CS537. Numerical Analysis
|
|
- Jacob Dixon
- 6 years ago
- Views:
Transcription
1 CS57 Numercl Alyss Lecture 4 System of Ler Equtos Professor Ju Zhg Deprtmet of Computer Scece Uversty of Ketucky Legto, KY Ferury, 6
2 System of Ler Equtos where j re coeffcets, re ukows, d re rght hd sdes. Wrtte compct form s The system c lso e wrtte mtr form where the coeffcet mtr s A A j j j,,, T T ],, [, ],,, [ d
3 A Upper Trgulr System The soluto of geerl ler system s ot esly vlle The soluto c e oted esly from upper trgulr system Ojectve: odfy geerl ler system to upper trgulr system Three elemetry opertos o ler system do ot chge ts soluto,,
4 Krl Fredrch Guss (Aprl, 777 Ferury, 855) Germ themtc d Scetst 4
5 Guss Elmto Ler systems re solved y Guss elmto, whch volves repeted procedure of multplyg row y umer d ddg t to other row to elmte cert vrle For prtculr step, ths mouts to k j j kj ( k j ) kk k k kk After ths step, the vrle k, s elmted the (k + ) th d the lter equtos The Guss elmto modfes mtr to upper trgulr form such tht j = for ll > j. The soluto of upper trgulr system s the esly oted y ck susttuto procedure 5
6 Illustrto of Guss Elmto 6
7 7
8 Bck Susttuto 8 The oted upper trgulr system s We c compute From the lst equto d susttute ts vlue other equtos d repet the process For =,,,,, j j j
9 9
10 Codto Numer d Error A qutty used to mesure the qulty of mtr s clled codto umer, defed s ( A) A A The codto umer mesures the trsfer of error from the mtr A to the rght hd sde vector. If A hs lrge codto umer, smll error A or my yeld lrge error the soluto = A -. Such mtr s clled ll codtoed The error e s defed s the dfferece etwee computed soluto d the ect soluto e ~ Sce the ect soluto s geerlly ukow, we mesure the resdul r A ~ As dctor of the sze of the error
11 Error Correcto If the error c e foud, the the ppromte soluto c e corrected: = + () You cot solve the resdul equto to get the error = I my moder tertve methods, we m to solve ppromte equto of them form = Where s good ppromte to A, so tht wll e good ppromte to the error e, we c correct the ppromte the soluto s: = + s etter ppromto the, d we c repet the process g
12 Smll Pvot for some smll ε. After the step of Guss elmto We hve For very smll ε, the computer result wll e = d =. The correct results re / /
13 Scled Prtl Pvotg We eed to choose elemet whch s lrge reltve to other elemets of the sme row s the pvot Let L = (l, l,, l ) e de rry of tegers. We frst compute rry of sclg fctor s S = (s, s,, s ) where s m j j ( ) The frst row s chose such tht the rto, /s s the gretest. Suppose ths de s l, the pproprte multplers of equto l re sutrcted from the other equtos to elmte from the other equtos Suppose tlly L = (l, l,, l ) = (,,, ), f our frst choce s l j, we wll terchge l j d l the de set, ot ctully terchge the frst d the l j rows, to vod movg dt roud the memory
14 Emple 4 Strghtforwrd Guss elmto does ot work well (ot roust) The sclg fctor wll e computed s S = {,}. I the frst step, the sclg fctor rto rry {ε,}. So the d row s the pvotg row After elmtg from the st equto, we hve It follows tht We computed correct results y usg scled prtl pvotg strtegy ) (
15 Guss Elmto wth Prtl Pvotg 5
16 Log Operto Cout We cout the umer of multplctos d dvsos, gore summtos d sutrctos The st step, fdg pvotg costs dvsos Addtol opertos re eeded to multply fctor to the pvotg row for ech of the elmtos. The cost s ( )opertos. The totl cost of ths step s opertos The computto s repeted o the remg ( )equtos. The totl costs of Guss elmto wth scled prtl pvotg s ( ) 4 ( )( ) 6 Bck susttuto costs ( )/opertos 6
