CS473-Algorithms I. Lecture 3. Solving Recurrences. Cevdet Aykanat - Bilkent University Computer Engineering Department
|
|
|
- Preston Knight
- 5 years ago
- Views:
Transcription
1 CS473-Algorthms I Lecture 3 Solvg Recurreces Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet
2 Solvg Recurreces The lyss of merge sort Lecture requred us to solve recurrece. Recurreces re lke solvg tegrls, dfferetl equtos, etc. Ler few trcks. Lecture 4 : Applctos of recurreces. Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 2
3 Recurreces Fucto expressed recursvely T T / 2 f = f > Solve for = 2 k Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 3
4 Recurreces Clmed swer: T = lg+ = lg Susttute clmed swer for T the recurrece Note: resultg equtos re true whe = 2 k.e. lg lg Tedous techclty: hve t show T = lg / 2 But, sce T s mootoclly o-decresg fucto of lg 2 2 lg T T lg 2 lg f =2 0 = f =2 k > lg lg Thus, celg dd t mtter much Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 4
5 Recurreces Techclly, should e creful out floors d celgs s the ook But, usully t s oky To gore floor/celg Just solve for exct powers of 2 or Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 5
6 Boudry Codtos Usully ssume T= Θ for smll Does ot usully ffect sol. f polyomlly ouded Exmple: Itl codto ffects sol. Expoetl T=T / 2 2 E.g., If T= c for costt c > 0, the T2 = T 2 =c 2, T4= T2 2 =c 4, T = Θc T 2 T 2 However 2 3 T 3 T 3 Dfferece sol. s more drmtc wth T T Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 6
7 Susttuto Method The most geerl method:. Guess the form of the soluto. 2. Verfy y ducto. 3. Solve for costts. Exmple: T = 4T/2 + [Assume tht T = Θ.] Guess O 3. Prove O d Ω seprtely. Assume tht Tk ck 3 for k <. Prove T c 3 y ducto. Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 7
8 Exmple of Susttuto T = 4T/2 + 4c /2 3 + = c/2 3 + = c 3 c/2 3 - desred resdul c 3 wheever c/2 3 0, for exmple, f c 2 d resdul Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 8
9 Exmple Cotued We must lso hdle the tl codtos, tht s, groud the ducto wth se cses. Bse: T = Θ for ll < 0, where 0 s sutle costt. For < 0, we hve Θ c 3, f we pck c g eough. Ths oud s ot tght! Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 9
10 A Tghter Upper Boud? We shll prove tht T = O 2 Assume tht Tk ck 2 for k < : T = 4T/2 + c 2 + = O 2 Wrog! We must prove the I.H. = c c 2 for o choce of c > 0. Lose! Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 0
11 A Tghter Upper Boud! IDEA: Stregthe the ductve hypothess. Sutrct low-order term. Iductve hypothess: Tk c k 2 c 2 k for k < T = 4T/2 + 4 c /2 2 - c 2 /2 + = c 2-2 c 2 + = c 2 - c 2 c 2 - c 2 - c 2 f c 2 > Pck c g eough to hdle the tl codtos Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet
12 Recurso-Tree Method A recurso tree models the costs tme of recursve executo of lgorthm. The recurso tree method s good for geertg guesses for the susttuto method. The recurso-tree method c e urelle. The recurso-tree method promotes tuto, however. Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 2
13 Exmple of Recurso Tree Solve T = T/4 + T/2 + 2 : Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 3
14 Exmple of Recurso Tree Solve T = T/4 + T/2 + 2 : T Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 4
15 Exmple of Recurso Tree Solve T = T/4 + T/2 + 2 : 2 T/4 T/2 Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 5
16 Exmple of Recurso Tree Solve T = T/4 + T/2 + 2 : 2 /4 2 /2 2 T/6 T/8 T/8 T/4 Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 6
17 Exmple of Recurso Tree Solve T = T/4 + T/2 + 2 : 2 /4 2 /2 2 /6 2 /8 2 /8 2 /4 2 Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 7
18 Exmple of Recurso Tree Solve T = T/4 + T/2 + 2 : 2 2 /4 2 /2 2 /6 2 /8 2 /8 2 /4 2 Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 8
19 Exmple of Recurso Tree Solve T = T/4 + T/2 + 2 : 2 2 /4 2 /2 2 5/6 2 /6 2 /8 2 /8 2 /4 2 Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 9
20 Exmple of Recurso Tree Solve T = T/4 + T/2 + 2 : 2 2 /4 2 /2 2 /6 2 /8 2 /8 2 /4 2 5/6 2 25/256 2 Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 20
21 Exmple of Recurso Tree Solve T = T/4 + T/2 + 2 : 2 2 /4 2 /2 2 /6 2 /8 2 /8 2 /4 2 5/6 2 25/256 2 Totl = 2 + 5/6 + 5/ / = 2 geometrc seres Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 2
22 The Mster Method The mster method pples to recurreces of the form T = T/ + f, where, >, d f s symptotclly postve. Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 22
23 Three Commo Cses Compre f wth :. f = O for some costt ε > 0. f grows polyomly slower th y ε fctor. Soluto: T = Θ. 2. f = Θ lg k for some costt k 0. f d grow t smlr rtes. Soluto: T = Θ lg k+. Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 23
24 Three Commo Cses Compre f wth : 3. f = for some costt ε > 0. f grows polyomlly fster th y ε fctor. d f stsfes the regulrty codto tht f / c f for some costt c < Soluto: T = Θ f. Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 24
