Numerical Matrix Analysis


 Poppy Gordon
 5 years ago
 Views:
Transcription
1 Numerical Matrix Analysis Lecture Notes #10 Conditioning and / Peter Blomgren, Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences Research Center San Diego State University San Diego, CA Spring 2016 Peter Blomgren, / (1/23)
2 Outline 1 Finite Precision IEEE Binary Floating Point (from Math 541) Nonrepresentable Values a Source of Errors 2 Theorem and Notation Fundamental Axiom of Example 3 Introduction: What is the correct answer? Accuracy Absolute and Relative Error, and Backward Peter Blomgren, / (2/23)
3 IEEE Binary Floating Point (from Math 541) Nonrepresentable Values a Source of Errors Finite Precision A 64bit real number, double The Binary Standard (IEEE The Institute for Electrical and Electronics Engineers) standard specified the following layout for a 64bit real number: Where sc 10 c 9... c 1 c 0 m 51 m m 1 m 0 Symbol Bits Description s 1 The sign bit 0=positive, 1=negative c 11 The characteristic (exponent) m 52 The mantissa r = ( 1) s 2 c 1023 (1+f), c = 10 k=0 c n 2 n, f = 51 k=0 m k 2 52 k Peter Blomgren, / (3/23)
4 IEEE Binary Floating Point (from Math 541) Nonrepresentable Values a Source of Errors IEEE Special Signals In order to be able to represent zero, ±, and NaN (notanumber), the following special signals are defined in the IEEE standard: Type S (1 bit) C (11 bits) M (52 bits) signaling NaN u 2047 (max).0uuuuu u (*) quiet NaN u 2047 (max).1uuuuu u negative infinity (max) positive infinity (max) negative zero positive zero (*) with at least one 1 bit. From Peter Blomgren, / (4/23)
5 IEEE Binary Floating Point (from Math 541) Nonrepresentable Values a Source of Errors Examples: Finite Precision r = ( 1) s 2 c 1023 (1+f), c = Example #1 10 k=0 c n 2 n, f = 51 k=0 m k 2 52 k 0, , ( r 1 = ( 1) ) = = 3.0 Example #2 (The Smallest Positive Real Number) 0, , r 2 = ( 1) (1+2 52) Peter Blomgren, / (5/23)
6 IEEE Binary Floating Point (from Math 541) Nonrepresentable Values a Source of Errors Examples: Finite Precision r = ( 1) s 2 c 1023 (1+f), c = 10 k=0 c n 2 n, f = 51 k=0 m k 2 52 k Example #3 (The Largest Positive Real Number) 0, , ( r 3 = ( 1) ) 2 52 ( = ) Peter Blomgren, / (6/23)
7 IEEE Binary Floating Point (from Math 541) Nonrepresentable Values a Source of Errors That s Quite a Range! In summary, we can represent { ±0, ± , ± , ±, NaN } and a whole bunch of numbers in ( , ) ( , ) Bottom line: Over or underflowing is usually not a problem in IEEE floating point arithmetic. The problem in scientific computing is what we cannot represent. Peter Blomgren, / (7/23)
8 IEEE Binary Floating Point (from Math 541) Nonrepresentable Values a Source of Errors Fun with Matlab......Integers ( ) 2 53 = 2 ( ) ( ) = 2 ( ) 2 53 = (2 53 1) = 1 realmax = realmin = eps = The smallest notexactlyrepresentable integer is ( ) = 9,007,199,254,740,993. Peter Blomgren, / (8/23)
9 IEEE Binary Floating Point (from Math 541) Nonrepresentable Values a Source of Errors Something is Missing Gaps in the Representation 1 of 3 There are gaps in the floatingpoint representation! Given the representation 0, , for the value v 1 = ( ), the next larger floatingpoint value is 0, , i.e. the value v 2 = ( ) The difference between these two values is = ( ). Any number in the interval (v 1,v 2 ) is not representable! Peter Blomgren, / (9/23)
10 IEEE Binary Floating Point (from Math 541) Nonrepresentable Values a Source of Errors Something is Missing Gaps in the Representation 2 of 3 A gap of doesn t seem too bad... However, the size of the gap depend on the value itself... Consider r = 3.0 0, , and the next value 0, , Here, the difference is = 2 51 ( ). In general, in the interval [2 n,2 n+1 ] the gap is 2 n 52. Peter Blomgren, / (10/23)
11 IEEE Binary Floating Point (from Math 541) Nonrepresentable Values a Source of Errors Something is Missing Gaps in the Representation 3 of 3 At the other extreme, the difference between 0, , and the next value 0, , is = That s a fairly significant gap!!! (A number large enough to comfortably count all the particles in the universe...) See, e.g. mrob/pub/math/numbers10.html Peter Blomgren, / (11/23)
