Solving DivideandConquer Recurrences


 Gertrude Welch
 2 years ago
 Views:
Transcription
1 Solvig DivideadCoquer Recurreces Victor Adamchik A divideadcoquer algorithm cosists of three steps: dividig a problem ito smaller subproblems solvig (recursively) each subproblem the combiig solutios to subproblems to get solutio to origial problem We use recurreces to aalyze the ruig time of such algorithms. Suppose T is the umber of steps i the worst case eeded to solve the problem of size. Let us split a problem ito a 1 subproblems, each of which is of the iput size where b>1. b Observe, that the umber of subproblems a is ot ecessarily equal to b. The total umber of steps T is obtaied by all steps eeded to solve smaller subproblems T êb plus the umber eeded to combie solutios ito a fial oe. The followig equatio is called divideadcoquer recurrece relatio T = a T êb + fhl As a example, cosider the mergesort: divide the iput i half recursively sort the two halves combie the two sorted subsequeces by mergig them. Let THL be worstcase rutime o a sequece of keys: If = 1, the THL = QH1L costat time If >1, the THL = 2 THê2L + QHL here Q() is time to do the merge. The
2 15451: Algorithm Desig ad Aalysis 2 T = 2 T ê2 +QHL Other examples of divide ad coquer algorithms: quicksort, iteger multiplicatio, matrix multiplicatio, fast Fourier trsform, fidig cover hull ad more. There are several techiques of solvig such recurrece equatios: the iteratio method the tree method the mastertheorem method guessadverify ü Tree method We could visualize the recursio as a tree, where each ode represets a recursive call. The root is the iitial call. Leaves correspod to the exit coditio. We ca ofte solve the recurrece by lookig at the structure of the tree. To illustrate, we take this example THL=2 TK 2 O+2 TH1L=1 Here is a recursio tree that diagrams the recursive fuctio calls T() T(/2) T(/2) T(/4) T(/4) T(/4) T(/4) T(1) Usig a recursio tree we ca model the time of a recursive executio by writig the size of the problem i each ode. T(1)
3 3 Usig a recursio tree we ca model the time of a recursive executio by writig the size of the problem i each ode. The last level correspods to the iitial coditio of the recurrece. Sice the work at each leaf is costat, the total work at all leaves is equal to the umber of leaves, which is 2 h = 2 log 2 = To fid the total time (for the whole tree), we must add up all the terms 1+log 2 THL= = + 2 The sum is easily computed by meas of the geometric series This yeilds h k=0 x k = xh+11 x1, x 1 k=0 1 2 k
4 15451: Algorithm Desig ad Aalysis 4 THL= = Check with Mathematica == 2 + 2, ä 1=, E H 1+2L<< Example. Solve the recurrece THL=3 TK 4 O+ The work at all levels is Sice the height is log 4, the tree has 3 log 4 leaves. Hece, the total work is give by 1+log 4 THL= k=0 3 4 k + 3 log 4 TH1L By meas of the geometric series ad takig ito accout 3 log 4 = log 4 3 the above sum yields THL=44 log log 4 3 TH1L = OHL ü The Master Theorem The master theorem solves recurreces of the form THL = a T b + fhl for a wide variety of fuctio fhl ad a 1, b>1. I this sectio we will outlie the
5 5 mai idea. Here is the recursive tree for the above equatio It is easy to see that the tree has a log b leaves. Ideed, sice the height is log b, ad the tree brachig factor is a, the umber of leaves is log a h = a log b a log = a a b Summig up values at each level, gives Therefore, the solutio is THL= fhl + a f b + a2 f = 1 log a b b 2 = log b a log b a TH1L 1+log b THL= log b a TH1L+ a k f k=0 b k Now we eed to compare the asymptotic behavior of fhl with log b a. There are three possible cases. QI log b a M if fhl=oi log b a M THL= QI log b log k+1 M if fhl=qi log b a log k M, k 0 QH fhll if fhl=wi log b a M The followig examples demostrate the theorem. Case 1. THL=4 TI M+ 2 We have fhl= ad log b a = log 2 4 = 2, therefore fhl=oi 2 M. The the solutio is THL= QI 2 M by case 1. Case 2. THL=4 TI 2 M+2 I this case fhl= 2 ad fhl=qi 2 M. The THL= QI 2 log M by case 2. Case 3. THL=4 TI 2 M+3 I this case fhl= 3 ad fhl=wi log b a M=WI 2 M. The THL= QI 3 M by case 3.