25 Exmples Ex: T = 4T/2 + =4, =2 = 2 ; f = CASE : f =O 2- ε for ε= T = Θ 2 Ex: T = 4T/2 + 2 =4, =2 = 2 ; f = 2 CASE 2: f = Θ 2 lg 0, tht s, k=0 T = Θ 2 lg Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 25
26 Exmples Ex: T = 4T/2 + 3 =4, =2 = 2 ; f = 3. CASE 3: f = 2+ ε for ε= d 4 c /2 3 c 3 reg. cod. for c=/2. T = Θ 3 Ex: T = 4T/2 + 2 / lg =4, =2 = 2 ; f = 2 / lg Mster method does ot pply. I prtculr, for every costt ε > 0, we hve ε = lg Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 26
27 Geerl Method Akr-Bzz T Let p e the uque soluto to k k The, the swers re the sme s for the T mster method, ut wth p sted of Akr d Bzz lso prove eve more geerl result. / / p f Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 27
28 Ide of Mster Theorem Recurso tree: f f / f / f / h= f / 2 f / 2 f / 2 f f / 2 f / 2 T #leves = h = = T Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 28
29 Ide of Mster Theorem Recurso tree: f f / f / f / h= f / 2 f / 2 f / 2 f f / 2 f / 2 T CASE : The weght creses geometrclly from the root to the leves. The leves hold costt T frcto of the totl weght. Θ Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 29
30 Ide of Mster Theorem Recurso tree: f f / f / f / h= f / 2 f / 2 f / 2 f f / 2 f / 2 T CASE 2 : k = 0 The weght s pproxmtely the sme o ech of the levels. Θ T lg Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 30
31 Ide of Mster Theorem Recurso tree: f f / f / f / h= f / 2 f / 2 f / 2 f f / 2 f / 2 T CASE 3 : The weght decreses geometrclly from the root to the leves. The root holds costt T frcto of the totl weght. Θ f Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 3
32 Proof of Mster Theorem: Cse d Cse 2 Recll from the recurso tree ote h = lg =tree heght T h f / 0 Lef cost No-lef cost = g Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 32
33 Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 33 Proof of Cse for some > 0 f O f O f f 0 0 / / h h O O g 0 / h O
34 Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 34 = A cresg geometrc seres sce > h h h O h h Cse cot
35 Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 35 O O O O g O g T Cse cot O Q.E.D.
36 Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 36 Proof of Cse 2 lmted to k= h h h f f f 0 / lg 0 / h g lg T lg lg 0 Q.E.D.
37 Cocluso Next tme: pplyg the mster method. Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet 37
Data Structures and Algorithms CMPSC 465
Dt Structures nd Algorithms CMPSC 465 LECTURE 10 Solving recurrences Mster theorem Adm Smith S. Rskhodnikov nd A. Smith; bsed on slides by E. Demine nd C. Leiserson Review questions Guess the solution
Sequences and summations
Lecture 0 Sequeces d summtos Istructor: Kgl Km CSE) E-ml: kkm0@kokuk.c.kr Tel. : 0-0-9 Room : New Mleum Bldg. 0 Lb : New Egeerg Bldg. 0 All sldes re bsed o CS Dscrete Mthemtcs for Computer Scece course
Similar idea to multiplication in N, C. Divide and conquer approach provides unexpected improvements. Naïve matrix multiplication
Next. Covered bsics of simple desig techique (Divided-coquer) Ch. of the text.. Next, Strsse s lgorithm. Lter: more desig d coquer lgorithms: MergeSort. Solvig recurreces d the Mster Theorem. Similr ide
Section IV.6: The Master Method and Applications
Sectio IV.6: The Mster Method d Applictios Defiitio IV.6.1: A fuctio f is symptoticlly positive if d oly if there exists rel umer such tht f(x) > for ll x >. A cosequece of this defiitio is tht fuctio
Algorithm Design and Analysis
Algorithm Design nd Anlysis LECTURE 12 Solving Recurrences Mster Theorem Adm Smith Review Question: Exponentition Problem: Compute b, where b N is n bits long. Question: How mny multiplictions? Nive lgorithm:
MTH 146 Class 7 Notes
7.7- Approxmte Itegrto Motvto: MTH 46 Clss 7 Notes I secto 7.5 we lered tht some defte tegrls, lke x e dx, cot e wrtte terms of elemetry fuctos. So, good questo to sk would e: How c oe clculte somethg
ITERATIVE METHODS FOR SOLVING SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS
Numercl Alyss for Egeers Germ Jord Uversty ITERATIVE METHODS FOR SOLVING SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS Numercl soluto of lrge systems of ler lgerc equtos usg drect methods such s Mtr Iverse, Guss
5. Solving recurrences
5. Solvig recurreces Time Complexity Alysis of Merge Sort T( ) 0 if 1 2T ( / 2) otherwise sortig oth hlves mergig Q. How to prove tht the ru-time of merge sort is O( )? A. 2 Time Complexity Alysis of Merge
Advanced Algorithmic Problem Solving Le 3 Arithmetic. Fredrik Heintz Dept of Computer and Information Science Linköping University
Advced Algorthmc Prolem Solvg Le Arthmetc Fredrk Hetz Dept of Computer d Iformto Scece Lköpg Uversty Overvew Arthmetc Iteger multplcto Krtsu s lgorthm Multplcto of polyomls Fst Fourer Trsform Systems of
Chapter 2 Intro to Math Techniques for Quantum Mechanics
Wter 3 Chem 356: Itroductory Qutum Mechcs Chpter Itro to Mth Techques for Qutum Mechcs... Itro to dfferetl equtos... Boudry Codtos... 5 Prtl dfferetl equtos d seprto of vrbles... 5 Itroducto to Sttstcs...