17 Trdgol d Bded Systems 7 Bded system hs coeffcet mtr such tht j = f j w. For trdgol system, w = Geerl elmto procedure The rry c s ot modfed. No ddtol ozero s creted tr c e stored three vector rrys d d c d d d c d c d c d c d
18 Trdgol Systems The ck susttuto s strghtforwrd d c d No pvotg s performed, otherwse the procedure wll e qute dfferet due to the fll (the rry c wll e modfed) Dgol domce: A mtr A = ( j ) s dgolly domt f j ( ) j, j For dgolly domt trdgol system, o pvotg s eeded,.e., o dvso y zero We wt to show Guss elmto preserves dgol domce,.e., d (,,) c 8
19 Trdgol Systems 9 The ew coeffcet mtr hs elemets t the s plces. The ew dgol elemets re determed recursvely s We ssume tht We wt to show tht > We use ducto to prove the equlty It s ovously true for =, s = ) ( d c d d d d c d
20 Trdgol Systems If we ssume tht > We prove for de I, s It follows tht the ew dgol etres wll ot e zero, the Guss elmto procedure c e crred out wthout y prolem c c d c d d c d d
21 Emple of petdgol mtr. It s erly trdgol. The mtr wth oly ozero etres o the m dgol, d The frst two dgols ove d elow t
22
23 LU Fctorzto As we showed efore, * system of ler equtos c e wrtte mtr form s A where the coeffcet mtr A hs the form A s the ukow vector d s rght hd sde kow vector We lso ssume A s of full rk, d most etres of A re ot zero
24 LU Fctorzto 4 There re two specl forms of mtrces. Oe s (ut) lower trgulr The other s upper trgulr We wt to fd pr of L d U mtrces, such tht l l l L A LU u u u u u u U
25 Emple 5 Tke system of ler equtos The Guss elmto process flly yelds upper trgulr system Ths could e cheved y multplyg the orgl system wth mtr, such tht A
26 Emple 6 We wt the mtr to e specl, so tht A s upper trgulr The questo s: c we fd such mtr? Look t the frst step of the Guss elmto Ths step c e cheved y multplyg the orgl system wth lower trgulr mtr A A U
27 Here the lower trgulr mtr s Emple Ths mtr s osgulr, ecuse t s lower trgulr wth m dgol cotg ll s. The ozero elemets the frst colums re the egtves of the multplers the postos where s were creted the orgl mtr A 7
28 Emple 8 If we cotue to the secod step of the Guss elmto, we hve Ths c e cheved wth the multplcto of other lower trgulr mtr Thus, we hve A
29 Emple 9 The lst step s wth other lower trgulr mtr Ad, we hve A
30 Emple If we defe The s lower trgulr d A s upper trgulr Sce AU the verse of lower trgulr s g lower trgulr mtr. The product of two lower trgulr mtrces s lower trgulr mtr A U U LU Ths shows tht we hve trsformed mtr A to the product of lower trgulr mtr d upper trgulr mtr Ths process s clled LU fctorzto or LU decomposto
31 Emple The lower trgulr mtr s whch the curret emple s Ad, we hve L L A LU
32 Illustrto of LU Fctorzto
33 The orgl system c e fctored to Emple The soluto process c e seprted to two phses. Frst, forwrd elmto wth termedte vrle z, s Secod, ck susttuto step A LU Lz U The dvtge of LU fctorzto or LU decomposto s tht severl ler systems wth the sme coeffcet mtr ut dfferet rght hd sde vectors c e solved more effcetly. We oly eed do the fctorzto oce. The forwrd elmto d ck susttuto steps re O( ) opertos z
34 The system of ler equtos c e symolclly solved s Compute the Iverse A A But the verse of A s seldom computed eplctly. Ths s ecuse performg Guss elmto s less epesve. If the eplct verse of mtr A s eeded, t c e oted y usg LU fctorzto of A, ote tht We c compute seres of ler systems s Ad LU[ ( ) () ( ) () () ( ), A,, AX I ] [ I [ (), (),, (, I ) ],, I ] 4
