Chapter Gauss-Seidel Method
|
|
- Aubrey Day
- 5 years ago
- Views:
Transcription
1 Chpter 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 the Guss-Sedel method lwys overges. Why do we eed other method to solve set of smulteous ler equtos? I ert ses, suh s whe system of equtos s lrge, tertve methods of solvg equtos re more dvtgeous. Elmto methods, suh s Guss elmto, re proe to lrge roud-off errors for lrge set of equtos. Itertve methods, suh s the Guss-Sedel method, gve the user otrol of the roud-off error. Also, f the physs of the problem re well kow, tl guesses eeded tertve methods be mde more udously ledg to fster overgee. Wht s the lgorthm for the Guss-Sedel method? Gve geerl set of equtos d ukows, we hve If the dgol elemets re o-zero, eh equto s rewrtte for the orrespodg ukow, tht s, the frst equto s rewrtte wth o the left hd sde, the seod equto s rewrtte wth o the left hd sde d so o s follows
2 Chpter 04.08,,,,,, These equtos be rewrtte summto form s...,, Hee for y row,.,,,, Now to fd s, oe ssumes tl guess for the s d the uses the rewrtte equtos to lulte the ew estmtes. Remember, oe lwys uses the most reet estmtes to lulte the et estmtes,. At the ed of eh terto, oe lultes the bsolute reltve ppromte error for eh s 00 ew old ew where ew s the reetly obted vlue of, d old s the prevous vlue of.
3 Guss-Sedel Method Whe the bsolute reltve ppromte error for eh s less th the pre-spefed tolere, the tertos re stopped. Emple The upwrd veloty of roket s gve t three dfferet tmes the followg tble Tble Veloty vs. tme dt. Tme, t (s) Veloty, v (m/s) The veloty dt s ppromted by polyoml s v t t t, 5 t Fd the vlues of,, d usg the Guss-Sedel method. Assume tl guess of the soluto s 5 d odut two tertos. Soluto The polyoml s gog through three dt pots, v, t, v, d t v bove tble t 5, v 06.8 t 8, v 77. t, v 79. Requrg tht v t t t v t t v t t v t t vt vt vt Substtutg the dt, v, t, v, d t v or, t where from the, psses through the three dt pots gves t gves
4 Chpter The oeffets,, d for the bove epresso re gve by Rewrtg the equtos gves Iterto # Gve the tl guess of the soluto vetor s 5 we get () (5) The bsolute reltve ppromte error for eh the s % % % At the ed of the frst terto, the estmte of the soluto vetor s
5 Guss-Sedel Method d the mmum bsolute reltve ppromte error s 5.47%. Iterto # The estmte of the soluto vetor t the ed of Iterto # s Now we get ( 55.6) ( 55.6) = The bsolute reltve ppromte error for eh the s % % % At the ed of the seod terto the estmte of the soluto vetor s d the mmum bsolute reltve ppromte error s %. Codutg more tertos gves the followg vlues for the soluto vetor d the orrespodg bsolute reltve ppromte errors.