Lecture 4 Recursive Algorithm Analysis. Merge Sort Solving Recurrences The Master Theorem
Lecture 4 Recursive Algorithm Alysis Merge Sort Solvig Recurreces The Mster Theorem Merge Sort MergeSortA, left, right) { if left < right) { mid floorleft right) / 2); MergeSortA, left, mid); MergeSortA,
ME 501A Seminar in Engineering Analysis Page 1
Mtr Trsformtos usg Egevectors September 8, Mtr Trsformtos Usg Egevectors Lrry Cretto Mechcl Egeerg A Semr Egeerg Alyss September 8, Outle Revew lst lecture Trsformtos wth mtr of egevectors: = - A ermt
In Calculus I you learned an approximation method using a Riemann sum. Recall that the Riemann sum is
Mth Sprg 08 L Approxmtg Dete Itegrls I Itroducto We hve studed severl methods tht llow us to d the exct vlues o dete tegrls However, there re some cses whch t s ot possle to evlute dete tegrl exctly I
Lecture 4 Recursive Algorithm Analysis. Merge Sort Solving Recurrences The Master Theorem
Lecture 4 Recursive Algorithm Alysis Merge Sort Solvig Recurreces The Mster Theorem Merge Sort MergeSortA, left, right) { if left < right) { mid = floorleft + right) / 2); MergeSortA, left, mid); MergeSortA,
Chapter 2 Intro to Math Techniques for Quantum Mechanics
Fll 4 Chem 356: Itroductory Qutum Mechcs Chpter Itro to Mth Techques for Qutum Mechcs... Itro to dfferetl equtos... Boudry Codtos... 5 Prtl dfferetl equtos d seprto of vrbles... 5 Itroducto to Sttstcs...
this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]
![this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]](/thumbs/78/76749440.jpg "this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]")
Atervtves The Itegrl Atervtves Ojectve: Use efte tegrl otto for tervtves. Use sc tegrto rules to f tervtves. Aother mportt questo clculus s gve ervtve f the fucto tht t cme from. Ths s the process kow
Available online through
Avlble ole through wwwmfo FIXED POINTS FOR NON-SELF MAPPINGS ON CONEX ECTOR METRIC SPACES Susht Kumr Moht* Deprtmet of Mthemtcs West Begl Stte Uverst Brst 4 PrgsNorth) Kolt 76 West Begl Id E-ml: smwbes@yhoo
POWERS OF COMPLEX PERSYMMETRIC ANTI-TRIDIAGONAL MATRICES WITH CONSTANT ANTI-DIAGONALS
IRRS 9 y 04 wwwrppresscom/volumes/vol9issue/irrs_9 05pdf OWERS OF COLE ERSERIC I-RIIGOL RICES WIH COS I-IGOLS Wg usu * Q e Wg Hbo & ue College of Scece versty of Shgh for Scece d echology Shgh Ch 00093
Area and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]
![Area and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]](/thumbs/83/87614710.jpg "Area and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]")
Are d the Defte Itegrl 1 Are uder Curve We wt to fd the re uder f (x) o [, ] y f (x) x The Prtto We eg y prttog the tervl [, ] to smller su-tervls x 0 x 1 x x - x -1 x 1 The Bsc Ide We the crete rectgles
Roberto s Notes on Integral Calculus Chapter 4: Definite integrals and the FTC Section 2. Riemann sums
Roerto s Notes o Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 2 Rem sums Wht you eed to kow lredy: The defto of re for rectgle. Rememer tht our curret prolem s how to compute the re of ple rego
Optimality of Strategies for Collapsing Expanded Random Variables In a Simple Random Sample Ed Stanek
Optmlt of Strteges for Collpsg Expe Rom Vrles Smple Rom Smple E Stek troucto We revew the propertes of prectors of ler comtos of rom vrles se o rom vrles su-spce of the orgl rom vrles prtculr, we ttempt
Presentation for use with the textbook, Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, Divide-and-Conquer
Presettio for use with the textook, Algorithm Desig d Applictios, y M. T. Goodrich d R. Tmssi, Wiley, 25 Divide-d-Coquer Divide-d-Coquer Divide-d coquer is geerl lgorithm desig prdigm: Divide: divide the
Rendering Equation. Linear equation Spatial homogeneous Both ray tracing and radiosity can be considered special case of this general eq.
