# Floating-Point Arithmetic Algorithms in the

Save this PDF as:

Size: px
Start display at page:

## Transcription

4 1 Sice the m M ki <m k 8 O the other had, whether sl = O or #0, ki k = M1S +1 ms + ms but hece This cotradicts (16). Hece m 8al+ m~~~~~ 1 8sl+ 8s >m ki k 8 SI = S. 8~~~ (16) Cosequetl, k, =k. This completes the proof of the theorem. For a real x we deote b fl (x) either of the two umbers of the form (13) which miimize fl(x) -x I, except that if the fl(lxx)= MMM<M I x I < MM (M+ m +l1) I1a(M + I MeM+, (si -1) +IMl)Me+,M0 (s=-l) The value fl(x) will be called a ormalized floatig-poit represetatio of x. We defme the rage of floatig-poit umbers to be the iterval D= [M-1 MEM, M- MEMl ~~ 1~j Let fl(x) = kmem. be the ormalized floatig-poit represetatio of x for x e D, Ixl > ((M/m,) + IW-E. The, sice m MeM < Ixl< MAme, (SOO) s MeM8. lxi< MMl M8, (S=#O) (17) IEEE TRANSACTIONS ON COMPUTERS, JANUARY 1974 it is easil see that e, s, ad k ca be determied i the followig maer: e = [logmmf Ixl -1. The iteger s is foud as follows. First compute 1x1 Me+l This quatit is used i the relatioship Gi_ M m. to obtai G,, G,, etc. This iterative procedure is cotiued util [Gi] = 0. If this occurs o the ith (i = 0, 1, ), the Fiall k is defied b s = i. k= ( \MeMS Here the R refers to correct roudig i fixed-poit arithmetic. If e is outside the iterval -E < e < E, we shall ol sa that a overflow or a uderflow has occurred ad shall ot proceed further. Coversio to the Residue Represetatio For the iteger k i (13) the least remaider i absolute value whe divided b mi ma be computed. This quatit, is referred to as the smmetric residue of k deoted b /k/mi, mod mi, ad the radices mi are called bases or modull. For a give set of moduli the residues of k ma be formed ito a -tuple Ik/ml J, klm Iklm This -tuple is called the smmetric residue represetatio of k. The iteger 1k/rm is called the ith smmetric residue digit -M-1 M-l to ma be uiquel represeted. Now cosider the coversio from a fixed-radix sstem such as decimal or biar to the residue sstem. The iteger k is specified i a fixed-radix sstem as k=d,r±+d, r-1 + +dir+do r is the radix ad 0 < di 6 r -1. The, takig this expressio modulo mi, the followig equatios are obtaied: km =/dllr'lm + d_1 Ir'-'/mi+ +d,irlmi+ do/ml (i= 1,,,).

9 KINOSHITA et al.: FLOATING-POINT ARITHMETIC 17 TABLE II TABLE COMPOSED OF LS = W/ms ) + 1) s z L S z 1 ((L1))5 ((L))5 3 ((L3))5 4 ((L4))5 TABLE III TABLE COMPOSED OF HiI=Zz++l m1 Z ((mm3m4m5 1)6 (( M34m5m1 ) 6 (m4m5m1m ))6 ((m5m1mm3 ))6 (( 1m m3m4 ))6 (( m3m4m5 ) 6 (Im4r5mr1 ) 6 ( m5m1m ) 6 r ml mr3 ) 6 ( mm3m4 ) 6 ((r3r4m5 6 (( m5r1 ) 6 r(( 1r ) 6 ( mr3 ) 6 r( 3m4 ) 6 4 (( m5 ( 6 ((m1 )6 ((r ( m 6 (( m3 ( ) 6 m((4 ( 6 4 TABLE IV TABLE COMPOSED OF MS s z i s z 1 ((ri1) F, 6 ((I ) 3 (( m3 )) 6 4 ((m4 )) 6 TABLE V TABLE COMPOSED OFA VALUES S X (( A )) ( 0 ( 0 m mm I m 1 (( r 3r4i5 ) 10 (( 10 ( r 3m4 5 ) 10 (t 3r4r5 )) 34m 10 (( 3 m m3m4m5 )) 10 ((-1mm ( 1(mlm3 4mb )) 10 1( i )) mS ) 10 (( r3r4 mii )5 10 ((m3i4ms ) 10 (( m14ff5 )) 10 r 1 r4r5 )) ) 10 4 (( m m - m m mm34 5 ) 10 (( m (( (( :4 A 10 _ (~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 3) If Ae =0, check whether sx = s. If so, set KR = k. Otherwise (if sx > s ), scale k b m +.m..+ m. ad let the result to be KR. 4) Compute kz = kx ± KR ad set ez = ex, sz = sx. 5) Fid the mixed-radix digits cal, a,.',a+, of kz which are associated with the smmetric residue sstem ml, m, m+l1 If all the digits a(i= 1,,, + 1) are zero, set ez = 0, sz = 0. 6) Otherwise, deote b au the most sigificat ozero digit. If u = + 1, scale kz b m,z+l ad icrease sz b 1. If sz has reached, set sz = 0 ad icrease ez b 1. 7) If u <, multipl k b HIzs=! fl+u+imi which ca be