6 15451: Algorithm Desig ad Aalysis 6 ü Multiplicatio of large itegers Karatsuba Algorithm The brute force approach ("grammar school" method) We say that multiplicatio of two digits itegers has time complexity at worst OI 2 M. We develop a algorithm that has better asymptotic complexity. The idea is based o divideadcoquer techique. Cosider the above itegers ad split each of them i two parts 123 = 12 * = 4 * ad the multiply them: 123*45 = (12*10 + 3)(4*10 + 5) = H L I geeral, the iteger which has digits ca be represeted as um = x * 10 m + y where m=floork 2 O x=ceiligk 2 O y=floork 2 O Example, = Cosider two digits umbers
7 7 um 1 = x 1 * 10 p + x 0 um 2 = y 1 * 10 p + y 0 Their product is um 1 * um 2 = x 1 * y 1 * 10 2 p +Hx 1 * y 0 + x 0 * y 1 L*10 p + x 0 * y 0 Just lookig at this geeral formula you ca say that just istead of oe multiplicatio we have 4. Where is the advatage? ü The worstcase complexity umbers x 1, x 0 ad y 1, y 0 have twice less digits. Let THL deote the umber of digit multiplicatios eeded to multiply two digits umbers. The recurrece (sice the algorithm does 4 multiplicatios o each step) THL=4 TI M+OHL, THcL = 1 2 Note, we igore multiplicatios by a base!!! Its solutio is give by The algorithm is still quadratic! THL = 4 log 2 = 2 ü The Karatsuba Algorithm 1962, Aatolii Karatsuba, Russia. um 1 * um 2 = x 1 * y 1 * 10 2 p +Hx 1 * y 0 + x 0 * y 1 L*10 p + x 0 * y 0 The goal is to decrease the umber of multiplicatios from 4 to 3. We ca do this by observig that It follows that Hx 1 + x 0 L*Hy 1 + y 0 L= x 1 * y 1 + x 0 * y 0 +Hx 1 * y 0 + x 0 * y 1 L um 1 * um 2 = x 1 * y 1 * 10 2 p +JHx 1 + x 0 L* Hy 1 + y 0 L x 1 * y 1  x 0 * y 0 N*10 p + x 0 * y 0 ad it is oly 3 multiplicatios (see it?). The total umber of multiplicatios is give by (we igore multiplicatios by a base) Its solutio is THL=3 TI M+OHL, T(c) = 1 2
8 15451: Algorithm Desig ad Aalysis 8 ü ToomCook 3Way Multiplicatio 1963, A. L. Toom, Russia. 1966, Cook, Harvard, Ph.D Thesis THL = 3 log 2 = log 2 3 = The key idea of the algorithm is to divide a large iteger ito 3 parts (rather tha 2) of size approximately ê 3 ad the multiply those parts. Here is the equatio of for the total umber of multiplicatios THL=9 T 3 + OHL, THcL = 1 ad the solutio TH L=9 log 3 = 2 Let us reduce the umber of multiplicatios by oe THL=8 T 3 + OHL THL=8 log 3 = log 3 8 = No advatage. This does ot improve the previous algorithm, that rus at OI M How may multiplicatio should we elimiate? Let us cosider that equatio i a geeral form, where parameter p>0 is arbitrary THL= p T 3 + OHL THL= p log 3 = log 3 p Therefore, the ew algoritm will be faster tha OI 1.58 M if we reduce the umber of multiplicatios to five This is a improvemet over Karatsuba. THL=5 log 3 = log 3 5 = Is it possible to reduce a umber of multiplicatios to 5? Yes, it follows from this system of equatios: where x 0 y 0 = Z 0 12Hx 1 y 0 + x 0 y 1 L=8 Z 1  Z 28 Z 3 + Z 4 24Hx 2 y 0 + x 1 y 1 + x 0 y 2 L=30 Z Z 1  Z Z 3  Z 4 12Hx 2 y 1 + x 1 y 2 L=2 Z 1 + Z Z 3  Z 4 24 x 2 y 2 = 6 Z 04 Z 1 + Z 24 Z 3 + Z 4
9 9 ü Further Geeralizatio Z 0 = x 0 y 0 Z 1 = Hx 0 + x 1 + x 2 LHy 0 + y 1 + y 2 L Z 2 = Hx x x 2 LHy y y 2 L Z 3 = Hx 0  x 1 + x 2 LHy 0  y 1 + y 2 L Z 4 = Hx 02 x x 2 LHy 02 y y 2 L It is possible to develop a faster algorithm by icreasig the umber of splits. Let us cosider a 4way splittig. How may multiplicatios should we have o each step so this algorithm will outperform the 3way splittig? We fid parameter p from which yields THL= p TK 4 O+OHL THL= p log 4 = log 4 p log 4 p log 3 5 p = 7 The followig table demostrates a relatioship betwee splits ad the umber of multiplicatios: Ituitively we see that the kway split requires 2 k  1 multiplicatios. This meas that istead of k 2 multiplicatios we do oly 2 k 1. The recurrece equatio for the total umber of multiplicatio is give by THL=H2 k 1L T k + OHL ad its solutio is THL=H2 k 1L log k = log k H2 k1l Here is the sequece of the kway splits whe k rus from 2 to 10:
