Product of polynomials. Introduction to Programming (in C++) Numerical algorithms. Product of polynomials. Product of polynomials
|
|
- Ruth Pearson
- 6 years ago
- Views:
Transcription
1 Product of polynomils Introduction to Progrmming (in C++) Numericl lgorithms Jordi Cortdell, Ricrd Gvldà, Fernndo Orejs Dept. of Computer Science, UPC Given two polynomils on one vrile nd rel coefficients, compute their product (we will decide lter how we represent polynomils) Exmple: given x 2 + 3x - 1 nd 2x - 5, otin 2x 3-5x 2 + 6x 2-15x - 2x + 5 = 2x 3 + x 2-17x + 5 Key point: Product of polynomils Given p(x) = n x n + n-1 x n x + 0 nd q(x) = m x m + m-1 x m x + 0, wht is the coefficient c i of x i in (p*q)(x)? Introduction to Progrmming Dept. CS, UPC 2 Product of polynomils Suppose we represent polynomil of degree n y vector of size n+1. Tht is, v[0..n] represents the polynomil v[n] x n + v[n-1] x n v[1] x + v[0] We otin x i+j whenever we multiply i x i j x j Ide: for every i nd j, dd i j to the (i+j)-th coefficient of the product polynomil. We wnt to mke sure tht v[v.size() - 1] 0 so tht degree(v) = v.size() - 1 The only exception is the constnt-0 polynomil. We ll represent it y vector of size 0. Introduction to Progrmming Dept. CS, UPC 3 Introduction to Progrmming Dept. CS, UPC 4
2 Product of polynomils Product of polynomils Polynomil product(const Polynomil& p, const Polynomil& q) { typedef vector<doule> Polynomil; // Pre: -- // Returns p q Polynomil product(const Polynomil& p, const Polynomil& q); // Specil cse for polynomil of size 0 if (p.size() == 0 or q.size() == 0) return Polynomil(0); else { int deg = p.size() 1 + q.size() - 1; // degree of p q Polynomil r(deg + 1, 0); for (int i = 0; i < p.size(); ++i) { for (int j = 0; j < q.size(); ++j) { r[i + j] = r[i + j] + p[i] q[j]; return r; // Invrint (of the outer loop): r = product p[0..i-1] q // (we hve used the coefficients p[0] p[i-1]) Introduction to Progrmming Dept. CS, UPC 5 Sum of polynomils Note tht over the rel numers, degree(p q) = degree(p) + degree(q) (except if p = 0 or q = 0). So we know the size of the result vector from the strt. This is not true for the polynomil sum, e.g. Introduction to Progrmming Dept. CS, UPC 6 Sum of polynomils // Pre: -- // Returns p+q Polynomil sum(const Polynomil& p, const Polynomil& q); int mxdeg = mx(p.size(), q.size()) - 1; int deg = -1; Polynomil r(mxdeg + 1, 0); // Inv r[0..i-1] = (p+q)[0..i-1] nd // deg = lrgest j s.t. r[j]!= 0 (or -1 if none exists) for (int i = 0; i <= mxdeg; ++i) { if (i >= p.size()) r[i] = q[i]; else if (i >= q.size()) r[i] = p[i]; else r[i] = p[i] + q[i]; if (r[i]!= 0) deg = i; degree((x + 5) + (-x - 1)) = 0 Polynomil rr(deg + 1); for (int i = 0; i <= deg; ++i) rr[i] = r[i]; return rr; Introduction to Progrmming Dept. CS, UPC 7 Introduction to Progrmming Dept. CS, UPC 8
3 Sum of sprse vectors Sum of sprse vectors In some cses, prolems must del with sprse vectors or mtrices (most of the elements re zero). Sprse vectors nd mtrices cn e represented more efficiently y only storing the non-zero elements. For exmple, vector cn e represented s vector of pirs (index, vlue), sorted in scending order of the indices. Exmple: [0,0,1,0,-3,0,0,0,2,0,0,4,0,0,0] cn e represented s Design function tht clcultes the sum of two sprse vectors, where ech non-zero vlue is represented y pir (index, vlue): struct Pir { int index; int vlue; typedef vector<pir> SprseVector; [(2,1),(4,-3),(8,2),(11,4)] Introduction to Progrmming Dept. CS, UPC 9 Sum of sprse vectors Introduction to Progrmming Dept. CS, UPC 10 Sum of sprse vectors // Pre: -- // Returns v1+v2 SprseVector sprse_sum(const SprseVector& v1, const SprseVector& v2); // Inv: p1 nd p2 will point to the first // non-treted elements of v1 nd v2. // vsum contins the elements of v1+v2 treted so fr. // psum points to the first free loction in vsum. Strtegy: Clculte the sum on sufficiently lrge vector. Copy the result on nother vector of pproprite size. SprseVector sprse_sum(const SprseVector& v1, const SprseVector& v2) { SprseVector vsum; int p1 = 0, p2 = 0; while (p1 < v1.size() nd p2 < v2.size()) { if (v1[p1].index < v2[p2].index) { // Element only in v1 vsum.push_ck(v1[p1]); ++p1; else if (v1[p1].index > v2[p2].index) { // Element only in v2 vsum.push_ck(v2[p2]); ++p2; else { // Element in oth Pir p; p.index = v1[p1].index; p.vlue = v1[p1].vlue + v2[p2].vlue; if (p.vlue!= 0) vsum.push_ck(p); ++p1; ++p2; Introduction to Progrmming Dept. CS, UPC 11 Introduction to Progrmming Dept. CS, UPC 12