35 Sgulr Vlue Decomposto (SVD) The egevlues d egevectors of mtr A, re I.e., the pplcto of A o the vector of s equvlet to sclg. Here s egevlue d s the ssocted egevector The sgulr vlues of mtr A re the oegtve squre roots of the egevlues of A T A By the Spectrl Theorem for trces, c e dgolzed y orthogol mtr, Q, s A T A QQ Where (orthogolty property) T T Q Q Q Q A I, A T A.e., Q T Q The dgol mtr cots the egevlues of o ts dgol A T A 5
36 Sgulr Vlue Decomposto II We c see tht A T AQQ So the colums of Q re egevectors of A T A If s egevlue of d s correspodg egevector, the A T A Ad T T T T A ( A) ( A) A A It follows tht the egevlue s rel d oegtve. Sce Q s orthogol mtr, ts colums form orthoorml se. They re ut egevectors of A T A A T A If vj s the jth colum of Q, we hve A T Av j j v j 6
37 For mtr, the SVD s Sgulr Vlue Decomposto III AUV Where U d V re orthogol mtrces d Σ s dgol mtr T 7
38 SVD Emple A = UΣV T emple: Eg ed dt fṛetrevl r lug =
39 SVD Emple A = UΣV T emple: dt fṛetrevl Eg Topcs ed Topcs Eg ed = r lug
40 SVD Emple A = UΣV T emple: Documet-to-Topcs Smlrty tr dt fṛetrevl Eg Topcs ed Topcs Eg ed = r lug KDD'9 Floutsos, ller, Tsourkks P 4
41 SVD Emple A = UΣV T emple: dt fṛetrevl r lug.8 Stregth of Eg Topcs Eg ed =
42 SVD Emple A = UΣV T emple: Eg ed dt fṛetrevl r lug = Word-to-Topcs Smlrty tr
43 Tructed SVD For mtr, the tructed rk k SVD s A k U Where U d V re orthogol mtrces d Σ s dgol mtr k k V T k 4
44 SVD Dmeso Reducto Why s t clled dmeso reducto? odfed dt: rk =
45 Gee H. Golu (Ferury 9, 9 Novemer 6, 7) Amerc themtc d Computer Scetst 45
46 46 Itertve Soluto ethods Ot ppromte soluto to ler system tertvely Crete sgle equto for ech vrle We ssg rtrry vlue to ech of the vrles the rght hd sde to strt terto ) ( ) ( ) (
47 Jco Iterto We repetedly use ech set of compoud vlues s the et tertes, d hope the process wll coverge to the true soluto. (y stop erly f ecessry.) Ths terto procedure s clled Jco terto method. Covergece s ot gurteed for geerl mtrces (k) (k) (k) ( ( ( (k) (k) (k) (k) (k) (k) (k) (k) ) (k) ) ) 47
48 Crl Gustv Jco (Decemer, 84 Ferury 8, 85) Pruss themtc 48
49 Guss Sedel Iterto If the terto process coverges, t my e fster f we use the ew vlues s soo s they re vlle. Ths gves the Guss Sedel terto method (k) (k) (k) ( ( ( (k) (k) (k) I geerl, Guss Sedel s fster th Jco, ut the lter s more ttrctve to ru o prllel computers. No covergece s gurteed (k) (k) (k) (k) (k) (k) ) ) ) 49
50 Phlpp Ludwg vo Sedel (Octoer, 8 August, 896) Germ themtc Sedel ws studet of Jco 5
51 SOR ethod Aroud 95, Dvd Youg t Hrvrd Uversty foud tht the terto c e ccelerted y usg weghted verge of the successve ppromte solutos. Gve some prmeter < ω <, we perform ( k ) ~ ( k ) ( k ) ( ),,, Ths step symolzes the egg of moder tertve methods Wth rtrry tl guess, covergece of these sc tertve methods s gurteed f.) The mtr s strctly dgolly domt;.) The mtr s symmetrc postve defte,.e., A = A T d T A > 5
52 Dvd. Youg, Jr. (9 8) Amerc themtc d Scetst 5
53 Splttg Coeffcet tr The system of ler equtos s We c splt the coeffcet mtr A s Such tht s osgulr d s esy to compute. So we hve d A A N ( N) N So ( N ) We c use ths reltoshp for terto scheme N N ( k ) ( k ) 5
54 54 Iterto tr The mtr N s clled the terto mtr For the terto process to e epesve to crry out, must e esy to vert Sce We hve If (k) coverges to the soluto, the It follows tht A A N A I ) ( N k k ) ( ) ( A I A k k k ) ( ) ( ) ( ) ( ) (