12 IEEE Binary Floating Point (from Math 541) Nonrepresentable Values a Source of Errors The Relative Gap It makes more sense to factor the exponent out of the discussion and talk about the relative gap: Exponent Gap Relative Gap (Gap/Exponent) Any difference between numbers smaller than the local gap is not representable, e.g. any number in the interval [ 3.0, ) 2 51 is represented by the value 3.0. Peter Blomgren, / (12/23)
13 Theorem and Notation Fundamental Axiom of Example The Floating Point Theorem ǫ mach Theorem Floating point numbers represent intervals! Notation: We let fl(x) denote the floating point representation of x R. The relative gap defines ǫ mach x R, there exists ǫ with ǫ ǫ mach, such that fl(x) = x(1+ǫ). In 64bit floating point arithmetic ǫ mach In matlab, the command eps returns this value. Let the symbols,,, and denote the floatingpoint operations: addition, subtraction, multiplication, and division. Peter Blomgren, / (13/23)
14 Theorem and Notation Fundamental Axiom of Example ǫ mach All floatingpoint operations are performed up to some precision, i.e. x y = fl(x +y), x y = fl(x y), x y = fl(x y), x y = fl(x/y) This paired with our definition of ǫ mach gives us Axiom (The Fundamental Axiom of ) For all x,y F (where F is the set of floating point numbers), there exists ǫ with ǫ ǫ mach, such that x y = (x +y)(1+ǫ), x y = (x y)(1+ǫ), x y = (x y)(1+ǫ), x y = (x/y)(1+ǫ) That is every operation of floating point arithmetic is exact up to a relative error of size at most ǫ mach. Peter Blomgren, / (14/23)
15 Theorem and Notation Fundamental Axiom of Example Example: Floating Point Error Scaled by Consider the following polynomial on the interval [1.92, 2.08]: p(x) = (x 2) 9 = x 9 18x x 7 672x x x x x x p(x) = x 9 18x x 7 672x x x x x x 512 p(x) as above p(x) = (x 2) Peter Blomgren, / (15/23)
16 Introduction: What is the correct answer? Accuracy Absolute and Relative Error, and Backward Peter Blomgren, / (16/23)
17 Introduction: What is the correct answer? Accuracy Absolute and Relative Error, and Backward : Introduction 1 of 3 With the knowledge that (floating point) errors happen, we have to redefine the concept of the right answer. Previously, in the context of conditioning we defined a mathematical problem as a map f : X Y where X C n is the set of data (input), and Y C m is the set of solutions. Peter Blomgren, / (17/23)
18 Introduction: What is the correct answer? Accuracy Absolute and Relative Error, and Backward : Introduction 2 of 3 We now define an implementation of an algorithm on a floatingpoint device, where F satisfies the fundamental axiom of floating point arithmetic as another map f : X Y i.e. f( x) Y is a numerical solution of the problem. WikiHistory: Pentium FDIV bug ( 1994) The Pentium FDIV bug was a bug in Intel s original Pentium FPU. Certain FP division operations performed with these processors would produce incorrect results. According to Intel, there were a few missing entries in the lookup table used by the divide operation algorithm. Although encountering the flaw was extremely rare in practice (Byte Magazine estimated that 1 in 9 billion FP divides with random parameters would produce inaccurate results), both the flaw and Intel s initial handling of the matter were heavily criticized. Intel ultimately recalled the defective processors. Peter Blomgren, / (18/23)
19 Introduction: What is the correct answer? Accuracy Absolute and Relative Error, and Backward : Introduction 3 of 3 The task at hand is to make useful statements about f( x). Even though f( x) is affected by many factors roundoff errors, convergence tolerances, competing processes on the computer, etc; we will be able to make (maybe surprisingly) clear statements about f( x). Note that depending on the memory model, the previous state of a memory location may affect the result in e.g. the case of cancellation errors: If we subtract two 16digit numbers with 13 common leading digits, we are left with 3 digits of valid information. We tend to view the remaining 13 digits as random. But really, there is nothing random about what happens inside the computer (we hope!) the randomness will depend on what happened previously... Peter Blomgren, / (19/23)
