Algorithms and Data Structures 2014 Exercises and Solutions Week 13
|
|
- Sheena Price
- 5 years ago
- Views:
Transcription
1 Algorithms ad Data Structures 204 Exercises ad Solutios Week 3 Toom-Cook (cotiued) Durig the last lecture, two polyomials A(x) a 0 + a x ad B(x) b 0 + b x both of degree were multiplied, first by evaluatig A ad B at three distict poits we picked 0, ad as cocrete values the these results were multiplied pairwise, ad fially the value represetatio of the resultig polyomial C(x) c 0 + c x + c 2 x 2 (of degree 2) was traslated back usig the iverse of the correspodig evaluatio matrix.. Repeat this procedure, but ow with 0,, as evaluatio poits. Evaluatig A at meas that we just take A s most leadig coefficiet a. Therefore, evaluatig C at (beig c 2 ) amouts to computig the product a b. 2. Explai the relatio of this procedure with Karatsuba s multiplicatio algorithm. 3. Istead of usig polyomials of degree we repeat the whole procedure with polyomials of degree 2. The the multiplicatio result will have degree 4, so we eed 5 evaluatio poits for iterpolatio (the traslatio back from values to coefficiets). Evaluate both A(x) a 0 + a x + a 2 x 2 ad B(x) b 0 + b x + b 2 x 2 at 0,,, 2,. Express the evaluatio of C(x) c 0 +c x+c 2 x 2 +c 3 x 3 +c 4 x 4 i these poits by meas of a 5 5 matrix, ad compute the iverse of this matrix. Agai, explai the relatio with the Toom-Cook3 algorithm (see exercise 4, week 2). Solutio. The evaluatio matrix correspodig to these evaluatio poits is M ad its iverse is give by Thus, c 0 c c 2 M C(0) C() 0 0 C( ) 0 0 A(0)B(0) A()B() 0 0 A( )B( ) 0 0 a 0 b 0 (a 0 + a )(b 0 + b ) 0 0 a b a 0 b 0 (a 0 + a )(b 0 + b ) a 0 b 0 a b. a b You were allowed to use a computer to obtai iverse matrices.
2 2. The resultig formulas for the coefficiets of C are exactly the oes used by Karatsuba s multiplicatio algorithm. 3. The evaluatio matrix is ow M ( ) 0 ( ) ( ) 2 ( ) 3 ( ) , with iverse M Now we have c C(0) c c 2 c C() C( ) C(2) c C( ) A(0)B(0) A()B() A( )B( ) A(2)B(2) A( )B( ) a 0 b (a 0 + a + a 2 )(b 0 + b + b 2 ) (a 0 a + a 2 )(b 0 b + b 2 ) (a 0 + 2a + 4a 2 )(b 0 + 2b + 4b 2 ) a 2 b 2 With some imagiatio, these are see to be precisely the calculatios performed by Toom- Cook3. 2 Fast Fourier Trasform 2. Evaluatio Give a algorithm that evaluates a polyomial A(x) a 0 + a x + a 2 x a x at poit x i O() time. Solutio : fuctio EVAL(A, x) 2: v 0 3: p 4: for i from 0 to do 5: v v + a i p 6: p x p 7: retur v
3 2.2 Basic properties of th roots of uity. What is the sum of the th roots of uity? 2. What is their product if is odd? If is eve? Solutio. For, the oly th root of uity is ω 0 cos 2π 0 + i si 2π 0. Now suppose >. Usig the formula for a geometric series, we see that ω k ω ω cos 2π 2π + i si ω cos(2π) + i si(2π) ω cos(2π) ω ω 0 ω Usig the laws show durig the lecture, we fid that 2.3 Iverse FFT ( ω k cos 2π k + i si 2π k ) ( ) ( ) 2π k 2π k cos + i si ( ) ( ) 2π 2π cos k + i si k ( ) ( ) 2π( ) 2π( ) cos + i si 2 2 cos(π( )) + i si(π( )) cos(π( )) { if is eve otherwise. Show that the matrices M(, ω, ω 2,..., ω ) ad M(, ω, ω 2,..., ω ( ) ) are ideed iverses of each other. Hit: Show that their product is the idetity matrix. Solutio We ame M M(, ω, ω 2,..., ω ) ad N M(, ω, ω 2,..., ω ( ) ). That is, M ij ω ij ad N ij ω ij if we umber the idices from 0 to. We wat to show that
4 MN I, where I is the idetity matrix. We have (MN) ij M ik N kj ω ik ω kj (ω i j ) k. Obviously, this yields if i j, givig us the desired diagoal. If i j, we use agai the geometric series formula to obtai (ω i j ) k (ωi j ) ω i j cos 2π(i j) + i si 2π(i j) ω i j cos(2π(i j)) + i si(2π(i j)) ω i j ω i j 0 ω i j Practice with the fast Fourier Trasform. What is the FFT of (, 0, 0, 0) What is the appropriate value of ω i this case? Ad of which sequece is (, 0, 0, 0) the FFT? 2. Repeat for (, 0,, ). Solutio. We are give 4 coefficiets, so ω cos 2π 4 + i si 2π 4 i. However, i this case the value of ω is irrelevat, sice FFT((, 0, 0, 0), ω) ω ω 2 ω 3 0 ω 2 ω 4 ω 6 0 ω 3 ω 6 ω 9 0.