10 15451: Algorithm Desig ad Aalysis , 1.46, 1.40, 1.36, 1.33, 1.31, 1.30, 1.28, We ca prove that asymptotically multiplicatio of two digits umbers requires OI 1+e M multiplicatios, where eø0. Note, we will NEVER get a liear performace (prove this!) Is it always possible to fid such 2 k 1 multiplicatios? Cosider two polyomials of k 1 degree poly 1 = a k1 x k1 + a k2 * x k a 1 * x+a 0 poly 2 = b k1 x k1 + b k2 * x k b 1 * x+b 0 whe we multiply them we get a polyomial of 2 k 2 degree poly 1 * poly 2 = a k1 b k1 * x 2 k Ha 1 b 0 + b 1 a 0 L* x + a 0 b 0 The above polyomial has exactly 2 k 1 coefficiets, therefore it's uiquely defied by 2 k 1 values.
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
More informationSoving 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
More informationPage 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
More informationSection 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
More informationCS 253: Algorithms. Chapter 4. DivideandConquer Recurrences Master Theorem. Credit: Dr. George Bebis
CS 5: Algorithms Chapter 4 DivideadCoquer Recurreces Master Theorem Credit: Dr. George Bebis Recurreces ad Ruig Time Recurreces arise whe a algorithm cotais recursive calls to itself Ruig time is represeted
More informationAlgorithms and Data Structures DV3. Arne Andersson
Algorithms ad Data Structures DV3 Are Adersso Today s lectures Textbook chapters 14, 5 Iformatiostekologi Itroductio Overview of Algorithmic Mathematics Recurreces Itroductio to Recurreces The Substitutio
More informationLecture Notes CMSC 251
We have this messy summatio to solve though First observe that the value remais costat throughout the sum, ad so we ca pull it out frot Also ote that we ca write 3 i / i ad (3/) i T () = log 3 (log ) 1
More informationA Gentle Introduction to Algorithms: Part II
A Getle Itroductio to Algorithms: Part II Cotets of Part I:. Merge: (to merge two sorted lists ito a sigle sorted list.) 2. Bubble Sort 3. Merge Sort: 4. The BigO, BigΘ, BigΩ otatios: asymptotic bouds
More informationDivide 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
More informationMath Background: Review & Beyond. Design & Analysis of Algorithms COMP 482 / ELEC 420. Solving for Closed Forms. Obtaining Recurrences
Math Backgroud: Review & Beyod. Asymptotic otatio Desig & Aalysis of Algorithms COMP 48 / ELEC 40 Joh Greier. Math used i asymptotics 3. Recurreces 4. Probabilistic aalysis To do: [CLRS] 4 # T() = O()
More informationHandout: How to calculate time complexity? CSE 101 Winter 2014
Hadout: How to calculate time complexity? CSE 101 Witer 014 Recipe (a) Kow algorithm If you are usig a modied versio of a kow algorithm, you ca piggyback your aalysis o the complexity of the origial algorithm
More informationAlgorithms Chapter 7 Quicksort
Algorithms Chapter 7 Quicksort Assistat Professor: Chig Chi Li 林清池助理教授 chigchi.li@gmail.com Departmet of Computer Sciece ad Egieerig Natioal Taiwa Ocea Uiversity Outlie Descriptio of Quicksort Performace
More informationS. Tanny MAT 344 Spring 1999. be the minimum number of moves required.