20 Introduction: What is the correct answer? Accuracy Absolute and Relative Error, and Backward Accuracy The absolute error of a computation is and the relative error is f( x) f( x) f( x) f( x) f( x) this latter quantity will be our standard measure of error. If f is a good algorithm, we expect the relative error to be small, of the order ǫ mach. We say that f is accurate if x X f( x) f( x) f( x) = O(ǫ mach ) Peter Blomgren, / (20/23)
21 Introduction: What is the correct answer? Accuracy Absolute and Relative Error, and Backward Interpretation: O(ǫ mach ) Since all floating point errors are functions of ǫ mach (the relative error in each operation is bounded by ǫ mach ), the relative error of the algorithm must be a function of ǫ mach : f( x) f( x) f( x) = e(ǫ mach ) The statement e(ǫ mach ) = O(ǫ mach ) means that C R + such that e(ǫ mach ) Cǫ mach, as ǫ mach 0 In practice ǫ mach is fixed, and the notation means that if we were to decrease ǫ mach, then our error would decrease at least proportionally to ǫ mach. Peter Blomgren, / (21/23)
22 Introduction: What is the correct answer? Accuracy Absolute and Relative Error, and Backward If the problem f : X Y is illconditioned, then the accuracy goal f( x) f( x) f( x) may be unreasonably ambitious. = O(ǫ mach ) Instead we aim for stability. We say that f is a stable algorithm if x X f( x) f( x) = O(ǫ mach ) f( x) for some x with x x x = O(ǫ mach ) A stable algorithm gives approximately the right answer, to approximately the right question. Peter Blomgren, / (22/23)
23 Introduction: What is the correct answer? Accuracy Absolute and Relative Error, and Backward Backward For many algorithms we can tighten this somewhat vague concept of stability. An algorithm f is backward stable if x X f( x) = f( x) for some x with x x x = O(ǫ mach ) A backward stable algorithm gives exactly the right answer, to approximately the right question. Next time: Examples of stable and unstable algorithms; of Householder triangularization. Peter Blomgren, / (23/23)
Measures of Error: for exact x and approximation x Absolute error e = x x. Relative error r = (x x )/x.
ERRORS and COMPUTER ARITHMETIC Types of Error in Numerical Calculations Initial Data Errors: from experiment, modeling, computer representation; problem dependent but need to know at beginning of calculation.
More informationECE 0142 Computer Organization. Lecture 3 Floating Point Representations
ECE 0142 Computer Organization Lecture 3 Floating Point Representations 1 Floatingpoint arithmetic We often incur floatingpoint programming. Floating point greatly simplifies working with large (e.g.,
More informationCS321. Introduction to Numerical Methods
CS3 Introduction to Numerical Methods Lecture Number Representations and Errors Professor Jun Zhang Department of Computer Science University of Kentucky Lexington, KY 405060633 August 7, 05 Number in
More informationCHAPTER 5 Roundoff errors
CHAPTER 5 Roundoff 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 in computers. Since any
More informationThis Unit: Floating Point Arithmetic. CIS 371 Computer Organization and Design. Readings. Floating Point (FP) Numbers
This Unit: Floating Point Arithmetic CIS 371 Computer Organization and Design Unit 7: Floating Point App App App System software Mem CPU I/O Formats Precision and range IEEE 754 standard Operations Addition
More informationDivide: Paper & Pencil. Computer Architecture ALU Design : Division and Floating Point. Divide algorithm. DIVIDE HARDWARE Version 1
Divide: Paper & Pencil Computer Architecture ALU Design : Division and Floating Point 1001 Quotient Divisor 1000 1001010 Dividend 1000 10 101 1010 1000 10 (or Modulo result) See how big a number can be
More informationBinary Number System. 16. Binary Numbers. Base 10 digits: 0 1 2 3 4 5 6 7 8 9. Base 2 digits: 0 1
Binary Number System 1 Base 10 digits: 0 1 2 3 4 5 6 7 8 9 Base 2 digits: 0 1 Recall that in base 10, the digits of a number are just coefficients of powers of the base (10): 417 = 4 * 10 2 + 1 * 10 1
More informationAttention: This material is copyright 19951997 Chris Hecker. All rights reserved.