5 The same holds for the iverse FFT, as FFT FFT((, 0, 0, 0), ω ) 0 ω ω 2 ω ω 2 ω 4 ω 6 0 ω 3 ω 6 ω Now we have (fillig i i for ω as obtaied i the previous part) FFT((, 0,, ), i) i i 2 i 3 0 i 2 i 4 i 6 i 3 i 6 i 9 i i 0 i i i 3. i Note that i i i 2 i i. Therefore, FFT 0 FFT((, 0,, ), i) 4 i i 2 i i 2 i 4 i 6 i 3 i 6 i 9 i i 0 4 i i i Practice with polyomial multiplicatio by FFT 3 i. Suppose that you wat to multiply the two polyomials x + ad x 2 + usig the FFT. Choose a appropriate power of two, fid the FFT of the two sequeces, multiply the results
6 compoetwise, ad compute the iverse FFT to get the fial result. 2. Repeat for the pair of polyomials + x + 2x 2 ad 2 + 3x. Solutio. Multiplicatio of a polyomial of degree with a polyomial of degree 2 yields a polyomial of degree 3. Obviously, 3 < 4, ad 4 is a power of 2. As i the previous exercise, ω i. The FFT of the first polyomial is FFT((,, 0, 0), i) i i 2 i 3 i 2 i 4 i 6 0 i 3 i 6 i 9 0 i i 0 i i i 0. i For the secod polyomial we have FFT((, 0,, 0), i) i i 2 i 3 0 i 2 i 4 i 6 i 3 i 6 i 9 0 i i 0 i i The coefficiets of the product polyomial are ow give by the iverse FFT of the poitwise product of these value represetatios. That is, 4 FFT FFT((4, 0, 0, 0), i ) 0 4 i i 2 i i 2 i 4 i 6 0 i 3 i 6 i 9 0. We coclude that the resultig polyomial is + x + x 2 + x 3.
7 2. We still have ω i. I this case, the FFT of the first polyomial is FFT((,, 2, 0), i) i i 2 i 3 i 2 i 4 i 6 2 i 3 i 6 i 9 0 i i 2 i i 0 4 i 2. i For the secod polyomial we have 2 FFT((2, 3, 0, 0), i) i i 2 i 3 3 i 2 i 4 i 6 0 i 3 i 6 i i i 3 0 i i i. 2 3i Now we obtai the coefficiets of the product polyomial as 20 FFT 5 i 2 4 FFT((20, 5 i, 2, i 5), i ) i 5 20 i i 2 i 3 5 i 4 i 2 i 4 i 6 2 i 3 i 6 i 9 i 5 20 i i 5 i 4 2 i i i , 6 so the product polyomial is 2 + 5x + 7x 2 + 6x 3.