4 Sum of sprse vectors // Copy the remining elements of v1 while (p1 < v1.size()) { vsum.push_ck(v1[p1]); ++p1; Bolzno s theorem: Let f e rel-vlued continuous function. Let nd e two vlues such tht < nd f() f() < 0. Then, there is vlue c [,] such tht f(c)=0. // Copy the remining elements of v2 while (p2 < v2.size()) { vsum.push_ck(v2[p2]); ++p2; return vsum; c Introduction to Progrmming Dept. CS, UPC 13 Design function tht finds root of continuous function f in the intervl [, ] ssuming the conditions of Bolzno s theorem re fulfilled. Given precision ( ), the function must return vlue c such tht the root of f is in the intervl [c, c+ ]. Introduction to Progrmming Dept. CS, UPC 14 Strtegy: nrrow the intervl [, ] y hlf, checking whether the vlue of f in the middle of the intervl is positive or negtive. Iterte until the width of the intervl is smller. c Introduction to Progrmming Dept. CS, UPC 15 Introduction to Progrmming Dept. CS, UPC 16
5 // Pre: f is continuous, < nd f() f() < 0. // Returns c [,] such tht root exists in the // intervl [c,c+ ]. // Inv: root of f exists in the intervl [,] doule root(doule, doule, doule epsilon) { while ( > epsilon) { doule c = ( + )/2; if (f() f(c) <= 0) = c; else = c; return ; Introduction to Progrmming Dept. CS, UPC 17 // A recursive version doule root(doule, doule, doule epsilon) { if ( <= epsilon) return ; doule c = ( + )/2; if (f() f(c) <= 0) return root(,c,epsilon); else return root(c,,epsilon); Introduction to Progrmming Dept. CS, UPC 18 Brcode A rcode is n opticl mchine-redle representtion of dt. One of the most populr encoding systems is the UPC (Universl Product Code). A UPC code hs 12 digits. Optionlly, check digit cn e dded. Introduction to Progrmming Dept. CS, UPC 19 Introduction to Progrmming Dept. CS, UPC 20
6 Brcode Brcode The check digit is clculted s follows: 1. Add the digits in odd-numered positions (first, third, fifth, etc.) nd multiply y Add the digits in the even-numered positions (second, fourth, sixth, etc.) to the result. 3. Clculte the result modulo If the result is not zero, sutrct the result from 10. Exmple: ( ) 3 = = 25 (30+25) mod 10 = = 5 Design progrm tht reds sequence of 12-digit numers tht represent UPCs without check digits nd writes the sme UPCs with the check digit. Question: do we need dt structure to store the UPCs? Answer: no, we only need few uxiliry vriles. Introduction to Progrmming Dept. CS, UPC 21 Brcode The progrm might hve loop treting UPC t ech itertion. The invrint could e s follows: // Inv: ll the UPCs of the treted codes // hve een written. At ech itertion, the progrm could red the UPC digits nd, t the sme time, write the UPC nd clculte the check digit. The invrint could e: // Inv: ll the treted digits hve een // written. The prtil clcultion of // the check digit hs een performed // sed on the treted digits. Introduction to Progrmming Dept. CS, UPC 22 Brcode // Pre: the input contins sequence of UPCs without check digits. // Post: the UPCs t the input hve een written with their check digits. int min() { chr c; while (cin >> c) { cout << c; int d = 3 (int(c) - int('0')); // first digit in n odd loction for (int i = 2; i <= 12; ++i) { cin >> c; cout << c; if (i%2 == 0) d = d + int(c) - int('0'); else d = d + 3 (int(c) - int('0')); d = d%10; if (d > 0) d = 10 d; cout << d << endl; Introduction to Progrmming Dept. CS, UPC 23 Introduction to Progrmming Dept. CS, UPC 24
Approximate computations
Living with floting-point numers Stndrd normlized representtion (sign + frction + exponent): Approximte computtions Rnges of vlues: Representtions for:, +, +0, 0, NN (not numer) Jordi Cortdell Deprtment
More informationFig.25: the Role of LEX
The Lnguge for Specifying Lexicl Anlyzer We shll now study how to uild lexicl nlyzer from specifiction of tokens in the form of list of regulr expressions The discussion centers round the design of n existing
More informationcisc1110 fall 2010 lecture VI.2 call by value function parameters another call by value example:
cisc1110 fll 2010 lecture VI.2 cll y vlue function prmeters more on functions more on cll y vlue nd cll y reference pssing strings to functions returning strings from functions vrile scope glol vriles
More informationDynamic Programming. Andreas Klappenecker. [partially based on slides by Prof. Welch] Monday, September 24, 2012
Dynmic Progrmming Andres Klppenecker [prtilly bsed on slides by Prof. Welch] 1 Dynmic Progrmming Optiml substructure An optiml solution to the problem contins within it optiml solutions to subproblems.