55 55 Covergece We defe the error vector t ech terto step s It follows tht If we wt the error to ecome smller d smller, we wt the terto mtr to e smll, some sese A I e k k ) ( ) ( ) ( ) )( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( k k k k k k e A I A I A I A I A I e
56 Iterto tr So we wt the mtr A to e close to the detty mtr. It s to sy tht we wt s close to A If fct, f ll egevelues of the terto mtr s smller th oe, the the terto e ( k ) coverges to zero d the orgl terto coverges. Ths requres ( I I ( I A) I.e., the spectrl rdus of the terto s smller th oe gurtees the covergece of the duced terto method A A) e ( k ) 56
57 Sttory Itertve ethods For the Jco terto A For the Guss Sedel ethod D For the SOR method, the terto s ( D L) ( L U ) A ( L D) ( k ) [ U U ( ) D] These tertve methods re clled sttory methods, or clssc tertve methods. They coverge slowly oder tertve methods, such s multgrd method d Krylov suspce methods coverge much more rpdly. Those methods wll e studed CS6 ( k ) 57
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
More informationITERATIVE 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 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 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 informationChapter 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
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 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 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 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 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 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 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 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 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 informationCS321. Numerical Analysis
CS3 Nuercl Alss Lecture 7 Lest Sures d Curve Fttg Professor Ju Zhg Deprtet of Coputer Scece Uverst of Ketuc Legto KY 456 633 Deceer 4 Method of Lest Sures Coputer ded dt collectos hve produced treedous
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 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 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 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 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 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 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 informationCS473-Algorithms I. Lecture 3. Solving Recurrences. Cevdet Aykanat - Bilkent University Computer Engineering Department
CS473-Algorthms I Lecture 3 Solvg Recurreces Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet Solvg Recurreces The lyss of merge sort Lecture requred us to solve recurrece. Recurreces re lke solvg
More informationPOWERS 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
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 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 informationOn 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
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 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 informationReview of Linear Algebra
PGE 30: Forulto d Soluto Geosstes Egeerg Dr. Blhoff Sprg 0 Revew of Ler Alger Chpter 7 of Nuercl Methods wth MATLAB, Gerld Recktewld Vector s ordered set of rel (or cople) uers rrged s row or colu sclr
More informationSUM 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
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 informationMATH2999 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.