8 3 FFT i modular arithmetic This exercise illustrates how to do the Fourier Trasform (FT) i modular arithmetic, for example, modulo 7.. There is a umber ω such that all the powers ω 0, ω,..., ω 5 are distict (modulo 7). Fid this ω ad show that ω 0 + ω + + ω 5 0 (modulo 7). 2. Usig the matrix form of the FT, produce the trasform of the sequece (0,,,, 5, 2) modulo 7; that is, multiply this vector by the matrix M(ω 0, ω,..., ω 5 ). I the matrix multiplicatio, all calculatios should be performed modulo Write dow the matrix ecessary to perform the iverse FT. Show that multiplyig by this matrix returs the origial sequece. (Agai all arithmetic should be performed modulo 7.) 4. Now show how to multiply the polyomials x 2 + x + ad x 3 + 2x usig the FT modulo 7.
Let A(x) and B(x) be two polynomials of degree n 1:
MI-EVY (2011/2012) J. Holub: 4. DFT, FFT ad Patter Matchig p. 2/42 Operatios o polyomials MI-EVY (2011/2012) J. Holub: 4. DFT, FFT ad Patter Matchig p. 4/42 Efficiet Patter Matchig (MI-EVY) 4. DFT, FFT
More informationThe Discrete Fourier Transform
The Discrete Fourier Trasform Complex Fourier Series Represetatio Recall that a Fourier series has the form a 0 + a k cos(kt) + k=1 b k si(kt) This represetatio seems a bit awkward, sice it ivolves two
More information1 Generating functions for balls in boxes
Math 566 Fall 05 Some otes o geeratig fuctios Give a sequece a 0, a, a,..., a,..., a geeratig fuctio some way of represetig the sequece as a fuctio. There are may ways to do this, with the most commo ways
More informationLecture 3: Divide and Conquer: Fast Fourier Transform
Lecture 3: Divide ad Coquer: Fast Fourier Trasform Polyomial Operatios vs. Represetatios Divide ad Coquer Algorithm Collapsig Samples / Roots of Uity FFT, IFFT, ad Polyomial Multiplicatio Polyomial operatios
More informationPolynomial Multiplication and Fast Fourier Transform
Polyomial Multiplicatio ad Fast Fourier Trasform Com S 477/577 Notes Ya-Bi Jia Sep 19, 2017 I this lecture we will describe the famous algorithm of fast Fourier trasform FFT, which has revolutioized digital
More informationQuantum Computing Lecture 7. Quantum Factoring
Quatum Computig Lecture 7 Quatum Factorig Maris Ozols Quatum factorig A polyomial time quatum algorithm for factorig umbers was published by Peter Shor i 1994. Polyomial time meas that the umber of gates
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationProperties and Tests of Zeros of Polynomial Functions
Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by
More informationECEN 644 HOMEWORK #5 SOLUTION SET
ECE 644 HOMEWORK #5 SOUTIO SET 7. x is a real valued sequece. The first five poits of its 8-poit DFT are: {0.5, 0.5 - j 0.308, 0, 0.5 - j 0.058, 0} To compute the 3 remaiig poits, we ca use the followig
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationPrinciple Of Superposition
ecture 5: PREIMINRY CONCEP O RUCUR NYI Priciple Of uperpositio Mathematically, the priciple of superpositio is stated as ( a ) G( a ) G( ) G a a or for a liear structural system, the respose at a give
More informationMAT 271 Project: Partial Fractions for certain rational functions
MAT 7 Project: Partial Fractios for certai ratioal fuctios Prerequisite kowledge: partial fractios from MAT 7, a very good commad of factorig ad complex umbers from Precalculus. To complete this project,
More informationMath 113, Calculus II Winter 2007 Final Exam Solutions
Math, Calculus II Witer 7 Fial Exam Solutios (5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute x x + dx The check your aswer usig the Evaluatio Theorem Solutio: I this
More informationEigenvalues and Eigenvectors
5 Eigevalues ad Eigevectors 5.3 DIAGONALIZATION DIAGONALIZATION Example 1: Let. Fid a formula for A k, give that P 1 1 = 1 2 ad, where Solutio: The stadard formula for the iverse of a 2 2 matrix yields
More information2 Geometric interpretation of complex numbers
2 Geometric iterpretatio of complex umbers 2.1 Defiitio I will start fially with a precise defiitio, assumig that such mathematical object as vector space R 2 is well familiar to the studets. Recall that
More informationMath 142, Final Exam. 5/2/11.