More information2 Computing all Intersections of a Set of Segments Line Segment Intersection
15-451/651: Design & Anlysis of Algorithms Novemer 14, 2016 Lecture #21 Sweep-Line nd Segment Intersection lst chnged: Novemer 8, 2017 1 Preliminries The sweep-line prdigm is very powerful lgorithmic design
More informationP(r)dr = probability of generating a random number in the interval dr near r. For this probability idea to make sense we must have
Rndom Numers nd Monte Crlo Methods Rndom Numer Methods The integrtion methods discussed so fr ll re sed upon mking polynomil pproximtions to the integrnd. Another clss of numericl methods relies upon using
More informationbox Boxes and Arrows 3 true 7.59 'X' An object is drawn as a box that contains its data members, for example:
Boxes nd Arrows There re two kinds of vriles in Jv: those tht store primitive vlues nd those tht store references. Primitive vlues re vlues of type long, int, short, chr, yte, oolen, doule, nd flot. References
More informationMath 142, Exam 1 Information.
Mth 14, Exm 1 Informtion. 9/14/10, LC 41, 9:30-10:45. Exm 1 will be bsed on: Sections 7.1-7.5. The corresponding ssigned homework problems (see http://www.mth.sc.edu/ boyln/sccourses/14f10/14.html) At
More informationZZ - Advanced Math Review 2017
ZZ - Advnced Mth Review Mtrix Multipliction Given! nd! find the sum of the elements of the product BA First, rewrite the mtrices in the correct order to multiply The product is BA hs order x since B is
More informationMa/CS 6b Class 1: Graph Recap
M/CS 6 Clss 1: Grph Recp By Adm Sheffer Course Detils Adm Sheffer. Office hour: Tuesdys 4pm. dmsh@cltech.edu TA: Victor Kstkin. Office hour: Tuesdys 7pm. 1:00 Mondy, Wednesdy, nd Fridy. http://www.mth.cltech.edu/~2014-15/2term/m006/
More information10.5 Graphing Quadratic Functions
0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions
More informationSimplifying Algebra. Simplifying Algebra. Curriculum Ready.
Simplifying Alger Curriculum Redy www.mthletics.com This ooklet is ll out turning complex prolems into something simple. You will e le to do something like this! ( 9- # + 4 ' ) ' ( 9- + 7-) ' ' Give this
More information1.5 Extrema and the Mean Value Theorem
.5 Extrem nd the Men Vlue Theorem.5. Mximum nd Minimum Vlues Definition.5. (Glol Mximum). Let f : D! R e function with domin D. Then f hs n glol mximum vlue t point c, iff(c) f(x) for ll x D. The vlue
More informationIn the last lecture, we discussed how valid tokens may be specified by regular expressions.
LECTURE 5 Scnning SYNTAX ANALYSIS We know from our previous lectures tht the process of verifying the syntx of the progrm is performed in two stges: Scnning: Identifying nd verifying tokens in progrm.
More informationMa/CS 6b Class 1: Graph Recap
M/CS 6 Clss 1: Grph Recp By Adm Sheffer Course Detils Instructor: Adm Sheffer. TA: Cosmin Pohot. 1pm Mondys, Wednesdys, nd Fridys. http://mth.cltech.edu/~2015-16/2term/m006/ Min ook: Introduction to Grph
More informationCOMPUTER SCIENCE 123. Foundations of Computer Science. 6. Tuples
COMPUTER SCIENCE 123 Foundtions of Computer Science 6. Tuples Summry: This lecture introduces tuples in Hskell. Reference: Thompson Sections 5.1 2 R.L. While, 2000 3 Tuples Most dt comes with structure
More informationUnion-Find Problem. Using Arrays And Chains. A Set As A Tree. Result Of A Find Operation
Union-Find Problem Given set {,,, n} of n elements. Initilly ech element is in different set. ƒ {}, {},, {n} An intermixed sequence of union nd find opertions is performed. A union opertion combines two
More informationSystems I. Logic Design I. Topics Digital logic Logic gates Simple combinational logic circuits
Systems I Logic Design I Topics Digitl logic Logic gtes Simple comintionl logic circuits Simple C sttement.. C = + ; Wht pieces of hrdwre do you think you might need? Storge - for vlues,, C Computtion
More informationCIS 1068 Program Design and Abstraction Spring2015 Midterm Exam 1. Name SOLUTION
CIS 1068 Progrm Design nd Astrction Spring2015 Midterm Exm 1 Nme SOLUTION Pge Points Score 2 15 3 8 4 18 5 10 6 7 7 7 8 14 9 11 10 10 Totl 100 1 P ge 1. Progrm Trces (41 points, 50 minutes) Answer the
More informationCOMP 423 lecture 11 Jan. 28, 2008
COMP 423 lecture 11 Jn. 28, 2008 Up to now, we hve looked t how some symols in n lphet occur more frequently thn others nd how we cn sve its y using code such tht the codewords for more frequently occuring
More informationVery sad code. Abstraction, List, & Cons. CS61A Lecture 7. Happier Code. Goals. Constructors. Constructors 6/29/2011. Selectors.