Attention: This material is copyright 19951997 Chris Hecker. All rights reserved. You have permission to read this article for your own education. You do not have permission to put it on your website
More informationBinary Division. Decimal Division. Hardware for Binary Division. Simple 16bit Divider Circuit
Decimal Division Remember 4th grade long division? 43 // quotient 12 521 // divisor dividend 480 4136 5 // remainder Shift divisor left (multiply by 10) until MSB lines up with dividend s Repeat until
More information2010/9/19. Binary number system. Binary numbers. Outline. Binary to decimal
2/9/9 Binary number system Computer (electronic) systems prefer binary numbers Binary number: represent a number in base2 Binary numbers 2 3 + 7 + 5 Some terminology Bit: a binary digit ( or ) Hexadecimal
More informationCorrectly Rounded Floatingpoint BinarytoDecimal and DecimaltoBinary Conversion Routines in Standard ML. By Prashanth Tilleti
Correctly Rounded Floatingpoint BinarytoDecimal and DecimaltoBinary Conversion Routines in Standard ML By Prashanth Tilleti Advisor Dr. Matthew Fluet Department of Computer Science B. Thomas Golisano
More informationMath 348: Numerical Methods with application in finance and economics. Boualem Khouider University of Victoria
Math 348: Numerical Methods with application in finance and economics Boualem Khouider University of Victoria Lecture notes: Updated January 2012 Contents 1 Basics of numerical analysis 9 1.1 Notion of
More informationOperation Count; Numerical Linear Algebra
10 Operation Count; Numerical Linear Algebra 10.1 Introduction Many computations are limited simply by the sheer number of required additions, multiplications, or function evaluations. If floatingpoint
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a nonempty
More informationFigure 1. A typical Laboratory Thermometer graduated in C.
SIGNIFICANT FIGURES, EXPONENTS, AND SCIENTIFIC NOTATION 2004, 1990 by David A. Katz. All rights reserved. Permission for classroom use as long as the original copyright is included. 1. SIGNIFICANT FIGURES
More informationComputer Science 281 Binary and Hexadecimal Review
Computer Science 281 Binary and Hexadecimal Review 1 The Binary Number System Computers store everything, both instructions and data, by using many, many transistors, each of which can be in one of two
More informationData Storage 3.1. Foundations of Computer Science Cengage Learning
3 Data Storage 3.1 Foundations of Computer Science Cengage Learning Objectives After studying this chapter, the student should be able to: List five different data types used in a computer. Describe how
More informationMonday January 19th 2015 Title: "Transmathematics  a survey of recent results on division by zero" Facilitator: TheNumberNullity / James Anderson, UK
Monday January 19th 2015 Title: "Transmathematics  a survey of recent results on division by zero" Facilitator: TheNumberNullity / James Anderson, UK It has been my pleasure to give two presentations
More informationComputers. Hardware. The Central Processing Unit (CPU) CMPT 125: Lecture 1: Understanding the Computer
Computers CMPT 125: Lecture 1: Understanding the Computer Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 3, 2009 A computer performs 2 basic functions: 1.
More informationATheoryofComplexity,Conditionand Roundoff
ATheoryofComplexity,Conditionand Roundoff Felipe Cucker Berkeley 2014 Background L. Blum, M. Shub, and S. Smale [1989] On a theory of computation and complexity over the real numbers: NPcompleteness,
More informationTo convert an arbitrary power of 2 into its English equivalent, remember the rules of exponential arithmetic:
Binary Numbers In computer science we deal almost exclusively with binary numbers. it will be very helpful to memorize some binary constants and their decimal and English equivalents. By English equivalents
More informationThis document has been written by Michaël Baudin from the Scilab Consortium. December 2010 The Scilab Consortium Digiteo. All rights reserved.