Math 4, Fial Exam 5// No otes, calculator, or text There are poits total Partial credit may be give Write your full ame i the upper right corer of page Number the pages i the upper right corer Do problem
More informationChimica Inorganica 3
himica Iorgaica Irreducible Represetatios ad haracter Tables Rather tha usig geometrical operatios, it is ofte much more coveiet to employ a ew set of group elemets which are matrices ad to make the rule
More informationLecture 23 Rearrangement Inequality
Lecture 23 Rearragemet Iequality Holde Lee 6/4/ The Iequalities We start with a example Suppose there are four boxes cotaiig $0, $20, $50 ad $00 bills, respectively You may take 2 bills from oe box, 3
More informationCALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationWorksheet on Generating Functions
Worksheet o Geeratig Fuctios October 26, 205 This worksheet is adapted from otes/exercises by Nat Thiem. Derivatives of Geeratig Fuctios. If the sequece a 0, a, a 2,... has ordiary geeratig fuctio A(x,
More informationMATH 31B: MIDTERM 2 REVIEW
MATH 3B: MIDTERM REVIEW JOE HUGHES. Evaluate x (x ) (x 3).. Partial Fractios Solutio: The umerator has degree less tha the deomiator, so we ca use partial fractios. Write x (x ) (x 3) = A x + A (x ) +
More informationSolutions to Homework 1
Solutios to Homework MATH 36. Describe geometrically the sets of poits z i the complex plae defied by the followig relatios /z = z () Re(az + b) >, where a, b (2) Im(z) = c, with c (3) () = = z z = z 2.
More informationPROBLEM SET 5 SOLUTIONS 126 = , 37 = , 15 = , 7 = 7 1.
Math 7 Sprig 06 PROBLEM SET 5 SOLUTIONS Notatios. Give a real umber x, we will defie sequeces (a k ), (x k ), (p k ), (q k ) as i lecture.. (a) (5 pts) Fid the simple cotiued fractio represetatios of 6
More informationLemma Let f(x) K[x] be a separable polynomial of degree n. Then the Galois group is a subgroup of S n, the permutations of the roots.
15 Cubics, Quartics ad Polygos It is iterestig to chase through the argumets of 14 ad see how this affects solvig polyomial equatios i specific examples We make a global assumptio that the characteristic
More informationSEQUENCES AND SERIES
Sequeces ad 6 Sequeces Ad SEQUENCES AND SERIES Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives rise to what is called a sequece. Sequeces
More information6.003 Homework #3 Solutions
6.00 Homework # Solutios Problems. Complex umbers a. Evaluate the real ad imagiary parts of j j. π/ Real part = Imagiary part = 0 e Euler s formula says that j = e jπ/, so jπ/ j π/ j j = e = e. Thus the
More informationFall 2011, EE123 Digital Signal Processing
Lecture 5 Miki Lustig, UCB September 14, 211 Miki Lustig, UCB Motivatios for Discrete Fourier Trasform Sampled represetatio i time ad frequecy umerical Fourier aalysis requires a Fourier represetatio that
More informationComplex Analysis Spring 2001 Homework I Solution
Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle
More informationBasic Sets. Functions. MTH299 - Examples. Example 1. Let S = {1, {2, 3}, 4}. Indicate whether each statement is true or false. (a) S = 4. (e) 2 S.
Basic Sets Example 1. Let S = {1, {2, 3}, 4}. Idicate whether each statemet is true or false. (a) S = 4 (b) {1} S (c) {2, 3} S (d) {1, 4} S (e) 2 S. (f) S = {1, 4, {2, 3}} (g) S Example 2. Compute the
More informationTMA4205 Numerical Linear Algebra. The Poisson problem in R 2 : diagonalization methods
TMA4205 Numerical Liear Algebra The Poisso problem i R 2 : diagoalizatio methods September 3, 2007 c Eiar M Røquist Departmet of Mathematical Scieces NTNU, N-749 Trodheim, Norway All rights reserved A
More informationGenerating Functions II
Geeratig Fuctios II Misha Lavrov ARML Practice 5/4/2014 Warm-up problems 1. Solve the recursio a +1 = 2a, a 0 = 1 by usig commo sese. 2. Solve the recursio b +1 = 2b + 1, b 0 = 1 by usig commo sese ad
More informationCarleton College, Winter 2017 Math 121, Practice Final Prof. Jones. Note: the exam will have a section of true-false questions, like the one below.