6 Chpter Iterto % % % As see the bove tble, the soluto estmtes re ot overgg to the true soluto of The bove system of equtos does ot seem to overge. Why? Well, ptfll of most tertve methods s tht they my or my ot overge. However, the soluto to ert lsses of systems of smulteous equtos does lwys overge usg the Guss-Sedel method. Ths lss of system of equtos s where the oeffet mtr [A] [ A][ X ] [ C] s dgolly domt, tht s for ll for t lest oe If system of equtos hs oeffet mtr tht s ot dgolly domt, t my or my ot overge. Fortutely, my physl systems tht result smulteous ler equtos hve dgolly domt oeffet mtr, whh the ssures overgee for tertve methods suh s the Guss-Sedel method of solvg smulteous ler equtos. Emple Fd the soluto to the followg system of equtos usg the Guss-Sedel method Use 0 s the tl guess d odut two tertos. Soluto The oeffet mtr
7 Guss-Sedel Method A 5 7 s dgolly domt s d the equlty s strtly greter th for t lest oe row. Hee, the soluto should overge usg the Guss-Sedel method. Rewrtg the equtos, we get Assumg tl guess of 0 Iterto # The bsolute reltve ppromte error t the ed of the frst terto s % % %
8 Chpter The mmum bsolute reltve ppromte error s 00.00% Iterto # At the ed of seod terto, the bsolute reltve ppromte error s % % % The mmum bsolute reltve ppromte error s 40.6%. Ths s greter th the vlue of 00.00% we obted the frst terto. Is the soluto dvergg? No, s you odut more tertos, the soluto overges s follows. Iterto % Ths s lose to the et soluto vetor of 4 Emple Gve the system of equtos 7 76 % %
9 Guss-Sedel Method fd the soluto usg the Guss-Sedel method. Use 0 s the tl guess. Soluto Rewrtg the equtos, we get Assumg tl guess of 0 the et s tertve vlues re gve the tble below. Iterto % % % You see tht ths soluto s ot overgg d the oeffet mtr s ot dgolly domt. The oeffet mtr 7 A 5 5 s ot dgolly domt s 7 0 Hee, the Guss-Sedel method my or my ot overge. However, t s the sme set of equtos s the prevous emple d tht overged. The oly dfferee s tht we ehged frst d the thrd equto wth eh other d tht mde the oeffet mtr ot dgolly domt.
10 Chpter Therefore, t s possble tht system of equtos be mde dgolly domt f oe ehges the equtos wth eh other. However, t s ot possble for ll ses. For emple, the followg set of equtos ot be rewrtte to mke the oeffet mtr dgolly domt. Key Terms: Guss-Sedel method Covergee of Guss-Sedel method Dgolly domt mtr
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
More information12 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
More informationThe linear system. The problem: solve
The ler syste The prole: solve Suppose A s vertle, the there ests uue soluto How to effetly opute the soluto uerlly??? A A A evew of dret ethods Guss elto wth pvotg Meory ost: O^ Coputtol ost: O^ C oly
More informationNumerical Analysis Topic 4: Least Squares Curve Fitting
Numerl Alss Top 4: Lest Squres Curve Fttg Red Chpter 7 of the tetook Alss_Numerk Motvto Gve set of epermetl dt: 3 5. 5.9 6.3 The reltoshp etwee d m ot e ler. Fd futo f tht est ft the dt 3 Alss_Numerk Motvto
More informationLecture 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
More information3/20/2013. Splines There are cases where polynomial interpolation is bad overshoot oscillations. Examplef x. Interpolation at -4,-3,-2,-1,0,1,2,3,4
// Sples There re ses where polyoml terpolto s d overshoot oslltos Emple l s Iterpolto t -,-,-,-,,,,,.... - - - Ide ehd sples use lower order polyomls to oet susets o dt pots mke oetos etwee djet sples
More informationChapter Simpson s 1/3 Rule of Integration. ( x)
Cpter 7. Smpso s / Rule o Itegrto Ater redg ts pter, you sould e le to. derve te ormul or Smpso s / rule o tegrto,. use Smpso s / rule t to solve tegrls,. develop te ormul or multple-segmet Smpso s / rule
More informationAdvanced 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
More informationChapter Unary Matrix Operations
Chpter 04.04 Ury trx Opertos After redg ths chpter, you should be ble to:. kow wht ury opertos mes,. fd the trspose of squre mtrx d t s reltoshp to symmetrc mtrces,. fd the trce of mtrx, d 4. fd the ermt
More informationNumerical Differentiation and Integration
Numerl Deretto d Itegrto Overvew Numerl Deretto Newto-Cotes Itegrto Formuls Trpezodl rule Smpso s Rules Guss Qudrture Cheyshev s ormul Numerl Deretto Forwrd te dvded deree Bkwrd te dvded deree Ceter te
More informationDATA 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
More informationChapter Newton-Raphson Method of Solving a Nonlinear Equation
Chpter.4 Newton-Rphson Method of Solvng Nonlner Equton After redng ths chpter, you should be ble to:. derve the Newton-Rphson method formul,. develop the lgorthm of the Newton-Rphson method,. use the Newton-Rphson
More informationChapter Linear Regression
Chpte 6.3 Le Regesso Afte edg ths chpte, ou should be ble to. defe egesso,. use sevel mmzg of esdul cte to choose the ght cteo, 3. deve the costts of le egesso model bsed o lest sques method cteo,. use
More information1 4 6 is symmetric 3 SPECIAL MATRICES 3.1 SYMMETRIC MATRICES. Defn: A matrix A is symmetric if and only if A = A, i.e., a ij =a ji i, j. Example 3.1.