SCILAB IS NOT NAIVE This document has been written by Michaël Baudin from the Scilab Consortium. December 2010 The Scilab Consortium Digiteo. All rights reserved. December 2010 Abstract Most of the time,
More informationData Storage. Chapter 3. Objectives. 31 Data Types. Data Inside the Computer. After studying this chapter, students should be able to:
Chapter 3 Data Storage Objectives After studying this chapter, students should be able to: List five different data types used in a computer. Describe how integers are stored in a computer. Describe how
More information1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal:
Exercises 1  number representations Questions 1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal: (a) 3012 (b)  435 2. For each of
More informationPrinciples of Scientific Computing. David Bindel and Jonathan Goodman
Principles of Scientific Computing David Bindel and Jonathan Goodman last revised February 2009, last printed March 6, 2009 2 Preface i ii PREFACE This book grew out of a one semester first course in Scientific
More informationTHE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0
THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0 RICHARD J. MATHAR Abstract. We count solutions to the RamanujanNagell equation 2 y +n = x 2 for fixed positive n. The computational
More informationNumerical Analysis I
Numerical Analysis I M.R. O Donohoe References: S.D. Conte & C. de Boor, Elementary Numerical Analysis: An Algorithmic Approach, Third edition, 1981. McGrawHill. L.F. Shampine, R.C. Allen, Jr & S. Pruess,
More informationSo let us begin our quest to find the holy grail of real analysis.
1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers
More informationUseful Number Systems
Useful Number Systems Decimal Base = 10 Digit Set = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Binary Base = 2 Digit Set = {0, 1} Octal Base = 8 = 2 3 Digit Set = {0, 1, 2, 3, 4, 5, 6, 7} Hexadecimal Base = 16 = 2
More informationLies My Calculator and Computer Told Me
Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing
More informationparent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL
parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL HS America s schools are working to provide higher quality instruction than ever before. The way we taught students in the past simply does
More informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More informationMath 115 Spring 2011 Written Homework 5 Solutions
. Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence
More informationLinear Equations and Inequalities
Linear Equations and Inequalities Section 1.1 Prof. Wodarz Math 109  Fall 2008 Contents 1 Linear Equations 2 1.1 Standard Form of a Linear Equation................ 2 1.2 Solving Linear Equations......................
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationReproducible Science and Modern Scientific Software Development
Reproducible Science and Modern Scientific Software Development 13 th evita Winter School in escience sponsored by Dr. Holms Hotel, Geilo, Norway January 2025, 2013 Advanced Topics Dr. André R. Brodtkorb,
More information1 Error in Euler s Method
1 Error in Euler s Method Experience with Euler s 1 method raises some interesting questions about numerical approximations for the solutions of differential equations. 1. What determines the amount of
More informationGrade 6 Mathematics Performance Level Descriptors
Limited Grade 6 Mathematics Performance Level Descriptors A student performing at the Limited Level demonstrates a minimal command of Ohio s Learning Standards for Grade 6 Mathematics. A student at this
More informationA new binary floatingpoint division algorithm and its software implementation on the ST231 processor
19th IEEE Symposium on Computer Arithmetic (ARITH 19) Portland, Oregon, USA, June 810, 2009 A new binary floatingpoint division algorithm and its software implementation on the ST231 processor ClaudePierre
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions  that is, algebraic fractions  and equations which contain them. The reader is encouraged to
More informationFAST INVERSE SQUARE ROOT
FAST INVERSE SQUARE ROOT CHRIS LOMONT Abstract. Computing reciprocal square roots is necessary in many applications, such as vector normalization in video games. Often, some loss of precision is acceptable
More informationChapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs
Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s
More informationThe string of digits 101101 in the binary number system represents the quantity