6/9/ Abstrction, List, & Cons CS6A Lecture 7-6-9 Colleen Lewis Very sd code (define (totl hnd) (if (empty? hnd) (+ (butlst (lst hnd)) (totl (butlst hnd))))) STk> (totl (h c d)) 7 STk> (totl (h ks d)) ;;;EEEK!
More informationIntroduction to Integration
Introduction to Integrtion Definite integrls of piecewise constnt functions A constnt function is function of the form Integrtion is two things t the sme time: A form of summtion. The opposite of differentition.
More informationGraphing Conic Sections
Grphing Conic Sections Definition of Circle Set of ll points in plne tht re n equl distnce, clled the rdius, from fixed point in tht plne, clled the center. Grphing Circle (x h) 2 + (y k) 2 = r 2 where
More informationThe Greedy Method. The Greedy Method
Lists nd Itertors /8/26 Presenttion for use with the textook, Algorithm Design nd Applictions, y M. T. Goodrich nd R. Tmssi, Wiley, 25 The Greedy Method The Greedy Method The greedy method is generl lgorithm
More informationASTs, Regex, Parsing, and Pretty Printing
ASTs, Regex, Prsing, nd Pretty Printing CS 2112 Fll 2016 1 Algeric Expressions To strt, consider integer rithmetic. Suppose we hve the following 1. The lphet we will use is the digits {0, 1, 2, 3, 4, 5,
More informationLecture 7: Integration Techniques
Lecture 7: Integrtion Techniques Antiderivtives nd Indefinite Integrls. In differentil clculus, we were interested in the derivtive of given rel-vlued function, whether it ws lgeric, eponentil or logrithmic.
More informationSlides for Data Mining by I. H. Witten and E. Frank
Slides for Dt Mining y I. H. Witten nd E. Frnk Simplicity first Simple lgorithms often work very well! There re mny kinds of simple structure, eg: One ttriute does ll the work All ttriutes contriute eqully
More informationMATH 2530: WORKSHEET 7. x 2 y dz dy dx =
MATH 253: WORKSHT 7 () Wrm-up: () Review: polr coordintes, integrls involving polr coordintes, triple Riemnn sums, triple integrls, the pplictions of triple integrls (especilly to volume), nd cylindricl
More informationLING/C SC/PSYC 438/538. Lecture 21 Sandiway Fong
LING/C SC/PSYC 438/538 Lecture 21 Sndiwy Fong Tody's Topics Homework 8 Review Optionl Homework 9 (mke up on Homework 7) Homework 8 Review Question1: write Prolog regulr grmmr for the following lnguge:
More informationSection 10.4 Hyperbolas
66 Section 10.4 Hyperbols Objective : Definition of hyperbol & hyperbols centered t (0, 0). The third type of conic we will study is the hyperbol. It is defined in the sme mnner tht we defined the prbol
More informationQuiz2 45mins. Personal Number: Problem 1. (20pts) Here is an Table of Perl Regular Ex
Long Quiz2 45mins Nme: Personl Numer: Prolem. (20pts) Here is n Tle of Perl Regulr Ex Chrcter Description. single chrcter \s whitespce chrcter (spce, t, newline) \S non-whitespce chrcter \d digit (0-9)
More informationRay surface intersections
Ry surfce intersections Some primitives Finite primitives: polygons spheres, cylinders, cones prts of generl qudrics Infinite primitives: plnes infinite cylinders nd cones generl qudrics A finite primitive
More informationSolution of Linear Algebraic Equations using the Gauss-Jordan Method
Solution of Liner Algebric Equtions using the Guss-Jordn Method Populr pproch for solving liner equtions The Guss Jordn method depends on two properties of liner equtions: Scling one or more of ny of the
More informationWhat are suffix trees?