SPECIAL MATRICES SYMMETRIC MATRICES Def: A mtr A s symmetr f d oly f A A, e,, Emple A s symmetr Def: A mtr A s skew symmetr f d oly f A A, e,, Emple A s skew symmetr Remrks: If A s symmetr or skew symmetr,
More informationMathematically, integration is just finding the area under a curve from one point to another. It is b
Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] CHAPTER VI Numerl Itegrto Tops - Rem sums - Trpezodl rule - Smpso s rule - Rrdso s etrpolto - Guss qudrture rule Mtemtlly, tegrto s just dg te re uder urve rom
More informationPubH 7405: REGRESSION ANALYSIS REGRESSION IN MATRIX TERMS
PubH 745: REGRESSION ANALSIS REGRESSION IN MATRIX TERMS A mtr s dspl of umbers or umercl quttes ld out rectgulr rr of rows d colums. The rr, or two-w tble of umbers, could be rectgulr or squre could be
More informationCS537. 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
More informationThe 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
More information2006 Jamie Trahan, Autar Kaw, Kevin Martin University of South Florida United States of America
SOLUTION OF SYSTEMS OF SIMULTANEOUS LINEAR EQUATIONS Gauss-Sedel Method 006 Jame Traha, Autar Kaw, Kev Mart Uversty of South Florda Uted States of Amerca kaw@eg.usf.edu Itroducto Ths worksheet demostrates
More informationChapter Newton-Raphson Method of Solving a Nonlinear Equation
Chpter 0.04 Newton-Rphson Method o Solvng Nonlner Equton Ater redng ths chpter, you should be ble to:. derve the Newton-Rphson method ormul,. develop the lgorthm o the Newton-Rphson method,. use the Newton-Rphson
More informationCS321. Introduction to Numerical Methods
CS Itroducto to Numercl Metods Lecture Revew Proessor Ju Zg Deprtmet o Computer Scece Uversty o Ketucky Legto, KY 6 6 Mrc 7, Number Coverso A geerl umber sould be coverted teger prt d rctol prt seprtely
More informationCURVE 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
More informationMatrix. Definition 1... a1 ... (i) where a. are real numbers. for i 1, 2,, m and j = 1, 2,, n (iii) A is called a square matrix if m n.
Mtrx Defto () s lled order of m mtrx, umer of rows ( 橫行 ) umer of olums ( 直列 ) m m m where j re rel umers () B j j for,,, m d j =,,, () s lled squre mtrx f m (v) s lled zero mtrx f (v) s lled detty mtrx
More informationChapter 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...