Carleto College, Witer 207 Math 2, Practice Fial Prof. Joes Note: the exam will have a sectio of true-false questios, like the oe below.. True or False. Briefly explai your aswer. A icorrectly justified
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationTopic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or
Topic : Sequeces ad Series A sequece is a ordered list of umbers, e.g.,,, 8, 6, or,,,.... A series is a sum of the terms of a sequece, e.g. + + + 8 + 6 + or... Sigma Notatio b The otatio f ( k) is shorthad
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More informationSolutions to Final Exam Review Problems
. Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the
More informationMATH 1910 Workshop Solution
MATH 90 Workshop Solutio Fractals Itroductio: Fractals are atural pheomea or mathematical sets which exhibit (amog other properties) self similarity: o matter how much we zoom i, the structure remais the
More informationSNAP Centre Workshop. Basic Algebraic Manipulation
SNAP Cetre Workshop Basic Algebraic Maipulatio 8 Simplifyig Algebraic Expressios Whe a expressio is writte i the most compact maer possible, it is cosidered to be simplified. Not Simplified: x(x + 4x)
More informationTR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT
TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the
More informationInverse Matrix. A meaning that matrix B is an inverse of matrix A.
Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix
More informationTHE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS
THE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS DEMETRES CHRISTOFIDES Abstract. Cosider a ivertible matrix over some field. The Gauss-Jorda elimiatio reduces this matrix to the idetity
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More informationThe Method of Least Squares. To understand least squares fitting of data.
The Method of Least Squares KEY WORDS Curve fittig, least square GOAL To uderstad least squares fittig of data To uderstad the least squares solutio of icosistet systems of liear equatios 1 Motivatio Curve
More informationNICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) =
AN INTRODUCTION TO SCHRÖDER AND UNKNOWN NUMBERS NICK DUFRESNE Abstract. I this article we will itroduce two types of lattice paths, Schröder paths ad Ukow paths. We will examie differet properties of each,
More informationMatrices and vectors
Oe Matrices ad vectors This book takes for grated that readers have some previous kowledge of the calculus of real fuctios of oe real variable It would be helpful to also have some kowledge of liear algebra
More informationANSWERS SOLUTIONS iiii i. and 1. Thus, we have. i i i. i, A.
013 ΜΑΘ Natioal Covetio ANSWERS (1) C A A A B (6) B D D A B (11) C D D A A (16) D B A A C (1) D B C B C (6) D C B C C 1. We have SOLUTIONS 1 3 11 61 iiii 131161 i 013 013, C.. The powers of i cycle betwee
More informationCSI 2101 Discrete Structures Winter Homework Assignment #4 (100 points, weight 5%) Due: Thursday, April 5, at 1:00pm (in lecture)
CSI 101 Discrete Structures Witer 01 Prof. Lucia Moura Uiversity of Ottawa Homework Assigmet #4 (100 poits, weight %) Due: Thursday, April, at 1:00pm (i lecture) Program verificatio, Recurrece Relatios
More informationCS:3330 (Prof. Pemmaraju ): Assignment #1 Solutions. (b) For n = 3, we will have 3 men and 3 women with preferences as follows: m 1 : w 3 > w 1 > w 2
Shiyao Wag CS:3330 (Prof. Pemmaraju ): Assigmet #1 Solutios Problem 1 (a) Cosider iput with me m 1, m,..., m ad wome w 1, w,..., w with the followig prefereces: All me have the same prefereces for wome:
More information11.6 Absolute Convergence and the Ratio and Root Tests
.6 Absolute Covergece ad the Ratio ad Root Tests The most commo way to test for covergece is to igore ay positive or egative sigs i a series, ad simply test the correspodig series of positive terms. Does
More informationMath 105 TOPICS IN MATHEMATICS REVIEW OF LECTURES VII. 7. Binomial formula. Three lectures ago ( in Review of Lectuires IV ), we have covered
Math 5 TOPICS IN MATHEMATICS REVIEW OF LECTURES VII Istructor: Lie #: 59 Yasuyuki Kachi 7 Biomial formula February 4 Wed) 5 Three lectures ago i Review of Lectuires IV ) we have covered / \ / \ / \ / \
More informationMath 110 Assignment #6 Due: Monday, February 10
Math Assigmet #6 Due: Moday, February Justify your aswers. Show all steps i your computatios. Please idicate your fial aswer by puttig a box aroud it. Please write eatly ad legibly. Illegible aswers will
More informationComplex Numbers Solutions
Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i
More informationINTEGRATION BY PARTS (TABLE METHOD)
INTEGRATION BY PARTS (TABLE METHOD) Suppose you wat to evaluate cos d usig itegratio by parts. Usig the u dv otatio, we get So, u dv d cos du d v si cos d si si d or si si d We see that it is ecessary
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 50-004 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig
More information[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms.