Suffix Trees 1 Wht re suffix trees? Allow lgorithm designers to store very lrge mount of informtion out strings while still keeping within liner spce Allow users to serch for new strings in the originl
More informationOrthogonal line segment intersection
Computtionl Geometry [csci 3250] Line segment intersection The prolem (wht) Computtionl Geometry [csci 3250] Orthogonl line segment intersection Applictions (why) Algorithms (how) A specil cse: Orthogonl
More informationMTH 146 Conics Supplement
105- Review of Conics MTH 146 Conics Supplement In this section we review conics If ou ne more detils thn re present in the notes, r through section 105 of the ook Definition: A prol is the set of points
More informationImproper Integrals. October 4, 2017
Improper Integrls October 4, 7 Introduction We hve seen how to clculte definite integrl when the it is rel number. However, there re times when we re interested to compute the integrl sy for emple 3. Here
More informationSubtracting Fractions
Lerning Enhncement Tem Model Answers: Adding nd Subtrcting Frctions Adding nd Subtrcting Frctions study guide. When the frctions both hve the sme denomintor (bottom) you cn do them using just simple dding
More informationSolving Problems by Searching. CS 486/686: Introduction to Artificial Intelligence Winter 2016
Solving Prolems y Serching CS 486/686: Introduction to Artificil Intelligence Winter 2016 1 Introduction Serch ws one of the first topics studied in AI - Newell nd Simon (1961) Generl Prolem Solver Centrl
More informationInformation Retrieval and Organisation
Informtion Retrievl nd Orgnistion Suffix Trees dpted from http://www.mth.tu.c.il/~himk/seminr02/suffixtrees.ppt Dell Zhng Birkeck, University of London Trie A tree representing set of strings { } eef d
More informationWhat do all those bits mean now? Number Systems and Arithmetic. Introduction to Binary Numbers. Questions About Numbers
Wht do ll those bits men now? bits (...) Number Systems nd Arithmetic or Computers go to elementry school instruction R-formt I-formt... integer dt number text chrs... floting point signed unsigned single
More informationMA1008. Calculus and Linear Algebra for Engineers. Course Notes for Section B. Stephen Wills. Department of Mathematics. University College Cork
MA1008 Clculus nd Liner Algebr for Engineers Course Notes for Section B Stephen Wills Deprtment of Mthemtics University College Cork s.wills@ucc.ie http://euclid.ucc.ie/pges/stff/wills/teching/m1008/ma1008.html
More informationCS201 Discussion 10 DRAWTREE + TRIES
CS201 Discussion 10 DRAWTREE + TRIES DrwTree First instinct: recursion As very generic structure, we could tckle this problem s follows: drw(): Find the root drw(root) drw(root): Write the line for the
More informationCompilers Spring 2013 PRACTICE Midterm Exam
Compilers Spring 2013 PRACTICE Midterm Exm This is full length prctice midterm exm. If you wnt to tke it t exm pce, give yourself 7 minutes to tke the entire test. Just like the rel exm, ech question hs
More informationUnit #9 : Definite Integral Properties, Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties, Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationWhat do all those bits mean now? Number Systems and Arithmetic. Introduction to Binary Numbers. Questions About Numbers
Wht do ll those bits men now? bits (...) Number Systems nd Arithmetic or Computers go to elementry school instruction R-formt I-formt... integer dt number text chrs... floting point signed unsigned single
More informationRepresentation of Numbers. Number Representation. Representation of Numbers. 32-bit Unsigned Integers 3/24/2014. Fixed point Integer Representation
Representtion of Numbers Number Representtion Computer represent ll numbers, other thn integers nd some frctions with imprecision. Numbers re stored in some pproximtion which cn be represented by fixed
More informationsuch that the S i cover S, or equivalently S
MATH 55 Triple Integrls Fll 16 1. Definition Given solid in spce, prtition of consists of finite set of solis = { 1,, n } such tht the i cover, or equivlently n i. Furthermore, for ech i, intersects i
More information9.1 apply the distance and midpoint formulas
9.1 pply the distnce nd midpoint formuls DISTANCE FORMULA MIDPOINT FORMULA To find the midpoint between two points x, y nd x y 1 1,, we Exmple 1: Find the distnce between the two points. Then, find the
More informationCS 241. Fall 2017 Midterm Review Solutions. October 24, Bits and Bytes 1. 3 MIPS Assembler 6. 4 Regular Languages 7.
CS 241 Fll 2017 Midterm Review Solutions Octoer 24, 2017 Contents 1 Bits nd Bytes 1 2 MIPS Assemly Lnguge Progrmming 2 3 MIPS Assemler 6 4 Regulr Lnguges 7 5 Scnning 9 1 Bits nd Bytes 1. Give two s complement
More informationTO REGULAR EXPRESSIONS
Suject :- Computer Science Course Nme :- Theory Of Computtion DA TO REGULAR EXPRESSIONS Report Sumitted y:- Ajy Singh Meen 07000505 jysmeen@cse.iit.c.in BASIC DEINITIONS DA:- A finite stte mchine where
More informationCS311H: Discrete Mathematics. Graph Theory IV. A Non-planar Graph. Regions of a Planar Graph. Euler s Formula. Instructor: Işıl Dillig
CS311H: Discrete Mthemtics Grph Theory IV Instructor: Işıl Dillig Instructor: Işıl Dillig, CS311H: Discrete Mthemtics Grph Theory IV 1/25 A Non-plnr Grph Regions of Plnr Grph The plnr representtion of
More informationThe Fundamental Theorem of Calculus
MATH 6 The Fundmentl Theorem of Clculus The Fundmentl Theorem of Clculus (FTC) gives method of finding the signed re etween the grph of f nd the x-xis on the intervl [, ]. The theorem is: FTC: If f is
More informationIf you are at the university, either physically or via the VPN, you can download the chapters of this book as PDFs.