More informationunder 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 ( )
More informationINTRODUCTION ( ) 1. Errors
INTRODUCTION Numercl lyss volves the study, developmet d lyss of lgorthms for obtg umercl solutos to vrous mthemtcl problems. Frequetly umercl lyss s clled the mthemtcs of scetfc computg. Numercl lyss
More informationIn 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
More informationAsynchronous Analogs of Iterative Methods and Their Efficiency for Parallel Computers
Itertol Coferece "Prllel d Dstrbuted Computg Systems" PDCS 23 (Ure, Khrv, Mrch 3-4, 23) sychroous logs of Itertve Methods d her Effcecy for Prllel Computers Msym Svcheo, leder Chemers Isttute for Modelg
More informationAutar Kaw Benjamin Rigsby. Transforming Numerical Methods Education for STEM Undergraduates
Autr Kw Bejmi Rigsby http://m.mthforcollege.com Trsformig Numericl Methods Eductio for STEM Udergrdutes http://m.mthforcollege.com . solve set of simulteous lier equtios usig Nïve Guss elimitio,. ler the
More informationSequences 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
More information10.2 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
More informationME 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
More informationMTH 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
More informationChapter 2. LOGARITHMS
Chpter. LOGARITHMS Dte: - 009 A. INTRODUCTION At the lst hpter, you hve studied bout Idies d Surds. Now you re omig to Logrithms. Logrithm is ivers of idies form. So Logrithms, Idies, d Surds hve strog
More informationMath 1313 Final Exam Review
Mth 33 Fl m Revew. The e Compy stlled ew mhe oe of ts ftores t ost of $0,000. The mhe s depreted lerly over 0 yers wth srp vlue of $,000. Fd the vlue of the mhe fter 5 yers.. mufturer hs mothly fed ost
More informationChapter 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
More informationComputer Programming
Computer Progrmmg I progrmmg, t s ot eough to be vetve d geous. Oe lso eeds to be dscpled d cotrolled order ot be become etgled oe's ow completes. Hrl D. Mlls, Forwrd to Progrmmg Proverbs b Her F. Ledgrd
More informationLevel-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
More informationChapter System of Equations
hpter 4.5 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vice-vers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee
More informationChapter 2 Solving Linear Equation
EE7 Computer odelg Techques Egeerg Chpter Solvg er Equto A ler equto represets the ler depedece of qutty φ o set of vrbles through d set of costt coeffcets α through α ; ts form s α α... α φ If we replce
More informationRecent Progresses on the Simplex Method
Reet Progresses o the Smple Method www.stford.edu/~yyye K.T. L Professor of Egeerg Stford Uversty d Itertol Ceter of Mgemet See d Egeerg Ng Uversty Outles Ler Progrmmg (LP) d the Smple Method Mrkov Deso
More informationSt 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
More informationCOMPLEX 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,
More informationRendering 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
More informationMATRIX AND VECTOR NORMS
Numercl lyss for Egeers Germ Jord Uversty MTRIX ND VECTOR NORMS vector orm s mesure of the mgtude of vector. Smlrly, mtr orm s mesure of the mgtude of mtr. For sgle comoet etty such s ordry umers, the
More informationChapter 12-b Integral Calculus - Extra
C - Itegrl Clulus Cpter - Itegrl Clulus - Etr Is Newto Toms Smpso BONUS Itroduto to Numerl Itegrto C - Itegrl Clulus Numerl Itegrto Ide s to do tegrl smll prts, lke te wy we preseted tegrto: summto. Numerl