[ 11 ] 1 1.1 Polyomial Fuctios 1 Algebra Ay fuctio f ( x) ax a1x... a1x a0 is a polyomial fuctio if ai ( i 0,1,,,..., ) is a costat which belogs to the set of real umbers ad the idices,, 1,...,1 are atural
More informationREVIEW 1, MATH n=1 is convergent. (b) Determine whether a n is convergent.
REVIEW, MATH 00. Let a = +. a) Determie whether the sequece a ) is coverget. b) Determie whether a is coverget.. Determie whether the series is coverget or diverget. If it is coverget, fid its sum. a)
More informationIntensive Algorithms Lecture 11. DFT and DP. Lecturer: Daniel A. Spielman February 20, f(n) O(g(n) log c g(n)).
Itesive Algorithms Lecture 11 DFT ad DP Lecturer: Daiel A. Spielma February 20, 2018 11.1 Itroductio The purpose of this lecture is to lear how use the Discrete Fourier Trasform to save space i Dyamic
More informationMath 4400/6400 Homework #7 solutions
MATH 4400 problems. Math 4400/6400 Homewor #7 solutios 1. Let p be a prime umber. Show that the order of 1 + p modulo p 2 is exactly p. Hit: Expad (1 + p) p by the biomial theorem, ad recall from MATH
More information1 Last time: similar and diagonalizable matrices
Last time: similar ad diagoalizable matrices Let be a positive iteger Suppose A is a matrix, v R, ad λ R Recall that v a eigevector for A with eigevalue λ if v ad Av λv, or equivaletly if v is a ozero
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More information1 Approximating Integrals using Taylor Polynomials
Seughee Ye Ma 8: Week 7 Nov Week 7 Summary This week, we will lear how we ca approximate itegrals usig Taylor series ad umerical methods. Topics Page Approximatig Itegrals usig Taylor Polyomials. Defiitios................................................
More informationMAXIMALLY FLAT FIR FILTERS
MAXIMALLY FLAT FIR FILTERS This sectio describes a family of maximally flat symmetric FIR filters first itroduced by Herrma [2]. The desig of these filters is particularly simple due to the availability
More informationSection 1 of Unit 03 (Pure Mathematics 3) Algebra
Sectio 1 of Uit 0 (Pure Mathematics ) Algebra Recommeded Prior Kowledge Studets should have studied the algebraic techiques i Pure Mathematics 1. Cotet This Sectio should be studied early i the course
More informationECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations
ECE-S352 Itroductio to Digital Sigal Processig Lecture 3A Direct Solutio of Differece Equatios Discrete Time Systems Described by Differece Equatios Uit impulse (sample) respose h() of a DT system allows
More informationPractical Spectral Anaysis (continue) (from Boaz Porat s book) Frequency Measurement
Practical Spectral Aaysis (cotiue) (from Boaz Porat s book) Frequecy Measuremet Oe of the most importat applicatios of the DFT is the measuremet of frequecies of periodic sigals (eg., siusoidal sigals),
More informationSolutions to Math 347 Practice Problems for the final
Solutios to Math 347 Practice Problems for the fial 1) True or False: a) There exist itegers x,y such that 50x + 76y = 6. True: the gcd of 50 ad 76 is, ad 6 is a multiple of. b) The ifiimum of a set is
More information1. By using truth tables prove that, for all statements P and Q, the statement
Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3
More information( 1) n (4x + 1) n. n=0
Problem 1 (10.6, #). Fid the radius of covergece for the series: ( 1) (4x + 1). For what values of x does the series coverge absolutely, ad for what values of x does the series coverge coditioally? Solutio.