Rederg quto Ler equto Sptl homogeeous oth ry trcg d rdosty c be cosdered specl cse of ths geerl eq. Relty ctul photogrph Rdosty Mus Rdosty Rederg quls the dfferece or error mge http://www.grphcs.corell.edu/ole/box/compre.html
Data Structures and Algorithm. Xiaoqing Zheng
Data Structures ad Algorithm Xiaoqig Zheg zhegxq@fudaeduc What are algorithms? A sequece of computatioal steps that trasform the iput ito the output Sortig problem: Iput: A sequece of umbers
12 Iterative Methods. Linear Systems: Gauss-Seidel Nonlinear Systems Case Study: Chemical Reactions
HK Km Slghtly moded //9 /8/6 Frstly wrtte t Mrch 5 Itertve Methods er Systems: Guss-Sedel Noler Systems Cse Study: Chemcl Rectos Itertve or ppromte methods or systems o equtos cosst o guessg vlue d the
CURVE FITTING LEAST SQUARES METHOD
Nuercl Alss for Egeers Ger Jord Uverst CURVE FITTING Although, the for of fucto represetg phscl sste s kow, the fucto tself ot be kow. Therefore, t s frequetl desred to ft curve to set of dt pots the ssued
i+1 by A and imposes Ax
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 09.9 NUMERICAL FLUID MECHANICS FALL 009 Mody, October 9, 009 QUIZ : SOLUTIONS Notes: ) Multple solutos
ICS141: Discrete Mathematics for Computer Science I
Uversty o Hw ICS: Dscrete Mthemtcs or Computer Scece I Dept. Iormto & Computer Sc., Uversty o Hw J Stelovsy bsed o sldes by Dr. Be d Dr. Stll Orgls by Dr. M. P. Fr d Dr. J.L. Gross Provded by McGrw-Hll
Asymptotic Dominance Problems. is not constant but for n 0, f ( n) 11. 0, so that for n N f
Asymptotc Domce Prolems Dsply ucto : N R tht s Ο( ) ut s ot costt 0 = 0 The ucto ( ) = > 0 s ot costt ut or 0, ( ) Dee the relto " " o uctos rom N to R y g d oly = Ο( g) Prove tht s relexve d trstve (Recll:
Merge Sort. Outline and Reading. Divide-and-Conquer. Divide-and-conquer paradigm ( 4.1.1) Merge-sort ( 4.1.1)
Merge Sort 7 2 9 4 2 4 7 9 7 2 2 7 9 4 4 9 7 7 2 2 9 9 4 4 Merge Sort versio 1.3 1 Outlie d Redig Divide-d-coquer prdigm ( 4.1.1 Merge-sort ( 4.1.1 Algorithm Mergig two sorted sequeces Merge-sort tree
6. Chemical Potential and the Grand Partition Function
6. Chemcl Potetl d the Grd Prtto Fucto ome Mth Fcts (see ppedx E for detls) If F() s lytc fucto of stte vrles d such tht df d pd the t follows: F F p lso sce F p F we c coclude: p I other words cross dervtves
DATA FITTING. Intensive Computation 2013/2014. Annalisa Massini
DATA FITTING Itesve Computto 3/4 Als Mss Dt fttg Dt fttg cocers the problem of fttg dscrete dt to obt termedte estmtes. There re two geerl pproches two curve fttg: Iterpolto Dt s ver precse. The strteg
Lecture 3-4 Solutions of System of Linear Equations
Lecture - Solutos of System of Ler Equtos Numerc Ler Alger Revew of vectorsd mtrces System of Ler Equtos Guss Elmto (drect solver) LU Decomposto Guss-Sedel method (tertve solver) VECTORS,,, colum vector
Chapter 3 Supplemental Text Material
S3-. The Defto of Fctor Effects Chpter 3 Supplemetl Text Mterl As oted Sectos 3- d 3-3, there re two wys to wrte the model for sglefctor expermet, the mes model d the effects model. We wll geerlly use
The z-transform. LTI System description. Prof. Siripong Potisuk
The -Trsform Prof. Srpog Potsuk LTI System descrpto Prevous bss fucto: ut smple or DT mpulse The put sequece s represeted s ler combto of shfted DT mpulses. The respose s gve by covoluto sum of the put
Preliminary Examinations: Upper V Mathematics Paper 1
relmr Emtos: Upper V Mthemtcs per Jul 03 Emer: G Evs Tme: 3 hrs Modertor: D Grgortos Mrks: 50 INSTRUCTIONS ND INFORMTION Ths questo pper sts of 0 pges, cludg swer Sheet pge 8 d Iformto Sheet pges 9 d 0
Stats & Summary
Stts 443.3 & 85.3 Summr The Woodbur Theorem BCD B C D B D where the verses C C D B, d est. Block Mtrces Let the m mtr m q q m be rttoed to sub-mtrces,,,, Smlrl rtto the m k mtr B B B mk m B B l kl Product