### Chapter Gaussian Elimination

Chapter 04.06 Gaussia Elimiatio After readig this chapter, you should be able to:. solve a set of simultaeous liear equatios usig Naïve Gauss elimiatio,. lear the pitfalls of the Naïve Gauss elimiatio

### THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,

### 1. MATHEMATICAL INDUCTION

1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1

### Complex Numbers. where x represents a root of Equation 1. Note that the ± sign tells us that quadratic equations will have

Comple Numbers I spite of Calvi s discomfiture, imagiar umbers (a subset of the set of comple umbers) eist ad are ivaluable i mathematics, egieerig, ad sciece. I fact, i certai fields, such as electrical

### 3. Greatest Common Divisor - Least Common Multiple

3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd

### ARITHMETIC AND GEOMETRIC PROGRESSIONS

Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives

### Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

### Modified Line Search Method for Global Optimization

Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o

### CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity

### I. Chi-squared Distributions

1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

### Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015

CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a

### Basic Elements of Arithmetic Sequences and Series

MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic

### Equation of a line. Line in coordinate geometry. Slope-intercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Point-slope form ( 點 斜 式 )

Chapter : Liear Equatios Chapter Liear Equatios Lie i coordiate geometr I Cartesia coordiate sstems ( 卡 笛 兒 坐 標 系 統 ), a lie ca be represeted b a liear equatio, i.e., a polomial with degree. But before

### 8.3 POLAR FORM AND DEMOIVRE S THEOREM

SECTION 8. POLAR FORM AND DEMOIVRE S THEOREM 48 8. POLAR FORM AND DEMOIVRE S THEOREM Figure 8.6 (a, b) b r a 0 θ Complex Number: a + bi Rectagular Form: (a, b) Polar Form: (r, θ) At this poit you ca add,

### Sequences and Series

CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

### Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is

0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values

### 4.1 Sigma Notation and Riemann Sums

0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

### Lesson 12. Sequences and Series

Retur to List of Lessos Lesso. Sequeces ad Series A ifiite sequece { a, a, a,... a,...} ca be thought of as a list of umbers writte i defiite order ad certai patter. It is usually deoted by { a } =, or

### In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

### NPTEL STRUCTURAL RELIABILITY

NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics

### .04. This means \$1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,

### One-step equations. Vocabulary

Review solvig oe-step equatios with itegers, fractios, ad decimals. Oe-step equatios Vocabulary equatio solve solutio iverse operatio isolate the variable Additio Property of Equality Subtractio Property

### Divide and Conquer, Solving Recurrences, Integer Multiplication Scribe: Juliana Cook (2015), V. Williams Date: April 6, 2016

CS 6, Lecture 3 Divide ad Coquer, Solvig Recurreces, Iteger Multiplicatio Scribe: Juliaa Cook (05, V Williams Date: April 6, 06 Itroductio Today we will cotiue to talk about divide ad coquer, ad go ito

### Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.

5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers

### The Field of Complex Numbers

The Field of Complex Numbers S. F. Ellermeyer The costructio of the system of complex umbers begis by appedig to the system of real umbers a umber which we call i with the property that i = 1. (Note that

### 8.1 Arithmetic Sequences

MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first

### when n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.

Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have

### Recursion and Recurrences

Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,

### Section IV.5: Recurrence Relations from Algorithms

Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or by

### SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

### CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all

### Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions

Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig

### Department of Computer Science, University of Otago

Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

### Learning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr.

Algorithms ad Data Structures Algorithm efficiecy Learig outcomes Able to carry out simple asymptotic aalysisof algorithms Prof. Dr. Qi Xi 2 Time Complexity Aalysis How fast is the algorithm? Code the

### Infinite Sequences and Series

CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

### Mathematical goals. Starting points. Materials required. Time needed

Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios

### Review for College Algebra Final Exam

Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 1-4. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i