More informationINTERPOLATION(2) ELM1222 Numerical Analysis. ELM1222 Numerical Analysis Dr Muharrem Mercimek
ELM Numerl Alss Dr Murrem Merme INTEROLATION ELM Numerl Alss Some of te otets re dopted from Luree V. Fusett Appled Numerl Alss usg MATLAB. rete Hll I. 999 ELM Numerl Alss Dr Murrem Merme Tod s leture
More informationES240 Solid Mechanics Z. Suo. Principal stress. . Write in the matrix notion, and we have
ES4 Sold Mehs Z Suo Prpl stress Prpl Stress Imge mterl prtle stte o stress The stte o stress s xed, but we represet the mterl prtle my wys by uttg ubes deret orettos For y gve stte o stress, t s lwys possble
More informationModeling uncertainty using probabilities
S 1571 Itroduto to I Leture 23 Modelg uertty usg probbltes Mlos Huskreht mlos@s.ptt.edu 5329 Seott Squre dmstrto Fl exm: Deember 11 2006 12:00-1:50pm 5129 Seott Squre Uertty To mke dgost feree possble
More informationIntroduction to Matrix Algebra
Itrodutio to Mtri Alger George H Olso, Ph D Dotorl Progrm i Edutiol Ledership Applhi Stte Uiversit Septemer Wht is mtri? Dimesios d order of mtri A p q dimesioed mtri is p (rows) q (olums) rr of umers,
More informationStats & 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
More informationAnalyzing Control Structures
Aalyzg Cotrol Strutures sequeg P, P : two fragmets of a algo. t, t : the tme they tae the tme requred to ompute P ;P s t t Θmaxt,t For loops for to m do P t: the tme requred to ompute P total tme requred
More informationBond Additive Modeling 5. Mathematical Properties of the Variable Sum Exdeg Index
CROATICA CHEMICA ACTA CCACAA ISSN 00-6 e-issn -7X Crot. Chem. Act 8 () (0) 9 0. CCA-5 Orgl Scetfc Artcle Bod Addtve Modelg 5. Mthemtcl Propertes of the Vrble Sum Edeg Ide Dmr Vukčevć Fculty of Nturl Sceces
More informationGraphing Review Part 3: Polynomials
Grphig Review Prt : Polomils Prbols Recll, tht the grph of f ( ) is prbol. It is eve fuctio, hece it is smmetric bout the bout the -is. This mes tht f ( ) f ( ). Its grph is show below. The poit ( 0,0)
More informationArea 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
More informationi+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
More informationSOLVING 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
More informationPreliminary 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
More informationDensity estimation II
CS 750 Mche Lerg Lecture 6 esty estmto II Mlos Husrecht mlos@tt.edu 539 Seott Squre t: esty estmto {.. } vector of ttrute vlues Ojectve: estmte the model of the uderlyg rolty dstruto over vrles X X usg
More informationGlobal Journal of Engineering and Technology Review
Glol Jourl of Egeerg d eholog Revew Jourl homepge: http://gjetr.org/ Glol Jourl of Egeerg d eholog Revew () 85 9 (06) Applto of Cojugte Grdet Method wth Cu No- Poloml Sple Sheme for Solvg wo-pot Boudr
More informationObjective of curve fitting is to represent a set of discrete data by a function (curve). Consider a set of discrete data as given in table.
CURVE FITTING Obectve curve ttg s t represet set dscrete dt b uct curve. Csder set dscrete dt s gve tble. 3 3 = T use the dt eectvel, curve epress s tted t the gve dt set, s = + = + + = e b ler uct plml
More informationPAIR OF STRAIGHT LINES. will satisfy L1 L2 0, and thus L1 L. 0 represent? It is obvious that any point lying on L 1
LOCUS 33 Seto - 3 PAIR OF STRAIGHT LINES Cosder two les L L Wht do ou thk wll L L represet? It s ovous tht pot lg o L d L wll stsf L L, d thus L L represets the set of pots osttutg oth the les,.e., L L
