 Fall 2017

Floating Point CSC207 Fall 2017 Ariane 5 Rocket Launch Ariane 5 rocket explosion In 1996, the European Space Agency s Ariane 5 rocket exploded 40 seconds after launch. During conversion of a 64-bit to

More information

### Floating-Point Arithmetic

Floating-Point Arithmetic if ((A + A) - A == A) { SelfDestruct() } Reading: Study Chapter 3. L12 Multiplication 1 Approximating Real Numbers on Computers Thus far, we ve entirely ignored one of the most

More information

### Numerical Methods in Physics. Lecture 1 Intro & IEEE Variable Types and Arithmetic

Variable types Numerical Methods in Physics Lecture 1 Intro & IEEE Variable Types and Arithmetic Pat Scott Department of Physics, Imperial College November 1, 2016 Slides available from http://astro.ic.ac.uk/pscott/

More information

### 4 Operations On Data 4.1. Foundations of Computer Science Cengage Learning

4 Operations On Data 4.1 Foundations of Computer Science Cengage Learning Objectives After studying this chapter, the student should be able to: List the three categories of operations performed on data.

More information

### Dr. Chuck Cartledge. 3 June 2015

Miscellanea 8224 Revisited Break 1.5 1.6 Conclusion References Backup slides CSC-205 Computer Organization Lecture #002 Section 1.5 Dr. Chuck Cartledge 3 June 2015 1/30 Table of contents I 5 1.6 1 Miscellanea

More information

### Floating Point. What can be represented in N bits? 0 to 2N-1. 9,349,398,989,787,762,244,859,087, x 1067

MIPS Floating Point Operations Cptr280 Dr Curtis Nelson Floating Point What can be represented in N bits? Unsigned 2 s Complement 0 to 2N-1-2N-1 to 2N-1-1 But, what about- Very large numbers? 9,349,398,989,787,762,244,859,087,678

More information

### The Size of the Cantor Set

The Size of the Cantor Set Washington University Math Circle November 6, 2016 In mathematics, a set is a collection of things called elements. For example, {1, 2, 3, 4}, {a,b,c,...,z}, and {cat, dog, chicken}

More information

### Computational Mathematics: Models, Methods and Analysis. Zhilin Li

Computational Mathematics: Models, Methods and Analysis Zhilin Li Chapter 1 Introduction Why is this course important (motivations)? What is the role of this class in the problem solving process using

More information

### 1. NUMBER SYSTEMS USED IN COMPUTING: THE BINARY NUMBER SYSTEM

1. NUMBER SYSTEMS USED IN COMPUTING: THE BINARY NUMBER SYSTEM 1.1 Introduction Given that digital logic and memory devices are based on two electrical states (on and off), it is natural to use a number

More information

### CHAPTER 5 Computer Arithmetic and Round-Off Errors

CHAPTER 5 Computer Arithmetic and Round-Off Errors In the two previous chapters we have seen how numbers can be represented in the binary numeral system and how this is the basis for representing numbers

More information

### Chapter 3. Fundamental Data Types

Chapter 3. Fundamental Data Types Byoung-Tak Zhang TA: Hanock Kwak Biointelligence Laboratory School of Computer Science and Engineering Seoul National Univertisy http://bi.snu.ac.kr Variable Declaration

More information

### Co-processor Math Processor. Richa Upadhyay Prabhu. NMIMS s MPSTME February 9, 2016

8087 Math Processor Richa Upadhyay Prabhu NMIMS s MPSTME richa.upadhyay@nmims.edu February 9, 2016 Introduction Need of Math Processor: In application where fast calculation is required Also where there

More information

### In this article, we present and analyze

[exploratory DSP] Manuel Richey and Hossein Saiedian Compressed Two s Complement Data s Provide Greater Dynamic Range and Improved Noise Performance In this article, we present and analyze a new family

More information

### VHDL IMPLEMENTATION OF IEEE 754 FLOATING POINT UNIT

VHDL IMPLEMENTATION OF IEEE 754 FLOATING POINT UNIT Ms. Anjana Sasidharan Student, Vivekanandha College of Engineering for Women, Namakkal, Tamilnadu, India. Abstract IEEE-754 specifies interchange and

More information

### The Sign consists of a single bit. If this bit is '1', then the number is negative. If this bit is '0', then the number is positive.

IEEE 754 Standard - Overview Frozen Content Modified by on 13-Sep-2017 Before discussing the actual WB_FPU - Wishbone Floating Point Unit peripheral in detail, it is worth spending some time to look at

More information

### Administrivia. CMSC 216 Introduction to Computer Systems Lecture 24 Data Representation and Libraries. Representing characters DATA REPRESENTATION

Administrivia CMSC 216 Introduction to Computer Systems Lecture 24 Data Representation and Libraries Jan Plane & Alan Sussman {jplane, als}@cs.umd.edu Project 6 due next Friday, 12/10 public tests posted

More information

### QUIZ ch.1. 1 st generation 2 nd generation 3 rd generation 4 th generation 5 th generation Rock s Law Moore s Law

QUIZ ch.1 1 st generation 2 nd generation 3 rd generation 4 th generation 5 th generation Rock s Law Moore s Law Integrated circuits Density of silicon chips doubles every 1.5 yrs. Multi-core CPU Transistors

More information

### CO Computer Architecture and Programming Languages CAPL. Lecture 15

CO20-320241 Computer Architecture and Programming Languages CAPL Lecture 15 Dr. Kinga Lipskoch Fall 2017 How to Compute a Binary Float Decimal fraction: 8.703125 Integral part: 8 1000 Fraction part: 0.703125

More information

### Arithmetic. Chapter 3 Computer Organization and Design

Arithmetic Chapter 3 Computer Organization and Design Addition Addition is similar to decimals 0000 0111 + 0000 0101 = 0000 1100 Subtraction (negate) 0000 0111 + 1111 1011 = 0000 0010 Over(under)flow For

More information

### Practical Numerical Methods in Physics and Astronomy. Lecture 1 Intro & IEEE Variable Types and Arithmetic

Practical Numerical Methods in Physics and Astronomy Lecture 1 Intro & IEEE Variable Types and Arithmetic Pat Scott Department of Physics, McGill University January 16, 2013 Slides available from http://www.physics.mcgill.ca/

More information