Data Representation Section 3.1 Data Types Registers contain either data or control information Control information is a bit or group of bits used to specify the sequence of command signals needed for
More informationInformation Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay
Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture  17 ShannonFanoElias Coding and Introduction to Arithmetic Coding
More informationThis 3digit ASCII string could also be calculated as n = (Data[2]0x30) +10*((Data[1]0x30)+10*(Data[0]0x30));
Introduction to Embedded Microcomputer Systems Lecture 5.1 2.9. Conversions ASCII to binary n = 100*(Data[0]0x30) + 10*(Data[1]0x30) + (Data[2]0x30); This 3digit ASCII string could also be calculated
More informationFast Arithmetic Coding (FastAC) Implementations
Fast Arithmetic Coding (FastAC) Implementations Amir Said 1 Introduction This document describes our fast implementations of arithmetic coding, which achieve optimal compression and higher throughput by
More information8 Square matrices continued: Determinants
8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You
More information1 The Brownian bridge construction
The Brownian bridge construction The Brownian bridge construction is a way to build a Brownian motion path by successively adding finer scale detail. This construction leads to a relatively easy proof
More informationNumber Representation
Number Representation CS10001: Programming & Data Structures Pallab Dasgupta Professor, Dept. of Computer Sc. & Engg., Indian Institute of Technology Kharagpur Topics to be Discussed How are numeric data
More informationWhat Every Computer Scientist Should Know About FloatingPoint Arithmetic
What Every Computer Scientist Should Know About FloatingPoint Arithmetic D Note This document is an edited reprint of the paper What Every Computer Scientist Should Know About FloatingPoint Arithmetic,
More informationStanford Math Circle: Sunday, May 9, 2010 SquareTriangular Numbers, Pell s Equation, and Continued Fractions
Stanford Math Circle: Sunday, May 9, 00 SquareTriangular Numbers, Pell s Equation, and Continued Fractions Recall that triangular numbers are numbers of the form T m = numbers that can be arranged in
More informationNumbering Systems. InThisAppendix...
G InThisAppendix... Introduction Binary Numbering System Hexadecimal Numbering System Octal Numbering System Binary Coded Decimal (BCD) Numbering System Real (Floating Point) Numbering System BCD/Binary/Decimal/Hex/Octal
More information1 Short Introduction to Time Series
ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The
More informationCS101 Lecture 11: Number Systems and Binary Numbers. Aaron Stevens 14 February 2011
CS101 Lecture 11: Number Systems and Binary Numbers Aaron Stevens 14 February 2011 1 2 1 3!!! MATH WARNING!!! TODAY S LECTURE CONTAINS TRACE AMOUNTS OF ARITHMETIC AND ALGEBRA PLEASE BE ADVISED THAT CALCULTORS
More informationTHE EXACT DOT PRODUCT AS BASIC TOOL FOR LONG INTERVAL ARITHMETIC
THE EXACT DOT PRODUCT AS BASIC TOOL FOR LONG INTERVAL ARITHMETIC ULRICH KULISCH AND VAN SNYDER Abstract. Computing with guarantees is based on two arithmetical features. One is fixed (double) precision
More informationModeling, Computers, and Error Analysis Mathematical Modeling and Engineering ProblemSolving
Next: Roots of Equations Up: Numerical Analysis for Chemical Previous: Contents Subsections Mathematical Modeling and Engineering ProblemSolving A Simple Mathematical Model Computers and Software The
More informationA Static Analyzer for Large SafetyCritical Software. Considered Programs and Semantics. Automatic Program Verification by Abstract Interpretation
PLDI 03 A Static Analyzer for Large SafetyCritical Software B. Blanchet, P. Cousot, R. Cousot, J. Feret L. Mauborgne, A. Miné, D. Monniaux,. Rival CNRS École normale supérieure École polytechnique Paris
More informationMA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity
MA4001 Engineering Mathematics 1 Lecture 10 Limits and Dr. Sarah Mitchell Autumn 2014 Infinite limits If f(x) grows arbitrarily large as x a we say that f(x) has an infinite limit. Example: f(x) = 1 x
More informationNotes on Factoring. MA 206 Kurt Bryan
The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationRoots of Polynomials
Roots of Polynomials (Com S 477/577 Notes) YanBin Jia Sep 24, 2015 A direct corollary of the fundamental theorem of algebra is that p(x) can be factorized over the complex domain into a product a n (x