S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + $ T is a sequece (f. o itegers). Solve for T? * is a recurrece,
More informationInfinite 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...
More informationRecursion 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,
More informationAsymptotic Notation, Recurrences: Substitution, Iteration, Master Method. Lecture 2
Asymptotic Notatio, Recurreces: Substitutio, Iteratio, Master Method Lecture 2 Solvig recurreces The aalysis of merge sort from Lecture 1 required us to solve a recurrece. Recurreces are like solvig itegrals,
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationChapter 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
More information6 Algorithm analysis
6 Algorithm aalysis Geerally, a algorithm has three cases Best case Average case Worse case. To demostrate, let us cosider the a really simple search algorithm which searches for k i the set A{a 1 a...
More informationData Structures. Outline
Data Structures Solvig Recurreces Tzachi (Isaac) Rose 1 Outlie Recurrece The Substitutio Method The Iteratio Method The Master Method Tzachi (Isaac) Rose 2 1 Recurrece A recurrece is a fuctio defied i
More information8.3 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,
More informationORDERS 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
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationFIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if
More informationModule 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
More informationFourier Series and the Wave Equation Part 2
Fourier Series ad the Wave Equatio Part There are two big ideas i our work this week. The first is the use of liearity to break complicated problems ito simple pieces. The secod is the use of the symmetries
More informationAlgebra 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
More informationx(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,...,
More informationAsymptotic 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
More informationMath Discrete Math Combinatorics MULTIPLICATION PRINCIPLE:
Math 355  Discrete Math 4.14.4 Combiatorics Notes MULTIPLICATION PRINCIPLE: If there m ways to do somethig ad ways to do aother thig the there are m ways to do both. I the laguage of set theory: Let
More information1. 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
More informationIn 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
More informationTrigonometric 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
More informationLearning 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
More informationwhen 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
More informationSECTION 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,
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationYour grandmother and her financial counselor
Sectio 10. Arithmetic Sequeces 963 Objectives Sectio 10. Fid the commo differece for a arithmetic sequece. Write s of a arithmetic sequece. Use the formula for the geeral of a arithmetic sequece. Use the
More informationMATH /2003. Assignment 4. Due January 8, 2003 Late penalty: 5% for each school day.
MATH 260 2002/2003 Assigmet 4 Due Jauary 8, 2003 Late pealty: 5% for each school day. 1. 4.6 #10. A croissat shop has plai croissats, cherry croissats, chocolate croissats, almod croissats, apple croissats
More informationSUMS OF nth POWERS OF ROOTS OF A GIVEN QUADRATIC EQUATION. N.A. Draim, Ventura, Calif., and Marjorie Bicknell Wilcox High School, Santa Clara, Calif.
SUMS OF th OWERS OF ROOTS OF A GIVEN QUADRATIC EQUATION N.A. Draim, Vetura, Calif., ad Marjorie Bickell Wilcox High School, Sata Clara, Calif. The quadratic equatio whose roots a r e the sum or differece
More informationLesson 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
More informationSequences 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
More informationRunning Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis
Ruig Time ( 3.) Aalysis of Algorithms Iput Algorithm Output A algorithm is a stepbystep procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.