More informationLinear 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
More information3. REVIEW OF PROPERTIES OF EIGENVALUES AND EIGENVECTORS
. REVIEW OF PROPERTIES OF EIGENVLUES ND EIGENVECTORS. EIGENVLUES ND EIGENVECTORS We hll ow revew ome bc fct from mtr theory. Let be mtr. clr clled egevlue of f there et ozero vector uch tht Emle: Let 9
More information5 - Determinants. r r. r r. r r. r s r = + det det det
5 - Detemts Assote wth y sque mtx A thee s ume lle the etemt of A eote A o et A. Oe wy to efe the etemt, ths futo fom the set of ll mtes to the set of el umes, s y the followg thee popetes. All mtes elow
More informationDiagonally Implicit Runge-Kutta Nystrom General Method Order Five for Solving Second Order IVPs
WSEAS TRANSACTIONS o MATHEMATICS Fudzh Isml Dgoll Implt Ruge-Kutt Nstrom Geerl Method Order Fve for Solvg Seod Order IVPs FUDZIAH ISMAIL Deprtmet of Mthemts Uverst Putr Mls Serdg Selgor MALAYSIA fudzh@mth.upm.edu.m
More information19 22 Evaluate the integral by interpreting it in terms of areas. (1 x ) dx. y Write the given sum or difference as a single integral in
SECTION. THE DEFINITE INTEGRAL. THE DEFINITE INTEGRAL A Clck here for swers. S Clck here for solutos. Use the Mdpot Rule wth the gve vlue of to pproxmte the tegrl. Roud the swer to four decml plces. 9
More information20 23 Evaluate the integral by interpreting it in terms of areas. (1 x ) dx. y 2 2
SECTION 5. THE DEFINITE INTEGRAL 5. THE DEFINITE INTEGRAL A Clck here for swers. S Clck here for solutos. 7 Use the Mdpot Rule wth the gve vlue of to pproxmte the tegrl. Roud the swer to four decml plces.
More informationTaylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best
Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =
More informationNumerical Methods. Lecture 5. Numerical integration. dr hab. inż. Katarzyna Zakrzewska, prof. AGH. Numerical Methods lecture 5 1
Numeril Methods Leture 5. Numeril itegrtio dr h. iż. Ktrzy Zkrzewsk, pro. AGH Numeril Methods leture 5 Outlie Trpezoidl rule Multi-segmet trpezoidl rule Rihrdso etrpoltio Romerg's method Simpso's rule
More informationChapter 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
More informationUNIVERSITI KEBANGSAAN MALAYSIA PEPERIKSAAN AKHIR SEMESTER I SESI AKADEMIK 2007/2008 IJAZAH SARJANAMUDA DENGAN KEPUJIAN NOVEMBER 2007 MASA : 3 JAM
UNIVERSITI KEBANGSAAN MALAYSIA PEPERIKSAAN AKHIR SEMESTER I SESI AKADEMIK 7/8 IJAZAH SARJANAMUDA DENGAN KEPUJIAN NOVEMBER 7 MASA : 3 JAM KOD KURSUS : KKKQ33/KKKF33 TAJUK : PENGIRAAN BERANGKA ARAHAN :.
More informationSPH3UW Unit 7.5 Snell s Law Page 1 of Total Internal Reflection occurs when the incoming refraction angle is
SPH3UW Uit 7.5 Sell s Lw Pge 1 of 7 Notes Physis Tool ox Refrtio is the hge i diretio of wve due to hge i its speed. This is most ommoly see whe wve psses from oe medium to other. Idex of refrtio lso lled
More informationDCDM BUSINESS SCHOOL NUMERICAL METHODS (COS 233-8) Solutions to Assignment 3. x f(x)
DCDM BUSINESS SCHOOL NUMEICAL METHODS (COS -8) Solutons to Assgnment Queston Consder the followng dt: 5 f() 8 7 5 () Set up dfference tble through fourth dfferences. (b) Wht s the mnmum degree tht n nterpoltng
More informationBand structures calculations
lultos reprol spe of -vetors, Brllou zoe seulr equto, vrtol method Ttle pge bd struture, perod potetl desty of sttes, Ferm eergy lmost free eletros method tght bdg method, MO-LCO, Bloh futo Lterture T..