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More information26 Integers: Multiplication, Division, and Order
26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue
More informationSome Functions Computable with a Fusedmac
Some Functions Computable with a Fusedmac Sylvie Boldo and JeanMichel Muller Laboratoire LIP (CNRS/ENS Lyon/Inria/Univ. lyon ), Projet Arénaire, 46 allée d Italie, 69364 Lyon Cedex 07, FRANCE Sylvie.Boldo@enslyon.fr,
More informationChapter 5. Binary, octal and hexadecimal numbers
Chapter 5. Binary, octal and hexadecimal numbers A place to look for some of this material is the Wikipedia page http://en.wikipedia.org/wiki/binary_numeral_system#counting_in_binary Another place that
More informationArithmetic Coding: Introduction
Data Compression Arithmetic coding Arithmetic Coding: Introduction Allows using fractional parts of bits!! Used in PPM, JPEG/MPEG (as option), Bzip More time costly than Huffman, but integer implementation
More informationLecture 2. Binary and Hexadecimal Numbers
Lecture 2 Binary and Hexadecimal Numbers Purpose: Review binary and hexadecimal number representations Convert directly from one base to another base Review addition and subtraction in binary representations
More informationFMA Implementations of the Compensated Horner Scheme
Reliable Implementation of Real Number Algorithms: Theory and Practice Dagstuhl Seminar 06021 January 0813, 2006 FMA Implementations of the Compensated Horner Scheme Stef GRAILLAT, Philippe LANGLOIS and
More informationCONTENTS 1. Peter Kahn. Spring 2007
CONTENTS 1 MATH 304: CONSTRUCTING THE REAL NUMBERS Peter Kahn Spring 2007 Contents 2 The Integers 1 2.1 The basic construction.......................... 1 2.2 Adding integers..............................
More informationBinary Numbering Systems
Binary Numbering Systems April 1997, ver. 1 Application Note 83 Introduction Binary numbering systems are used in virtually all digital systems, including digital signal processing (DSP), networking, and
More informationChapter 1. Binary, octal and hexadecimal numbers
Chapter 1. Binary, octal and hexadecimal numbers This material is covered in the books: Nelson Magor Cooke et al, Basic mathematics for electronics (7th edition), Glencoe, Lake Forest, Ill., 1992. [Hamilton
More informationLSN 2 Number Systems. ECT 224 Digital Computer Fundamentals. Department of Engineering Technology
LSN 2 Number Systems Department of Engineering Technology LSN 2 Decimal Number System Decimal number system has 10 digits (09) Base 10 weighting system... 10 5 10 4 10 3 10 2 10 1 10 0. 101 102 103
More informationDNA Data and Program Representation. Alexandre David 1.2.05 adavid@cs.aau.dk
DNA Data and Program Representation Alexandre David 1.2.05 adavid@cs.aau.dk Introduction Very important to understand how data is represented. operations limits precision Digital logic built on 2valued
More information1 Review of Least Squares Solutions to Overdetermined Systems
cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares
More informationSystems I: Computer Organization and Architecture
Systems I: Computer Organization and Architecture Lecture 2: Number Systems and Arithmetic Number Systems  Base The number system that we use is base : 734 = + 7 + 3 + 4 = x + 7x + 3x + 4x = x 3 + 7x
More informationAdvanced Tutorials. Numeric Data In SAS : Guidelines for Storage and Display Paul Gorrell, Social & Scientific Systems, Inc., Silver Spring, MD
Numeric Data In SAS : Guidelines for Storage and Display Paul Gorrell, Social & Scientific Systems, Inc., Silver Spring, MD ABSTRACT Understanding how SAS stores and displays numeric data is essential
More informationNumber Systems and Radix Conversion
Number Systems and Radix Conversion Sanjay Rajopadhye, Colorado State University 1 Introduction These notes for CS 270 describe polynomial number systems. The material is not in the textbook, but will
More informationComputer Arithmetic Aliasing issues: Call by reference, Pointer programs
Computer Arithmetic Aliasing issues: Call by reference, Pointer programs Claude Marché Cours MPRI 2361 Preuve de Programme 24 janvier 2014 Outline Exercises from last lecture Computer Arithmetic Handling
More informationThis unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
More informationReading 13 : Finite State Automata and Regular Expressions
CS/Math 24: Introduction to Discrete Mathematics Fall 25 Reading 3 : Finite State Automata and Regular Expressions Instructors: Beck Hasti, Gautam Prakriya In this reading we study a mathematical model
More informationSolution of Linear Systems
Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start
More informationA numerically adaptive implementation of the simplex method
A numerically adaptive implementation of the simplex method József Smidla, Péter Tar, István Maros Department of Computer Science and Systems Technology University of Pannonia 17th of December 2014. 1
More information2.3. Finding polynomial functions. An Introduction:
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
More informationRNcoding of Numbers: New Insights and Some Applications
RNcoding of Numbers: New Insights and Some Applications Peter Kornerup Dept. of Mathematics and Computer Science SDU, Odense, Denmark & JeanMichel Muller LIP/Arénaire (CRNSENS LyonINRIAUCBL) Lyon,
More informationPreliminary Mathematics
Preliminary Mathematics The purpose of this document is to provide you with a refresher over some topics that will be essential for what we do in this class. We will begin with fractions, decimals, and
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationFLOATINGPOINT ARITHMETIC IN AMD PROCESSORS MICHAEL SCHULTE AMD RESEARCH JUNE 2015
FLOATINGPOINT ARITHMETIC IN AMD PROCESSORS MICHAEL SCHULTE AMD RESEARCH JUNE 2015 AGENDA The Kaveri Accelerated Processing Unit (APU) The Graphics Core Next Architecture and its FloatingPoint Arithmetic
More informationCSI 333 Lecture 1 Number Systems
CSI 333 Lecture 1 Number Systems 1 1 / 23 Basics of Number Systems Ref: Appendix C of Deitel & Deitel. Weighted Positional Notation: 192 = 2 10 0 + 9 10 1 + 1 10 2 General: Digit sequence : d n 1 d n 2...
More informationToday. Binary addition Representing negative numbers. Andrew H. Fagg: Embedded Real Time Systems: Binary Arithmetic
Today Binary addition Representing negative numbers 2 Binary Addition Consider the following binary numbers: 0 0 1 0 0 1 1 0 0 0 1 0 1 0 1 1 How do we add these numbers? 3 Binary Addition 0 0 1 0 0 1 1
More informationAMATH 352 Lecture 3 MATLAB Tutorial Starting MATLAB Entering Variables
AMATH 352 Lecture 3 MATLAB Tutorial MATLAB (short for MATrix LABoratory) is a very useful piece of software for numerical analysis. It provides an environment for computation and the visualization. Learning
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationData Storage: Each time you create a variable in memory, a certain amount of memory is allocated for that variable based on its data type (or class).
Data Storage: Computers are made of many small parts, including transistors, capacitors, resistors, magnetic materials, etc. Somehow they have to store information in these materials both temporarily (RAM,
More informationOutline. hardware components programming environments. installing Python executing Python code. decimal and binary notations running Sage
Outline 1 Computer Architecture hardware components programming environments 2 Getting Started with Python installing Python executing Python code 3 Number Systems decimal and binary notations running
More informationLevent EREN levent.eren@ieu.edu.tr A306 Office Phone:4889882 INTRODUCTION TO DIGITAL LOGIC
Levent EREN levent.eren@ieu.edu.tr A306 Office Phone:4889882 1 Number Systems Representation Positive radix, positional number systems A number with radix r is represented by a string of digits: A n
More informationZuse's Z3 Square Root Algorithm Talk given at Fall meeting of the Ohio Section of the MAA October 1999  College of Wooster
Zuse's Z3 Square Root Algorithm Talk given at Fall meeting of the Ohio Section of the MAA October 1999  College of Wooster Abstract Brian J. Shelburne Dept of Math and Comp Sci Wittenberg University In
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationOct: 50 8 = 6 (r = 2) 6 8 = 0 (r = 6) Writing the remainders in reverse order we get: (50) 10 = (62) 8
ECE Department Summer LECTURE #5: Number Systems EEL : Digital Logic and Computer Systems Based on lecture notes by Dr. Eric M. Schwartz Decimal Number System: Our standard number system is base, also
More informationInternational Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013
FACTORING CRYPTOSYSTEM MODULI WHEN THE COFACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II MohammediaCasablanca,
More informationSession 29 Scientific Notation and Laws of Exponents. If you have ever taken a Chemistry class, you may have encountered the following numbers:
Session 9 Scientific Notation and Laws of Exponents If you have ever taken a Chemistry class, you may have encountered the following numbers: There are approximately 60,4,79,00,000,000,000,000 molecules
More information