A Brief Introduction to Olympiad Inequalities
Ev Che Aprl 0, 04 The gol of ths documet s to provde eser troducto to olympd equltes th the stdrd exposto Olympd Iequltes, by Thoms Mldorf I ws motvted to wrte t by feelg gulty for gettg free 7 s o problems
Regression. By Jugal Kalita Based on Chapter 17 of Chapra and Canale, Numerical Methods for Engineers
Regresso By Jugl Klt Bsed o Chpter 7 of Chpr d Cle, Numercl Methods for Egeers Regresso Descrbes techques to ft curves (curve fttg) to dscrete dt to obt termedte estmtes. There re two geerl pproches two
On Several Inequalities Deduced Using a Power Series Approach
It J Cotemp Mth Sceces, Vol 8, 203, o 8, 855-864 HIKARI Ltd, wwwm-hrcom http://dxdoorg/02988/jcms2033896 O Severl Iequltes Deduced Usg Power Seres Approch Lored Curdru Deprtmet of Mthemtcs Poltehc Uversty
Chapter 7. Bounds for weighted sums of Random Variables
Chpter 7. Bouds for weghted sums of Rdom Vrbles 7. Itroducto Let d 2 be two depedet rdom vrbles hvg commo dstrbuto fucto. Htczeko (998 d Hu d L (2000 vestgted the Rylegh dstrbuto d obted some results bout
Numerical Solution of Second order Integro- Differential Equations(Ides) with Different Four Polynomials Bases Functions
Numercl Soluto of Secod order Itegro- Dfferetl Equtos(Ides) wth Dfferet Four Polyomls Bses Fuctos wo O A R M Deprtmet of Mthemtcs Uersty of Ilor Deprtmet of Mthemtcs d Sttstcs he Poly Id Astrct: - I ths
Analytical Approach for the Solution of Thermodynamic Identities with Relativistic General Equation of State in a Mixture of Gases
Itertol Jourl of Advced Reserch Physcl Scece (IJARPS) Volume, Issue 5, September 204, PP 6-0 ISSN 2349-7874 (Prt) & ISSN 2349-7882 (Ole) www.rcourls.org Alytcl Approch for the Soluto of Thermodymc Idettes
Data Structures and Algorithms CMPSC 465
Data Structures and Algorithms CMPSC 465 LECTURE 9 Solving recurrences Substitution method Adam Smith S. Raskhodnikova and A. Smith; based on slides by E. Demaine and C. Leiserson Review question Draw
COMPLEX NUMBERS AND DE MOIVRE S THEOREM
COMPLEX NUMBERS AND DE MOIVRE S THEOREM OBJECTIVE PROBLEMS. s equl to b d. 9 9 b 9 9 d. The mgr prt of s 5 5 b 5. If m, the the lest tegrl vlue of m s b 8 5. The vlue of 5... s f s eve, f s odd b f s eve,
Patterns of Continued Fractions with a Positive Integer as a Gap
IOSR Jourl of Mthemtcs (IOSR-JM) e-issn: 78-578, -ISSN: 39-765X Volume, Issue 3 Ver III (My - Ju 6), PP -5 wwwosrjourlsorg Ptters of Cotued Frctos wth Postve Iteger s G A Gm, S Krth (Mthemtcs, Govermet
Differential Method of Thin Layer for Retaining Wall Active Earth Pressure and Its Distribution under Seismic Condition Li-Min XU, Yong SUN
Itertol Coferece o Mechcs d Cvl Egeerg (ICMCE 014) Dfferetl Method of Th Lyer for Retg Wll Actve Erth Pressure d Its Dstrbuto uder Sesmc Codto L-M XU, Yog SUN Key Lbortory of Krst Evromet d Geologcl Hzrd
Soo King Lim Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Figure 9: Figure 10: Figure 11:
Soo Kg Lm 1.0 Nested Fctorl Desg... 1.1 Two-Fctor Nested Desg... 1.1.1 Alss of Vrce... Exmple 1... 5 1.1. Stggered Nested Desg for Equlzg Degree of Freedom... 7 1.1. Three-Fctor Nested Desg... 8 1.1..1
Linear Open Loop Systems
Colordo School of Me CHEN43 Trfer Fucto Ler Ope Loop Sytem Ler Ope Loop Sytem... Trfer Fucto for Smple Proce... Exmple Trfer Fucto Mercury Thermometer... 2 Derblty of Devto Vrble... 3 Trfer Fucto for Proce
SUM PROPERTIES FOR THE K-LUCAS NUMBERS WITH ARITHMETIC INDEXES
Avlble ole t http://sc.org J. Mth. Comput. Sc. 4 (04) No. 05-7 ISSN: 97-507 SUM PROPERTIES OR THE K-UCAS NUMBERS WITH ARITHMETIC INDEXES BIJENDRA SINGH POOJA BHADOURIA AND OMPRAKASH SIKHWA * School of
Answer: First, I ll show how to find the terms analytically then I ll show how to use the TI to find them.