More informationSOME REMARKS ON HORIZONTAL, SLANT, PARABOLIC AND POLYNOMIAL ASYMPTOTE
D I D A C T I C S O F A T H E A T I C S No (4) 3 SOE REARKS ON HORIZONTAL, SLANT, PARABOLIC AND POLYNOIAL ASYPTOTE Tdeusz Jszk Abstct I the techg o clculus, we cosde hozotl d slt symptote I ths ppe the
More informationProblem Set 4 Solutions
4 Eoom Altos of Gme Theory TA: Youg wg /08/0 - Ato se: A A { B, } S Prolem Set 4 Solutos - Tye Se: T { α }, T { β, β} Se Plyer hs o rte formto, we model ths so tht her tye tke oly oe lue Plyer kows tht
More informationThe Algebraic Least Squares Fitting Of Ellipses
IOSR Jourl of Mthets (IOSR-JM) e-issn: 78-578 -ISSN: 39-765 Volue 4 Issue Ver II (Mr - Ar 8) PP 74-83 wwwosrjourlsorg he Algebr Lest Squres Fttg Of Ellses Abdelltf Betteb Dertet of Geerl Studes Jubl Idustrl
More informationCHAPTER 7 SOLVING FUZZY LINEAR FRACTIONAL PROGRAMMING PROBLEM BY USING TAYLOR'S SERIES METHOD
67 CHAPTER 7 SOLVING FUZZY LINEAR FRACTIONAL PROGRAMMING PROBLEM BY USING TAYLOR'S SERIES METHOD 7. INTRODUCTION The eso mers the setors le fl ororte lg routo lg mretg me seleto uversty lg stuet mssos
More informationCHAPTER 5 Vectors and Vector Space
HAPTE 5 Vetors d Vetor Spe 5. Alger d eometry of Vetors. Vetor A ordered trple,,, where,, re rel umers. Symol:, B,, A mgtude d dreto.. Norm of vetor,, Norm =,, = = mgtude. Slr multplto Produt of slr d
More informationChapter 17. Least Square Regression
The Islmc Uvest of Gz Fcult of Egeeg Cvl Egeeg Deptmet Numecl Alss ECIV 336 Chpte 7 Lest que Regesso Assocte Pof. Mze Abultef Cvl Egeeg Deptmet, The Islmc Uvest of Gz Pt 5 - CURVE FITTING Descbes techques
More informationA 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
More informationDescribes techniques to fit curves (curve fitting) to discrete data to obtain intermediate estimates.
CURVE FITTING Descbes techques to ft cuves (cuve fttg) to dscete dt to obt temedte estmtes. Thee e two geel ppoches fo cuve fttg: Regesso: Dt ehbt sgfct degee of sctte. The stteg s to deve sgle cuve tht
More information4 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;
More informationContent: Essential Calculus, Early Transcendentals, James Stewart, 2007 Chapter 1: Functions and Limits., in a set B.
Review Sheet: Chpter Cotet: Essetil Clculus, Erly Trscedetls, Jmes Stewrt, 007 Chpter : Fuctios d Limits Cocepts, Defiitios, Lws, Theorems: A fuctio, f, is rule tht ssigs to ech elemet i set A ectly oe
More informationAn Indian Journal FULL PAPER ABSTRACT KEYWORDS. Trade Science Inc. Matrix eigenvalues of jacobi iterative method for solving
[Type tet] [Type tet] [Type tet] ISSN : 974-745 Volume Issue BoTechology 4 Id Jourl FU PPER BTIJ 4 [77-75] tr egevlues of co tertve method for solvg Yuhu Cu Jgguo Qu Qggog College Hee Uted Uversty Tgsh
More informationIntroduction to mathematical Statistics
More ttistics tutoril t wwwdumblittledoctorcom Itroductio to mthemticl ttistics Chpter 7 ypothesis Testig Exmple (ull hypothesis) : the verge height is 5 8 or more ( : μ 5'8" ) (ltertive hypothesis) :
More informationRoberto 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
More informationGENERALIZED OPERATIONAL RELATIONS AND PROPERTIES OF FRACTIONAL HANKEL TRANSFORM
S. Res. Chem. Commu.: (3 8-88 ISSN 77-669 GENERLIZED OPERTIONL RELTIONS ND PROPERTIES OF FRCTIONL NKEL TRNSFORM R. D. TYWDE *. S. GUDDE d V. N. MLLE b Pro. Rm Meghe Isttute o Teholog & Reserh Bder MRVTI
More informationDecompression diagram sampler_src (source files and makefiles) bin (binary files) --- sh (sample shells) --- input (sample input files)
. Iroduco Probblsc oe-moh forecs gudce s mde b 50 esemble members mproved b Model Oupu scs (MO). scl equo s mde b usg hdcs d d observo d. We selec some prmeers for modfg forecs o use mulple regresso formul.