### hp calculators HP 30S Base Conversions Numbers in Different Bases Practice Working with Numbers in Different Bases

Numbers i Differet Bases Practice Workig with Numbers i Differet Bases Numbers i differet bases Our umber system (called Hidu-Arabic) is a decimal system (it s also sometimes referred to as deary system)

### Confidence Intervals and Sample Size

8/7/015 C H A P T E R S E V E N Cofidece Itervals ad Copyright 015 The McGraw-Hill Compaies, Ic. Permissio required for reproductio or display. 1 Cofidece Itervals ad Outlie 7-1 Cofidece Itervals for the

### Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio

### Alternatives To Pearson s and Spearman s Correlation Coefficients

Alteratives To Pearso s ad Spearma s Correlatio Coefficiets Floreti Smaradache Chair of Math & Scieces Departmet Uiversity of New Mexico Gallup, NM 8730, USA Abstract. This article presets several alteratives

### Asymptotic Growth of Functions

CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

### The Euler Totient, the Möbius and the Divisor Functions

The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship

### Factors of sums of powers of binomial coefficients

ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the

### Solving Divide-and-Conquer Recurrences

Solvig Divide-ad-Coquer Recurreces Victor Adamchik A divide-ad-coquer algorithm cosists of three steps: dividig a problem ito smaller subproblems solvig (recursively) each subproblem the combiig solutios

Chapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity

### Section 9.2 Series and Convergence

Sectio 9. Series ad Covergece Goals of Chapter 9 Approximate Pi Prove ifiite series are aother importat applicatio of limits, derivatives, approximatio, slope, ad cocavity of fuctios. Fid challegig atiderivatives

### x(x 1)(x 2)... (x k + 1) = [x] k n+m 1

1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,

### Section 6.1 Radicals and Rational Exponents

Sectio 6.1 Radicals ad Ratioal Expoets Defiitio of Square Root The umber b is a square root of a if b The priciple square root of a positive umber is its positive square root ad we deote this root by usig

### COMP 251 Assignment 2 Solutions

COMP 251 Assigmet 2 Solutios Questio 1 Exercise 8.3-4 Treat the umbers as 2-digit umbers i radix. Each digit rages from 0 to 1. Sort these 2-digit umbers ith the RADIX-SORT algorithm preseted i Sectio

### Algebra Work Sheets. Contents

The work sheets are grouped accordig to math skill. Each skill is the arraged i a sequece of work sheets that build from simple to complex. Choose the work sheets that best fit the studet s eed ad will

### BENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets

BENEIT-CST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal - Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts

### Chapter 7 Methods of Finding Estimators

Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

### Systems Design Project: Indoor Location of Wireless Devices

Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 698-5295 Email: bcm1@cec.wustl.edu Supervised

### Linear Algebra II. 4 Determinants. Notes 4 1st November Definition of determinant

MTH6140 Liear Algebra II Notes 4 1st November 2010 4 Determiats The determiat is a fuctio defied o square matrices; its value is a scalar. It has some very importat properties: perhaps most importat is

### Soving Recurrence Relations

Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

### ME 101 Measurement Demonstration (MD 1) DEFINITIONS Precision - A measure of agreement between repeated measurements (repeatability).

INTRODUCTION This laboratory ivestigatio ivolves makig both legth ad mass measuremets of a populatio, ad the assessig statistical parameters to describe that populatio. For example, oe may wat to determie

### Confidence Intervals for One Mean

Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a

### The Field Q of Rational Numbers

Chapter 3 The Field Q of Ratioal Numbers I this chapter we are goig to costruct the ratioal umber from the itegers. Historically, the positive ratioal umbers came first: the Babyloias, Egyptias ad Grees

### ORDERS OF GROWTH KEITH CONRAD

ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus

### Page 2 of 14 = T(-2) + 2 = [ T(-3)+1 ] + 2 Substitute T(-3)+1 for T(-2) = T(-3) + 3 = [ T(-4)+1 ] + 3 Substitute T(-4)+1 for T(-3) = T(-4) + 4 After i

Page 1 of 14 Search C455 Chapter 4 - Recursio Tree Documet last modified: 02/09/2012 18:42:34 Uses: Use recursio tree to determie a good asymptotic boud o the recurrece T() = Sum the costs withi each level

### Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.

Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory

### BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients

652 (12-26) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you

### Vladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT

Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee

### SEQUENCES AND SERIES

Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say

### 13 Fast Fourier Transform (FFT)

13 Fast Fourier Trasform FFT) The fast Fourier trasform FFT) is a algorithm for the efficiet implemetatio of the discrete Fourier trasform. We begi our discussio oce more with the cotiuous Fourier trasform.

### NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS

### Confidence Intervals for One Mean with Tolerance Probability

Chapter 421 Cofidece Itervals for Oe Mea with Tolerace Probability Itroductio This procedure calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) with

### Automatic Tuning for FOREX Trading System Using Fuzzy Time Series

utomatic Tuig for FOREX Tradig System Usig Fuzzy Time Series Kraimo Maeesilp ad Pitihate Soorasa bstract Efficiecy of the automatic currecy tradig system is time depedet due to usig fixed parameters which

### arxiv:1012.1336v2 [cs.cc] 8 Dec 2010

Uary Subset-Sum is i Logspace arxiv:1012.1336v2 [cs.cc] 8 Dec 2010 1 Itroductio Daiel M. Kae December 9, 2010 I this paper we cosider the Uary Subset-Sum problem which is defied as follows: Give itegers

### 5 Boolean Decision Trees (February 11)

5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected

### A CHARACTERIZATION OF MINIMAL ZERO-SEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A27 A CHARACTERIZATION OF MINIMAL ZERO-SEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS Scott T. Chapma 1 Triity Uiversity, Departmet

### Algebra Vocabulary List (Definitions for Middle School Teachers)

Algebra Vocabulary List (Defiitios for Middle School Teachers) A Absolute Value Fuctio The absolute value of a real umber x, x is xifx 0 x = xifx < 0 http://www.math.tamu.edu/~stecher/171/f02/absolutevaluefuctio.pdf

### 2.7 Sequences, Sequences of Sets

2.7. SEQUENCES, SEQUENCES OF SETS 67 2.7 Sequeces, Sequeces of Sets 2.7.1 Sequeces Defiitio 190 (sequece Let S be some set. 1. A sequece i S is a fuctio f : K S where K = { N : 0 for some 0 N}. 2. For

### Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

### Module 4: Mathematical Induction

Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate

Radicals ad Roots Radicals ad Fractioal Expoets I math, may problems will ivolve what is called the radical symbol, X is proouced the th root of X, where is or greater, ad X is a positive umber. What it

### SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

### Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley

Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Cosider a legth- sequece x[ with a -poit DFT X[ where Represet the idices ad as +, +, Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Usig these

### 7. Sample Covariance and Correlation

1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y

### 3. Covariance and Correlation

Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics

### Gregory Carey, 1998 Linear Transformations & Composites - 1. Linear Transformations and Linear Composites

Gregory Carey, 1998 Liear Trasformatios & Composites - 1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio

### Approximating the Sum of a Convergent Series

Approximatig the Sum of a Coverget Series Larry Riddle Ages Scott College Decatur, GA 30030 lriddle@agesscott.edu The BC Calculus Course Descriptio metios how techology ca be used to explore covergece

### Chapter 5: Inner Product Spaces

Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples

### Chapter 9: Correlation and Regression: Solutions

Chapter 9: Correlatio ad Regressio: Solutios 9.1 Correlatio I this sectio, we aim to aswer the questio: Is there a relatioship betwee A ad B? Is there a relatioship betwee the umber of emploee traiig hours

### Determining the sample size

Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors

### Geometric Sequences and Series. Geometric Sequences. Definition of Geometric Sequence. such that. a2 4

3330_0903qxd /5/05 :3 AM Page 663 Sectio 93 93 Geometric Sequeces ad Series 663 Geometric Sequeces ad Series What you should lear Recogize, write, ad fid the th terms of geometric sequeces Fid th partial

### Incremental calculation of weighted mean and variance

Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

### + 1= x + 1. These 4 elements form a field.

Itroductio to fiite fields II Fiite field of p elemets F Because we are iterested i doig computer thigs it would be useful for us to costruct fields havig elemets. Let s costruct a field of elemets; we

### Research Article Sign Data Derivative Recovery

Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov

### Grade 7. Strand: Number Specific Learning Outcomes It is expected that students will:

Strad: Number Specific Learig Outcomes It is expected that studets will: 7.N.1. Determie ad explai why a umber is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, ad why a umber caot be divided by 0. [C, R] [C]

### A probabilistic proof of a binomial identity

A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

### Section 11.3: The Integral Test

Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

### A Guide to the Pricing Conventions of SFE Interest Rate Products

A Guide to the Pricig Covetios of SFE Iterest Rate Products SFE 30 Day Iterbak Cash Rate Futures Physical 90 Day Bak Bills SFE 90 Day Bak Bill Futures SFE 90 Day Bak Bill Futures Tick Value Calculatios

### INVESTMENT PERFORMANCE COUNCIL (IPC)

INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks

### where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return

EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The