. CHAPTER 0 SEQUENCE, SERIES, d INDUCTION Secto 0. Seqece A lst of mers specfc order. E / Fd the frst terms : of the gve seqece: Aswer: Frst, I ll show how to fd the terms ltcll the I ll show how to se
Definite Integral. The Left and Right Sums
Clculus Li Vs Defiite Itegrl. The Left d Right Sums The defiite itegrl rises from the questio of fidig the re betwee give curve d x-xis o itervl. The re uder curve c be esily clculted if the curve is give
On a class of analytic functions defined by Ruscheweyh derivative
Lfe Scece Jourl ;9( http://wwwlfescecestecom O clss of lytc fuctos defed by Ruscheweyh dervtve S N Ml M Arf K I Noor 3 d M Rz Deprtmet of Mthemtcs GC Uversty Fslbd Pujb Pst Deprtmet of Mthemtcs Abdul Wl
14.2 Line Integrals. determines a partition P of the curve by points Pi ( xi, y
4. Le Itegrls I ths secto we defe tegrl tht s smlr to sgle tegrl except tht sted of tegrtg over tervl [ ] we tegrte over curve. Such tegrls re clled le tegrls lthough curve tegrls would e etter termology.
The limit comparison test
Roerto s Notes o Ifiite Series Chpter : Covergece tests Sectio 4 The limit compriso test Wht you eed to kow lredy: Bsics of series d direct compriso test. Wht you c ler here: Aother compriso test tht does
UNIT I. Definition and existence of Riemann-Stieltjes
1 UNIT I Defto d exstece of Rem-Steltjes Itroducto: The reder wll recll from elemetry clculus tht to fd the re of the rego uder the grph of postve fucto f defed o [, ], we sudvde the tervl [, ] to fte
4 Linear Homogeneous Recurrence Relation 4-1 Fibonacci Rabbits. 组合数学 Combinatorics
4 Ler Homogeeous Recurrece Relto 4- bocc Rbbts 组合数学 ombtorcs The delt of the th moth d - th moth s gve brth by the rbbts - moth. o = - + - Moth Moth Moth Moth 4 I the frst moth there s pr of ewly-bor rbbts;
CS583 Lecture 02. Jana Kosecka. some materials here are based on E. Demaine, D. Luebke slides
CS583 Lecture 02 Jaa Kosecka some materials here are based o E. Demaie, D. Luebke slides Previously Sample algorithms Exact ruig time, pseudo-code Approximate ruig time Worst case aalysis Best case aalysis
Introduction to Group Theory
Introduction to Group Theory Let G be n rbitrry set of elements, typiclly denoted s, b, c,, tht is, let G = {, b, c, }. A binry opertion in G is rule tht ssocites with ech ordered pir (,b) of elements
Section 6.3: Geometric Sequences
40 Chpter 6 Sectio 6.: Geometric Sequeces My jobs offer ul cost-of-livig icrese to keep slries cosistet with ifltio. Suppose, for exmple, recet college grdute fids positio s sles mger erig ul slry of $6,000.
The Basic Properties of the Integral
The Bsic Properties of the Itegrl Whe we compute the derivtive of complicted fuctio, like x + six, we usully use differetitio rules, like d [f(x)+g(x)] d f(x)+ d g(x), to reduce the computtio dx dx dx
Algorithms and Data Structures Lecture IV
Algorithms ad Data Structures Lecture IV Simoas Šalteis Aalborg Uiversity simas@cs.auc.dk September 5, 00 1 This Lecture Aalyzig the ruig time of recursive algorithms (such as divide-ad-coquer) Writig
Chapter Gauss-Seidel Method
Chpter 04.08 Guss-Sedel Method After redg ths hpter, you should be ble to:. solve set of equtos usg the Guss-Sedel method,. reogze the dvtges d ptflls of the Guss-Sedel method, d. determe uder wht odtos
Approximate Integration
Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio:
Review of Sections
Review of Sectios.-.6 Mrch 24, 204 Abstrct This is the set of otes tht reviews the mi ides from Chpter coverig sequeces d series. The specific sectios tht we covered re s follows:.: Sequces..2: Series,
St John s College. UPPER V Mathematics: Paper 1 Learning Outcome 1 and 2. Examiner: GE Marks: 150 Moderator: BT / SLS INSTRUCTIONS AND INFORMATION
St Joh s College UPPER V Mthemtcs: Pper Lerg Outcome d ugust 00 Tme: 3 hours Emer: GE Mrks: 50 Modertor: BT / SLS INSTRUCTIONS ND INFORMTION Red the followg structos crefull. Ths questo pper cossts of
On Solution of Min-Max Composition Fuzzy Relational Equation
U-Sl Scece Jourl Vol.4()7 O Soluto of M-Mx Coposto Fuzzy eltol Equto N.M. N* Dte of cceptce /5/7 Abstrct I ths pper, M-Mx coposto fuzzy relto equto re studed. hs study s geerlzto of the works of Ohsto
This Lecture. Divide and Conquer. Merge Sort: Algorithm. Merge Sort Algorithm. MergeSort (Example) - 1. MergeSort (Example) - 2
This Lecture Divide-ad-coquer techique for algorithm desig. Example the merge sort. Writig ad solvig recurreces Divide ad Coquer Divide-ad-coquer method for algorithm desig: Divide: If the iput size is
under the curve in the first quadrant.