More informationAssignment 2 Solutions SOLUTION. ϕ 1 Â = 3 ϕ 1 4i ϕ 2. The other case can be dealt with in a similar way. { ϕ 2 Â} χ = { 4i ϕ 1 3 ϕ 2 } χ.
PHYSICS 34 QUANTUM PHYSICS II (25) Assigmet 2 Solutios 1. With respect to a pair of orthoormal vectors ϕ 1 ad ϕ 2 that spa the Hilbert space H of a certai system, the operator  is defied by its actio
More informationx x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0,
Math Activity 9( Due with Fial Eam) Usig first ad secod Taylor polyomials with remaider, show that for, 8 Usig a secod Taylor polyomial with remaider, fid the best costat C so that for, C 9 The th Derivative
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationPractice Test Problems for Test IV, with Solutions
Practice Test Problems for Test IV, with Solutios Dr. Holmes May, 2008 The exam will cover sectios 8.2 (revisited) to 8.8. The Taylor remaider formula from 8.9 will ot be o this test. The fact that sums,
More informationMATH1035: Workbook Four M. Daws, 2009
MATH1035: Workbook Four M. Daws, 2009 Roots of uity A importat result which ca be proved by iductio is: De Moivre s theorem atural umber case: Let θ R ad N. The cosθ + i siθ = cosθ + i siθ. Proof: The
More informationMath 5705: Enumerative Combinatorics, Fall 2018: Homework 3
Uiversity of Miesota, School of Mathematics Math 5705: Eumerative Combiatorics, all 2018: Homewor 3 Darij Griberg October 15, 2018 due date: Wedesday, 10 October 2018 at the begiig of class, or before
More informationSolutions to Tutorial 3 (Week 4)
The Uiversity of Sydey School of Mathematics ad Statistics Solutios to Tutorial Week 4 MATH2962: Real ad Complex Aalysis Advaced Semester 1, 2017 Web Page: http://www.maths.usyd.edu.au/u/ug/im/math2962/
More informationMatrix Algebra 2.2 THE INVERSE OF A MATRIX Pearson Education, Inc.
2 Matrix Algebra 2.2 THE INVERSE OF A MATRIX MATRIX OPERATIONS A matrix A is said to be ivertible if there is a matrix C such that CA = I ad AC = I where, the idetity matrix. I = I I this case, C is a
More informationSolutions to Problem Set 8
8.78 Solutios to Problem Set 8. We ow that ( ) ( + x) x. Now we plug i x, ω, ω ad add the three equatios. If 3 the we ll get a cotributio of + ω + ω + ω + ω 0, whereas if 3 we ll get a cotributio of +
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 informationMath 116 Second Midterm November 13, 2017
Math 6 Secod Midterm November 3, 7 EXAM SOLUTIONS. Do ot ope this exam util you are told to do so.. Do ot write your ame aywhere o this exam. 3. This exam has pages icludig this cover. There are problems.
More information6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.
6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio
More informationLinear Programming and the Simplex Method
Liear Programmig ad the Simplex ethod Abstract This article is a itroductio to Liear Programmig ad usig Simplex method for solvig LP problems i primal form. What is Liear Programmig? Liear Programmig is
More informationProblem Set 2 Solutions
CS271 Radomess & Computatio, Sprig 2018 Problem Set 2 Solutios Poit totals are i the margi; the maximum total umber of poits was 52. 1. Probabilistic method for domiatig sets 6pts Pick a radom subset S
More informationThe Binomial Theorem
The Biomial Theorem Robert Marti Itroductio The Biomial Theorem is used to expad biomials, that is, brackets cosistig of two distict terms The formula for the Biomial Theorem is as follows: (a + b ( k
More information(a) (b) All real numbers. (c) All real numbers. (d) None. to show the. (a) 3. (b) [ 7, 1) (c) ( 7, 1) (d) At x = 7. (a) (b)
Chapter 0 Review 597. E; a ( + )( + ) + + S S + S + + + + + + S lim + l. D; a diverges by the Itegral l k Test sice d lim [(l ) ], so k l ( ) does ot coverge absolutely. But it coverges by the Alteratig
More informationIn number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.
Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers
More informationPolynomial Functions and Their Graphs
Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively
More informationFinite-length Discrete Transforms. Chapter 5, Sections
Fiite-legth Discrete Trasforms Chapter 5, Sectios 5.2-50 5.0 Dr. Iyad djafar Outlie The Discrete Fourier Trasform (DFT) Matrix Represetatio of DFT Fiite-legth Sequeces Circular Covolutio DFT Symmetry Properties
More informationMath 210A Homework 1
Math 0A Homework Edward Burkard Exercise. a) State the defiitio of a aalytic fuctio. b) What are the relatioships betwee aalytic fuctios ad the Cauchy-Riema equatios? Solutio. a) A fuctio f : G C is called
More informationRecursive Algorithms. Recurrences. Recursive Algorithms Analysis
Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects
More informationMachine Learning for Data Science (CS 4786)
Machie Learig for Data Sciece CS 4786) Lecture 9: Pricipal Compoet Aalysis The text i black outlies mai ideas to retai from the lecture. The text i blue give a deeper uderstadig of how we derive or get
More informationCS 270 Algorithms. Oliver Kullmann. Growth of Functions. Divide-and- Conquer Min-Max- Problem. Tutorial. Reading from CLRS for week 2
Geeral remarks Week 2 1 Divide ad First we cosider a importat tool for the aalysis of algorithms: Big-Oh. The we itroduce a importat algorithmic paradigm:. We coclude by presetig ad aalysig two examples.
More informationMAS160: Signals, Systems & Information for Media Technology. Problem Set 5. DUE: November 3, (a) Plot of u[n] (b) Plot of x[n]=(0.
MAS6: Sigals, Systems & Iformatio for Media Techology Problem Set 5 DUE: November 3, 3 Istructors: V. Michael Bove, Jr. ad Rosalid Picard T.A. Jim McBride Problem : Uit-step ad ruig average (DSP First
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR. ANTHONY BROWN 8. Statistics 8.1. Measures of Cetre: Mea, Media ad Mode. If we have a series of umbers the
More informationLecture 3. Digital Signal Processing. Chapter 3. z-transforms. Mikael Swartling Nedelko Grbic Bengt Mandersson. rev. 2016
Lecture 3 Digital Sigal Processig Chapter 3 z-trasforms Mikael Swartlig Nedelko Grbic Begt Madersso rev. 06 Departmet of Electrical ad Iformatio Techology Lud Uiversity z-trasforms We defie the z-trasform
More informationMath 451: Euclidean and Non-Euclidean Geometry MWF 3pm, Gasson 204 Homework 3 Solutions
Math 451: Euclidea ad No-Euclidea Geometry MWF 3pm, Gasso 204 Homework 3 Solutios Exercises from 1.4 ad 1.5 of the otes: 4.3, 4.10, 4.12, 4.14, 4.15, 5.3, 5.4, 5.5 Exercise 4.3. Explai why Hp, q) = {x
More informationA Proof of Birkhoff s Ergodic Theorem
A Proof of Birkhoff s Ergodic Theorem Joseph Hora September 2, 205 Itroductio I Fall 203, I was learig the basics of ergodic theory, ad I came across this theorem. Oe of my supervisors, Athoy Quas, showed
More informationEstimation for Complete Data
Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of
More informationFrequency Domain Filtering
Frequecy Domai Filterig Raga Rodrigo October 19, 2010 Outlie Cotets 1 Itroductio 1 2 Fourier Represetatio of Fiite-Duratio Sequeces: The Discrete Fourier Trasform 1 3 The 2-D Discrete Fourier Trasform
More informationGenerating Functions. 1 Operations on generating functions
Geeratig Fuctios The geeratig fuctio for a sequece a 0, a,..., a,... is defied to be the power series fx a x. 0 We say that a 0, a,... is the sequece geerated by fx ad a is the coefficiet of x. Example
More information