More informationWinter Camp 2012 Sequences Alexander Remorov. Sequences. Alexander Remorov
Witer Camp 202 Sequeces Alexader Remorov Sequeces Alexader Remorov alexaderrem@gmail.com Warmup Problem : Give a positive iteger, cosider a sequece of real umbers a 0, a,..., a defied as a 0 = 2 ad =
More information1 Notes on Little s Law (l = λw)
Copyright c 29 by Karl Sigma Notes o Little s Law (l λw) We cosider here a famous ad very useful law i queueig theory called Little s Law, also kow as l λw, which asserts that the time average umber of
More informationCooleyTukey. Tukey FFT Algorithms. FFT Algorithms. Cooley
Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Cosider a legth sequece x[ with a poit DFT X[ where Represet the idices ad as +, +, Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Usig these
More informationChapter Suppose you wish to use the Principle of Mathematical Induction to prove that 1 1! + 2 2! + 3 3! n n! = (n + 1)! 1 for all n 1.
Chapter 4. Suppose you wish to prove that the followig is true for all positive itegers by usig the Priciple of Mathematical Iductio: + 3 + 5 +... + ( ) =. (a) Write P() (b) Write P(7) (c) Write P(73)
More informationMath 115 HW #4 Solutions
Math 5 HW #4 Solutios From 2.5 8. Does the series coverge or diverge? ( ) 3 + 2 = Aswer: This is a alteratig series, so we eed to check that the terms satisfy the hypotheses of the Alteratig Series Test.
More informationChapter 7. In the questions below, describe each sequence recursively. Include initial conditions and assume that the sequences begin with a 1.
Use the followig to aswer questios 6: Chapter 7 I the questios below, describe each sequece recursively Iclude iitial coditios ad assume that the sequeces begi with a a = 5 As: a = 5a,a = 5 The Fiboacci
More informationEquation of a line. Line in coordinate geometry. Slopeintercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Pointslope 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
More informationLiteral Equations and Formulas
. Literal Equatios ad Formulas. OBJECTIVE 1. Solve a literal equatio for a specified variable May problems i algebra require the use of formulas for their solutio. Formulas are simply equatios that express
More informationBinet Formulas for Recursive Integer Sequences
Biet Formulas for Recursive Iteger Sequeces Homer W. Austi Jatha W. Austi Abstract May iteger sequeces are recursive sequeces ad ca be defied either recursively or explicitly by use of Biettype formulas.
More informationLinear Algebra II. Notes 6 25th November 2010
MTH6140 Liear Algebra II Notes 6 25th November 2010 6 Quadratic forms A lot of applicatios of mathematics ivolve dealig with quadratic forms: you meet them i statistics (aalysis of variace) ad mechaics
More informationThe 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
More information23 The Remainder and Factor Theorems
 The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRECALCULUS 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
More informationCOMP 251 Assignment 2 Solutions
COMP 251 Assigmet 2 Solutios Questio 1 Exercise 8.34 Treat the umbers as 2digit umbers i radix. Each digit rages from 0 to 1. Sort these 2digit umbers ith the RADIXSORT algorithm preseted i Sectio
More informationTo get the next Fibonacci number, you add the previous two. numbers are defined by the recursive formula. F 1 F n+1
Liear Algebra Notes Chapter 6 FIBONACCI NUMBERS The Fiboacci umbers are F, F, F 2, F 3 2, F 4 3, F, F 6 8, To get the ext Fiboacci umber, you add the previous two umbers are defied by the recursive formula
More informationSearching Algorithm Efficiencies
Efficiecy of Liear Search Searchig Algorithm Efficiecies Havig implemeted the liear search algorithm, how would you measure its efficiecy? A useful measure (or metric) should be geeral, applicable to ay
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationGregory 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
More informationARITHMETIC 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
More informationThe 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
More information4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then
SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or pseries (the Compariso Test), but of
More information5 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
More informationREACHABILITY AND OBSERVABILITY
Updated: Saturday October 4 8 Copyright F.L. Lewis ll rights reserved RECHILIY ND OSERVILIY Cosider a liear statespace system give by x x + u y Cx + Du m with x t R the iteral state u t R the cotrol iput
More informationSection 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
More informationReview for College Algebra Final Exam
Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 14. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive
More informationNPTEL 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
More informationSection 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