Lecture 5 Wlks, Trils, Pths nd Connectedness Reding: Some of the mteril in this lecture comes from Section 1.2 of Dieter Jungnickel (2008), Grphs, Networks nd Algorithms, 3rd edition, which is ville online
More informationBefore We Begin. Introduction to Spatial Domain Filtering. Introduction to Digital Image Processing. Overview (1): Administrative Details (1):
Overview (): Before We Begin Administrtive detils Review some questions to consider Winter 2006 Imge Enhncement in the Sptil Domin: Bsics of Sptil Filtering, Smoothing Sptil Filters, Order Sttistics Filters
More informationGeorge Boole. IT 3123 Hardware and Software Concepts. Switching Algebra. Boolean Functions. Boolean Functions. Truth Tables
George Boole IT 3123 Hrdwre nd Softwre Concepts My 28 Digitl Logic The Little Mn Computer 1815 1864 British mthemticin nd philosopher Mny contriutions to mthemtics. Boolen lger: n lger over finite sets
More informationCMPSC 470: Compiler Construction
CMPSC 47: Compiler Construction Plese complete the following: Midterm (Type A) Nme Instruction: Mke sure you hve ll pges including this cover nd lnk pge t the end. Answer ech question in the spce provided.
More informationTries. Yufei Tao KAIST. April 9, Y. Tao, April 9, 2013 Tries
Tries Yufei To KAIST April 9, 2013 Y. To, April 9, 2013 Tries In this lecture, we will discuss the following exct mtching prolem on strings. Prolem Let S e set of strings, ech of which hs unique integer
More informationAlignment of Long Sequences. BMI/CS Spring 2012 Colin Dewey
Alignment of Long Sequences BMI/CS 776 www.biostt.wisc.edu/bmi776/ Spring 2012 Colin Dewey cdewey@biostt.wisc.edu Gols for Lecture the key concepts to understnd re the following how lrge-scle lignment
More informationSolving Problems by Searching. CS 486/686: Introduction to Artificial Intelligence
Solving Prolems y Serching CS 486/686: Introduction to Artificil Intelligence 1 Introduction Serch ws one of the first topics studied in AI - Newell nd Simon (1961) Generl Prolem Solver Centrl component
More informationINTRODUCTION TO SIMPLICIAL COMPLEXES
INTRODUCTION TO SIMPLICIAL COMPLEXES CASEY KELLEHER AND ALESSANDRA PANTANO 0.1. Introduction. In this ctivity set we re going to introduce notion from Algebric Topology clled simplicil homology. The min
More informationCSCI1950 Z Computa4onal Methods for Biology Lecture 2. Ben Raphael January 26, hhp://cs.brown.edu/courses/csci1950 z/ Outline
CSCI1950 Z Comput4onl Methods for Biology Lecture 2 Ben Rphel Jnury 26, 2009 hhp://cs.brown.edu/courses/csci1950 z/ Outline Review of trees. Coun4ng fetures. Chrcter bsed phylogeny Mximum prsimony Mximum
More informationLecture Overview. Knowledge-based systems in Bioinformatics, 1MB602. Procedural abstraction. The sum procedure. Integration as a procedure
Lecture Overview Knowledge-bsed systems in Bioinformtics, MB6 Scheme lecture Procedurl bstrction Higher order procedures Procedures s rguments Procedures s returned vlues Locl vribles Dt bstrction Compound
More information1 Quad-Edge Construction Operators
CS48: Computer Grphics Hndout # Geometric Modeling Originl Hndout #5 Stnford University Tuesdy, 8 December 99 Originl Lecture #5: 9 November 99 Topics: Mnipultions with Qud-Edge Dt Structures Scribe: Mike
More informationA Tautology Checker loosely related to Stålmarck s Algorithm by Martin Richards
A Tutology Checker loosely relted to Stålmrck s Algorithm y Mrtin Richrds mr@cl.cm.c.uk http://www.cl.cm.c.uk/users/mr/ University Computer Lortory New Museum Site Pemroke Street Cmridge, CB2 3QG Mrtin
More informationToday. CS 188: Artificial Intelligence Fall Recap: Search. Example: Pancake Problem. Example: Pancake Problem. General Tree Search.
CS 88: Artificil Intelligence Fll 00 Lecture : A* Serch 9//00 A* Serch rph Serch Tody Heuristic Design Dn Klein UC Berkeley Multiple slides from Sturt Russell or Andrew Moore Recp: Serch Exmple: Pncke
More informationLU Decomposition. Mechanical Engineering Majors. Authors: Autar Kaw
LU Decomposition Mechnicl Engineering Mjors Authors: Autr Kw Trnsforming Numericl Methods Eduction for STEM Undergrdutes // LU Decomposition LU Decomposition LU Decomposition is nother method to solve
More informationQuestions About Numbers. Number Systems and Arithmetic. Introduction to Binary Numbers. Negative Numbers?
Questions About Numbers Number Systems nd Arithmetic or Computers go to elementry school How do you represent negtive numbers? frctions? relly lrge numbers? relly smll numbers? How do you do rithmetic?