NOTES 5: INTEGRALS Nme: Dte: Perod: LESSON 5. AREAS AND DISTANCES Are uder the curve Are uder f( ), ove the -s, o the dom., Prctce Prolems:. f ( ). Fd the re uder the fucto, ove the - s, etwee,.. f ( )
Sect Simplifying Radical Expressions. We can use our properties of exponents to establish two properties of radicals: and
128 Sect 10.3 - Simplifyig Rdicl Expressios Cocept #1 Multiplictio d Divisio Properties of Rdicls We c use our properties of expoets to estlish two properties of rdicls: () 1/ 1/ 1/ & ( Multiplictio d
MATLAB PROGRAM FOR THE NUMERICAL SOLUTION OF DUHAMEL CONVOLUTION INTEGRAL
Bullet of te Trslv Uversty of Brşov Vol 5 (54-1 Seres 1: Specl Issue No 1 MATLAB PROGRAM FOR THE NUMERICAL SOLUTION OF DUHAMEL CONVOLUTION INTEGRAL M BOTIŞ 1 Astrct: I te ler lyss of structures troug modl
CS 301. Lecture 04 Regular Expressions. Stephen Checkoway. January 29, 2018
CS 301 Lecture 04 Regulr Expressions Stephen Checkowy Jnury 29, 2018 1 / 35 Review from lst time NFA N = (Q, Σ, δ, q 0, F ) where δ Q Σ P (Q) mps stte nd n lphet symol (or ) to set of sttes We run n NFA
Math 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
Unit 1. Extending the Number System. 2 Jordan School District
Uit Etedig the Number System Jord School District Uit Cluster (N.RN. & N.RN.): Etedig Properties of Epoets Cluster : Etedig properties of epoets.. Defie rtiol epoets d eted the properties of iteger epoets
Solutions Manual for Polymer Science and Technology Third Edition
Solutos ul for Polymer Scece d Techology Thrd Edto Joel R. Fred Uer Sddle Rver, NJ Bosto Idols S Frcsco New York Toroto otrel Lodo uch Prs drd Cetow Sydey Tokyo Sgore exco Cty Ths text s ssocted wth Fred/Polymer
The Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
PATTERNS IN CONTINUED FRACTION EXPANSIONS
PATTERNS IN CONTINUED FRACTION EXPANSIONS A THESIS SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY SAMUEL WAYNE JUDNICK IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
13: Diffusion in 2 Energy Groups
3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September Contents We study the diffusion eqution in two energy groups
( a n ) converges or diverges.
Chpter Ifiite Series Pge of Sectio E Rtio Test Chpter : Ifiite Series By the ed of this sectio you will be ble to uderstd the proof of the rtio test test series for covergece by pplyig the rtio test pprecite
Level-2 BLAS. Matrix-Vector operations with O(n 2 ) operations (sequentially) BLAS-Notation: S --- single precision G E general matrix M V --- vector
evel-2 BS trx-vector opertos wth 2 opertos sequetlly BS-Notto: S --- sgle precso G E geerl mtrx V --- vector defes SGEV, mtrx-vector product: r y r α x β r y ther evel-2 BS: Solvg trgulr system x wth trgulr
SOLVING SYSTEMS OF EQUATIONS, DIRECT METHODS
ELM Numecl Alyss D Muhem Mecmek SOLVING SYSTEMS OF EQUATIONS DIRECT METHODS ELM Numecl Alyss Some of the cotets e dopted fom Luee V. Fusett Appled Numecl Alyss usg MATLAB. Petce Hll Ic. 999 ELM Numecl
A Technique for Constructing Odd-order Magic Squares Using Basic Latin Squares
Itertol Jourl of Scetfc d Reserch Publctos, Volume, Issue, My 0 ISSN 0- A Techque for Costructg Odd-order Mgc Squres Usg Bsc Lt Squres Tomb I. Deprtmet of Mthemtcs, Mpur Uversty, Imphl, Mpur (INDIA) tombrom@gml.com
f(bx) dx = f dx = dx l dx f(0) log b x a + l log b a 2ɛ log b a.