More informationDerivation of the Poisson distribution
Gle Cowa RHUL Physics 1 December, 29 Derivatio of the Poisso distributio I this ote we derive the fuctioal form of the Poisso distributio ad ivestigate some of its properties. Cosider a time t i which
More informationContinued Fractions continued. 3. Best rational approximations
Cotiued Fractios cotiued 3. Best ratioal approximatios We hear so much about π beig approximated by 22/7 because o other ratioal umber with deomiator < 7 is closer to π. Evetually 22/7 is defeated by 333/06
More informationMath 475, Problem Set #6: Solutions
Math 475, Problem Set #6: Solutios A (a) For each poit (a, b) with a, b oegative itegers satisfyig ab 8, cout the paths from (0,0) to (a, b) where the legal steps from (i, j) are to (i 2, j), (i, j 2),
More informationSection 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
More information1 The Binomial Theorem: Another Approach
The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets
More informationHW 1 Solutions Math 115, Winter 2009, Prof. Yitzhak Katznelson
HW Solutios Math 5, Witer 2009, Prof. Yitzhak Katzelso.: Prove 2 + 2 2 +... + 2 = ( + )(2 + ) for all atural umbers. The proof is by iductio. Call the th propositio P. The basis for iductio P is the statemet
More informationConcept: Types of algorithms
Discrete Math for Bioiformatics WS 10/11:, by A. Bockmayr/K. Reiert, 18. Oktober 2010, 21:22 1001 Cocept: Types of algorithms The expositio is based o the followig sources, which are all required readig:
More information2.7 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
More information{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers
. Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,
More informationRiemann Sums y = f (x)
Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, oegative fuctio o the closed iterval [a, b] Fid
More informationLecture 2: Karger s Min Cut Algorithm
priceto uiv. F 3 cos 5: Advaced Algorithm Desig Lecture : Karger s Mi Cut Algorithm Lecturer: Sajeev Arora Scribe:Sajeev Today s topic is simple but gorgeous: Karger s mi cut algorithm ad its extesio.
More information.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,
More information13 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.
More informationExample 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
More information1 StateSpace Canonical Forms
StateSpace Caoical Forms For ay give system, there are essetially a ifiite umber of possible state space models that will give the idetical iput/output dyamics Thus, it is desirable to have certai stadardized
More information8.5 Alternating infinite series
65 8.5 Alteratig ifiite series I the previous two sectios we cosidered oly series with positive terms. I this sectio we cosider series with both positive ad egative terms which alterate: positive, egative,
More informationChapter 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
More informationNUMBERS COMMON TO TWO POLYGONAL SEQUENCES
NUMBERS COMMON TO TWO POLYGONAL SEQUENCES DIANNE SMITH LUCAS Chia Lake, Califoria a iteger, The polygoal sequece (or sequeces of polygoal umbers) of order r (where r is r > 3) may be defied recursively
More informationSecond Order Linear Partial Differential Equations. Part III
Secod Order iear Partial Differetial Equatios Part III Oedimesioal Heat oductio Equatio revisited; temperature distributio of a bar with isulated eds; ohomogeeous boudary coditios; temperature distributio
More informationMeasurable Functions
Measurable Fuctios Dug Le 1 1 Defiitio It is ecessary to determie the class of fuctios that will be cosidered for the Lebesgue itegratio. We wat to guaratee that the sets which arise whe workig with these
More informationHere 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
More informationAlgebra 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
More information8.1 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
More information1.3 Binomial Coefficients
18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to
More information7.1 Finding Rational Solutions of Polynomial Equations
4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio?
More informationWHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?
WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This
More informationReview: Classification Outline
Data Miig CS 341, Sprig 2007 Decisio Trees Neural etworks Review: Lecture 6: Classificatio issues, regressio, bayesia classificatio Pretice Hall 2 Data Miig Core Techiques Classificatio Clusterig Associatio
More informationhp calculators HP 12C Platinum Statistics  correlation coefficient The correlation coefficient HP12C Platinum correlation coefficient
HP 1C Platium Statistics  correlatio coefficiet The correlatio coefficiet HP1C Platium correlatio coefficiet Practice fidig correlatio coefficiets ad forecastig HP 1C Platium Statistics  correlatio coefficiet
More informationLinear 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
More information