More informationMethods to Invert a Matrix
Lecture 3: Determts & trx Iverso ethods to Ivert trx The pproches vlble to fd the verse of mtrx re extesve d dverse. ll methods seek to solve ler system of equtos tht c be expressed mtrx formt s for the
More informationChapter 1 Vector Spaces
Chpter Vetor pes - Vetor pes Ler Comtos Vetor spe V V s set over fel F f V F! + V. Eg. R s vetor spe. For R we hek -4=-4-4R -7=-7-7R et. Eg. how tht the set of ll polomls PF wth oeffets from F s vetor
More informationThe Z-Transform in DSP Lecture Andreas Spanias
The Z-Trsform DSP eture - Adres Ss ss@su.edu 6 Coyrght 6 Adres Ss -- Poles d Zeros of I geerl the trsfer futo s rtol; t hs umertor d deomtor olyoml. The roots of the umertor d deomtor olyomls re lled the
More informationCurrent Programmed Control (i.e. Peak Current-Mode Control) Lecture slides part 2 More Accurate Models
Curret Progred Cotrol.e. Pek Curret-Mode Cotrol eture lde prt More Aurte Model ECEN 5807 Drg Mkovć Sple Frt-Order CPM Model: Sury Aupto: CPM otroller operte delly, Ueful reult t low frequee, well uted
More informationImplicit Runge-Kutta method for Van der pol problem
Appled d Computtol Mthemts 5; 4(-: - Publshed ole Jul, 4 (http://www.seepublshggroup.om//m do:.48/.m.s.54. ISSN: 8-55 (Prt; ISSN: 8-5 (Ole Implt Ruge-Kutt method for V der pol problem Jfr Bzr *, Mesm Nvd
More informationICS141: 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
More informationG S Power Flow Solution
G S Power Flow Soluto P Q I y y * 0 1, Y y Y 0 y Y Y 1, P Q ( k) ( k) * ( k 1) 1, Y Y PQ buses * 1 P Q Y ( k1) *( k) ( k) Q Im[ Y ] 1 P buses & Slack bus ( k 1) *( k) ( k) Y 1 P Re[ ] Slack bus 17 Calculato
More informationAnalele Universităţii din Oradea, Fascicula: Protecţia Mediului, Vol. XIII, 2008
Alele Uverstăţ d Orde Fsul: Proteţ Medulu Vol. XIII 00 THEORETICAL AND COMPARATIVE STUDY REGARDING THE MECHANICS DISPLASCEMENTS UNDER THE STATIC LOADINGS FOR THE SQUARE PLATE MADE BY WOOD REFUSE AND MASSIF
More informationA Mean- maximum Deviation Portfolio Optimization Model
A Mea- mamum Devato Portfolo Optmzato Model Wu Jwe Shool of Eoom ad Maagemet, South Cha Normal Uversty Guagzhou 56, Cha Tel: 86-8-99-6 E-mal: wujwe@9om Abstrat The essay maes a thorough ad systemat study
More informationChapter Simpson s 1/3 Rule of Integration. ( x)
Cper 7. Smpso s / Rule o Iegro Aer redg s per, you sould e le o. derve e ormul or Smpso s / rule o egro,. use Smpso s / rule o solve egrls,. develop e ormul or mulple-segme Smpso s / rule o egro,. use
More information