Eercise 5 For y < A < B, we hve B A f fb B d = = A B A f d f d For y ɛ >, there re N > δ >, such tht d The for y < A < δ d B > N, we hve ba f d f A bb f d l By ba A A B A bb ba fb d f d = ba < m{, b}δ
1 Error Analysis of Simple Rules for Numerical Integration
cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion
Review of the Riemann Integral
Chpter 1 Review of the Riem Itegrl This chpter provides quick review of the bsic properties of the Riem itegrl. 1.0 Itegrls d Riem Sums Defiitio 1.0.1. Let [, b] be fiite, closed itervl. A prtitio P of
POWER SERIES R. E. SHOWALTER
POWER SERIES R. E. SHOWALTER. sequeces We deote by lim = tht the limit of the sequece { } is the umber. By this we me tht for y ε > 0 there is iteger N such tht < ε for ll itegers N. This mkes precise
MATH2999 Directed Studies in Mathematics Matrix Theory and Its Applications
MATH999 Drected Studes Mthemtcs Mtr Theory d Its Applctos Reserch Topc Sttory Probblty Vector of Hgher-order Mrkov Ch By Zhg Sho Supervsors: Prof. L Ch-Kwog d Dr. Ch Jor-Tg Cotets Abstrct. Itroducto: Bckgroud.
Introduction to mathematical Statistics
Itroducto to mthemtcl ttstcs Fl oluto. A grou of bbes ll of whom weghed romtely the sme t brth re rdomly dvded to two grous. The bbes smle were fed formul A; those smle were fed formul B. The weght gs
Chapter 7 Infinite Series
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2
Math 2414 Activity 16 (Due by end of class August 13) 1. Let f be a positive, continuous, decreasing function for x 1, and suppose that
Mth Actvty 6 (Due y ed of clss August ). Let f e ostve, cotuous, decresg fucto for x, d suose tht f. If the seres coverges to s, d we cll the th rtl sum of the seres the the remder doule equlty r 0 s,
Itō Calculus (An Abridged Overview)
Itō Clculus (A Abrdged Overvew) Arturo Ferdez Uversty of Clfor, Berkeley Sttstcs 157: Topcs I Stochstc Processes Semr Aprl 14, 211 1 Itroducto I my prevous set of otes, I troduced the cocept of Stochstc
Connected-components. Summary of lecture 9. Algorithms and Data Structures Disjoint sets. Example: connected components in graphs
Prm University, Mth. Deprtment Summry of lecture 9 Algorithms nd Dt Structures Disjoint sets Summry of this lecture: (CLR.1-3) Dt Structures for Disjoint sets: Union opertion Find opertion Mrco Pellegrini
Lecture 20: Numerical Integration III
cs4: introduction to numericl nlysis /8/0 Lecture 0: Numericl Integrtion III Instructor: Professor Amos Ron Scribes: Mrk Cowlishw, Yunpeng Li, Nthnel Fillmore For the lst few lectures we hve discussed
10.2 Series. , we get. which is called an infinite series ( or just a series) and is denoted, for short, by the symbol. i i n
0. Sere I th ecto, we wll troduce ere tht wll be dcug for the ret of th chpter. Wht ere? If we dd ll term of equece, we get whch clled fte ere ( or jut ere) d deoted, for hort, by the ymbol or Doe t mke
Strategies for the AP Calculus Exam
Strteges for the AP Clculus Em Strteges for the AP Clculus Em Strtegy : Kow Your Stuff Ths my seem ovous ut t ees to e metoe. No mout of cochg wll help you o the em f you o t kow the mterl. Here s lst
Repeated Root and Common Root
Repeted Root d Commo Root 1 (Method 1) Let α, β, γ e the roots of p(x) x + x + 0 (1) The α + β + γ 0, αβ + βγ + γα, αβγ - () (α - β) (α + β) - αβ (α + β) [ (βγ + γα)] + [(α + β) + γ (α + β)] +γ (α + β)
The Selection Problem - Variable Size Decrease/Conquer (Practice with algorithm analysis)
We have covered: Selecto, Iserto, Mergesort, Bubblesort, Heapsort Next: Selecto the Qucksort The Selecto Problem - Varable Sze Decrease/Coquer (Practce wth algorthm aalyss) Cosder the problem of fdg the
Mathematical models for computer systems behaviour
Mthemtcl models for comuter systems ehvour Gols : redct comuter system ehvours - erformces mesuremets, - comrso of systems, - dmesog, Methodology : - modellg evromet (stochstc rocess) - modellg system
CS537. Numerical Analysis
CS57 Numercl Alyss Lecture 4 System of Ler Equtos Professor Ju Zhg Deprtmet of Computer Scece Uversty of Ketucky Legto, KY 456 6 Ferury, 6 System of Ler Equtos where j re coeffcets, re ukows, d re rght
Abstrct Pell equto s mportt reserch object elemetry umber theory of defte equto ts form s smple, but t s rch ture My umber theory problems ce trsforme
The solvblty of egtve Pell equto Jq Wg, Lde C Metor: Xgxue J Bejg Ntol Dy School No66 Yuqu Rod Hd Dst Bejg Ch PC December 8, 013 1 / 4 Pge - 38 Abstrct Pell equto s mportt reserch object elemetry umber