More informationCS 241 Week 4 Tutorial Solutions
CS 4 Week 4 Tutoril Solutions Writing n Assemler, Prt & Regulr Lnguges Prt Winter 8 Assemling instrutions utomtilly. slt $d, $s, $t. Solution: $d, $s, nd $t ll fit in -it signed integers sine they re 5-it
More informationBasics of Logic Design Arithmetic Logic Unit (ALU)
Bsics of Logic Design Arithmetic Logic Unit (ALU) CPS 4 Lecture 9 Tody s Lecture Homework #3 Assigned Due Mrch 3 Project Groups ssigned & posted to lckord. Project Specifiction is on We Due April 9 Building
More informationLexical Analysis. Amitabha Sanyal. (www.cse.iitb.ac.in/ as) Department of Computer Science and Engineering, Indian Institute of Technology, Bombay
Lexicl Anlysis Amith Snyl (www.cse.iit.c.in/ s) Deprtment of Computer Science nd Engineering, Indin Institute of Technology, Bomy Septemer 27 College of Engineering, Pune Lexicl Anlysis: 2/6 Recp The input
More informationSection 5.3 : Finding Area Between Curves
MATH 9 Section 5. : Finding Are Between Curves Importnt: In this section we will lern just how to set up the integrls to find re etween curves. The finl nswer for ech emple in this hndout is given for
More informationCSc 453. Compilers and Systems Software. 4 : Lexical Analysis II. Department of Computer Science University of Arizona
CSc 453 Compilers nd Systems Softwre 4 : Lexicl Anlysis II Deprtment of Computer Science University of Arizon collerg@gmil.com Copyright c 2009 Christin Collerg Implementing Automt NFAs nd DFAs cn e hrd-coded
More informationUninformed Search. Hal Daumé III. Computer Science University of Maryland CS 421: Introduction to Artificial Intelligence 31 Jan 2012
1 Hl Dumé III (me@hl3.nme) Uninformed Serch Hl Dumé III Comuter Science University of Mrylnd me@hl3.nme CS 421: Introduction to Artificil Intelligence 31 Jn 2012 Mny slides courtesy of Dn Klein, Sturt
More informationMid-term exam. Scores. Fall term 2012 KAIST EE209 Programming Structures for EE. Thursday Oct 25, Student's name: Student ID:
Fll term 2012 KAIST EE209 Progrmming Structures for EE Mid-term exm Thursdy Oct 25, 2012 Student's nme: Student ID: The exm is closed book nd notes. Red the questions crefully nd focus your nswers on wht
More informationEngineer To Engineer Note
Engineer To Engineer Note EE-186 Technicl Notes on using Anlog Devices' DSP components nd development tools Contct our technicl support by phone: (800) ANALOG-D or e-mil: dsp.support@nlog.com Or visit
More informationPhysics 208: Electricity and Magnetism Exam 1, Secs Feb IMPORTANT. Read these directions carefully:
Physics 208: Electricity nd Mgnetism Exm 1, Secs. 506 510 11 Feb. 2004 Instructor: Dr. George R. Welch, 415 Engineering-Physics, 845-7737 Print your nme netly: Lst nme: First nme: Sign your nme: Plese
More informationEXPONENTIAL & POWER GRAPHS
Eponentil & Power Grphs EXPONENTIAL & POWER GRAPHS www.mthletics.com.u Eponentil EXPONENTIAL & Power & Grphs POWER GRAPHS These re grphs which result from equtions tht re not liner or qudrtic. The eponentil
More informationSummer Review Packet For Algebra 2 CP/Honors
Summer Review Pcket For Alger CP/Honors Nme Current Course Mth Techer Introduction Alger uilds on topics studied from oth Alger nd Geometr. Certin topics re sufficientl involved tht the cll for some review
More informationAgilent Mass Hunter Software
Agilent Mss Hunter Softwre Quick Strt Guide Use this guide to get strted with the Mss Hunter softwre. Wht is Mss Hunter Softwre? Mss Hunter is n integrl prt of Agilent TOF softwre (version A.02.00). Mss
More informationImplementing Automata. CSc 453. Compilers and Systems Software. 4 : Lexical Analysis II. Department of Computer Science University of Arizona
Implementing utomt Sc 5 ompilers nd Systems Softwre : Lexicl nlysis II Deprtment of omputer Science University of rizon collerg@gmil.com opyright c 009 hristin ollerg NFs nd DFs cn e hrd-coded using this
More informationApproximation by NURBS with free knots
pproximtion by NURBS with free knots M Rndrinrivony G Brunnett echnicl University of Chemnitz Fculty of Computer Science Computer Grphics nd Visuliztion Strße der Ntionen 6 97 Chemnitz Germny Emil: mhrvo@informtiktu-chemnitzde
More informationPointers and Arrays. More Pointer Examples. Pointers CS 217
Pointers nd Arrs CS 21 1 2 Pointers More Pointer Emples Wht is pointer A vrile whose vlue is the ddress of nother vrile p is pointer to vrile v Opertions &: ddress of (reference) *: indirection (dereference)
More informationControl-Flow Analysis and Loop Detection
! Control-Flow Anlysis nd Loop Detection!Lst time! PRE!Tody! Control-flow nlysis! Loops! Identifying loops using domintors! Reducibility! Using loop identifiction to identify induction vribles CS553 Lecture
More information10/9/2012. Operator is an operation performed over data at runtime. Arithmetic, Logical, Comparison, Assignment, Etc. Operators have precedence
/9/22 P f Performing i Si Simple l Clcultions C l l ti with ith C#. Opertors in C# nd Opertor Precedence 2. Arithmetic Opertors 3. Logicl Opertors 4. Bitwise Opertors 5. Comprison Opertors 6. Assignment
More informationAgenda & Reading. Class Exercise. COMPSCI 105 SS 2012 Principles of Computer Science. Arrays
COMPSCI 5 SS Principles of Computer Science Arrys & Multidimensionl Arrys Agend & Reding Agend Arrys Creting & Using Primitive & Reference Types Assignments & Equlity Pss y Vlue & Pss y Reference Copying
More informationSection 3.1: Sequences and Series
Section.: Sequences d Series Sequences Let s strt out with the definition of sequence: sequence: ordered list of numbers, often with definite pttern Recll tht in set, order doesn t mtter so this is one
More informationTree Structured Symmetrical Systems of Linear Equations and their Graphical Solution
Proceedings of the World Congress on Engineering nd Computer Science 4 Vol I WCECS 4, -4 October, 4, Sn Frncisco, USA Tree Structured Symmetricl Systems of Liner Equtions nd their Grphicl Solution Jime
More informationIf f(x, y) is a surface that lies above r(t), we can think about the area between the surface and the curve.
Line Integrls The ide of line integrl is very similr to tht of single integrls. If the function f(x) is bove the x-xis on the intervl [, b], then the integrl of f(x) over [, b] is the re under f over the
More informationLily Yen and Mogens Hansen
SKOLID / SKOLID No. 8 Lily Yen nd Mogens Hnsen Skolid hs joined Mthemticl Myhem which is eing reformtted s stnd-lone mthemtics journl for high school students. Solutions to prolems tht ppered in the lst
More informationSIMPLIFYING ALGEBRA PASSPORT.
SIMPLIFYING ALGEBRA PASSPORT www.mthletics.com.u This booklet is ll bout turning complex problems into something simple. You will be ble to do something like this! ( 9- # + 4 ' ) ' ( 9- + 7-) ' ' Give
More informationMath 464 Fall 2012 Notes on Marginal and Conditional Densities October 18, 2012
Mth 464 Fll 2012 Notes on Mrginl nd Conditionl Densities klin@mth.rizon.edu October 18, 2012 Mrginl densities. Suppose you hve 3 continuous rndom vribles X, Y, nd Z, with joint density f(x,y,z. The mrginl
More informationCSCI 104. Rafael Ferreira da Silva. Slides adapted from: Mark Redekopp and David Kempe
CSCI 0 fel Ferreir d Silv rfsilv@isi.edu Slides dpted from: Mrk edekopp nd Dvid Kempe LOG STUCTUED MEGE TEES Series Summtion eview Let n = + + + + k $ = #%& #. Wht is n? n = k+ - Wht is log () + log ()
More informationMath/CS 467/667 Programming Assignment 01. Adaptive Gauss Quadrature. q(x)p 4 (x) = 0
Adptive Guss Qudrture 1. Find n orthogonl polynomil p 4 of degree 4 such tht 1 1 q(x)p 4 (x) = 0 for every polynomil q(x) of degree 3 or less. You my use Mple nd the Grm Schmidt process s done in clss.
More informationMatrices and Systems of Equations
Mtrices Mtrices nd Sstems of Equtions A mtri is rectngulr rr of rel numbers. CHAT Pre-Clculus Section 8. m m m............ n n n mn We will use the double subscript nottion for ech element of the mtri.
More informationPARALLEL AND DISTRIBUTED COMPUTING
PARALLEL AND DISTRIBUTED COMPUTING 2009/2010 1 st Semester Teste Jnury 9, 2010 Durtion: 2h00 - No extr mteril llowed. This includes notes, scrtch pper, clcultor, etc. - Give your nswers in the ville spce
More informationCOMBINATORIAL PATTERN MATCHING
COMBINATORIAL PATTERN MATCHING Genomic Repets Exmple of repets: ATGGTCTAGGTCCTAGTGGTC Motivtion to find them: Genomic rerrngements re often ssocited with repets Trce evolutionry secrets Mny tumors re chrcterized
More informationMath 4 Review for Quarter 2 Cumulative Test
Mth 4 Review for Qurter 2 Cumultive Test Nme: I. Right Tringle Trigonometry (3.1-3.3) Key Fcts Pythgoren Theorem - In right tringle, 2 + b 2 = c 2 where c is the hypotenuse s shown below. c b Trigonometric
More information4/29/18 FIBONACCI NUMBERS GOLDEN RATIO, RECURRENCES. Fibonacci function. Fibonacci (Leonardo Pisano) ? Statue in Pisa Italy
/9/8 Fioncci (Leonrdo Pisno) -? Sttue in Pis Itly FIBONACCI NUERS GOLDEN RATIO, RECURRENCES Lecture CS Spring 8 Fioncci function fi() fi() fi(n) fi(n-) + fi(n-) for n,,,,,, 8,,, In his ook in titled Lier
More informationLecture 10 Evolutionary Computation: Evolution strategies and genetic programming
Lecture 10 Evolutionry Computtion: Evolution strtegies nd genetic progrmming Evolution strtegies Genetic progrmming Summry Negnevitsky, Person Eduction, 2011 1 Evolution Strtegies Another pproch to simulting
More information