More informationOptimality 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
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 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 informationthis 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
More informationPhysics 220: Worksheet5 Name
ocepts: pctce, delectrc costt, resstce, seres/prllel comtos () coxl cle cossts of sultor of er rdus wth chrge/legth +λ d outer sultg cylder of rdus wth chrge/legth -λ. () Fd the electrc feld everywhere
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 informationAn Alternative Method to Find the Solution of Zero One Integer Linear Fractional Programming Problem with the Help of -Matrix
Itertol Jourl of Scetfc d Reserch Pulctos, Volue 3, Issue 6, Jue 3 ISSN 5-353 A Altertve Method to Fd the Soluto of Zero Oe Iteger Ler Frctol Progrg Prole wth the Help of -Mtr VSeeregsy *, DrKJeyr ** *
More informationLinear Algebra Concepts
Ler Algebr Cocepts Ke Kreutz-Delgdo (Nuo Vscocelos) ECE 75A Wter 22 UCSD Vector spces Defto: vector spce s set H where ddto d sclr multplcto re defed d stsf: ) +( + ) (+ )+ 5) l H 2) + + H 6) 3) H, + 7)
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 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 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 informationA 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
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 informationSoo 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
More informationMath 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,
More informationThe total number of permutations of S is n!. We denote the set of all permutations of S by
DETERMINNTS. DEFINITIONS Def: Let S {,,, } e the set of itegers from to, rrged i scedig order. rerrgemet jjj j of the elemets of S is clled permuttio of S. S. The totl umer of permuttios of S is!. We deote
More informationAnswer: 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
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 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 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 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 informationThe definite Riemann integral
Roberto s Notes o Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 4 The defte Rem tegrl Wht you eed to kow lredy: How to ppromte the re uder curve by usg Rem sums. Wht you c ler here: How to use
More informationCS321. Numerical Analysis
CS Numercl Alyss Lecture 4 Numercl Itegrto Proessor Ju Zg Deprtmet o Computer Scece Uversty o Ketucky Legto, KY 456 6 Octoer 6, 5 Dete Itegrl A dete tegrl s tervl or tegrto. For ed tegrto tervl, te result
More informationFibonacci and Lucas Numbers as Tridiagonal Matrix Determinants
Rochester Isttute of echology RI Scholr Wors Artcles 8-00 bocc d ucs Nubers s rdgol trx Deterts Nth D. Chll Est Kod Copy Drre Nry Rochester Isttute of echology ollow ths d ddtol wors t: http://scholrwors.rt.edu/rtcle
More informationLinear Algebra Concepts
Ler Algebr Cocepts Nuo Vscocelos (Ke Kreutz-Delgdo) UCSD Vector spces Defto: vector spce s set H where ddto d sclr multplcto re defed d stsf: ) +( + ) = (+ )+ 5) H 2) + = + H 6) = 3) H, + = 7) ( ) = (
More informationSystems of second order ordinary differential equations
Ffth order dgolly mplct Ruge-Kutt Nystrom geerl method solvg secod Order IVPs Fudzh Isml Astrct A dgolly mplct Ruge-Kutt-Nystróm Geerl (SDIRKNG) method of ffth order wth explct frst stge for the tegrto
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 information6. 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
More informationInternational Journal of Scientific and Research Publications, Volume 3, Issue 5, May ISSN
Itertol Jourl of Scetfc d Reserch Pulctos, Volue 3, Issue 5, My 13 1 A Effcet Method for Esy Coputto y Usg - Mtr y Cosderg the Iteger Vlues for Solvg Iteger Ler Frctol Progrg Proles VSeeregsy *, DrKJeyr
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 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 informationChapter 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...
More informationAlgorithms Theory, Solution for Assignment 2
Juor-Prof. Dr. Robert Elsässer, Marco Muñz, Phllp Hedegger WS 2009/200 Algorthms Theory, Soluto for Assgmet 2 http://lak.formatk.u-freburg.de/lak_teachg/ws09_0/algo090.php Exercse 2. - Fast Fourer Trasform
More informationmeans the first term, a2 means the term, etc. Infinite Sequences: follow the same pattern forever.
9.4 Sequeces ad Seres Pre Calculus 9.4 SEQUENCES AND SERIES Learg Targets:. Wrte the terms of a explctly defed sequece.. Wrte the terms of a recursvely defed sequece. 3. Determe whether a sequece s arthmetc,
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 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 informationOn 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
More informationIntroduction to Numerical Integration Part II
Introducton to umercl Integrton Prt II CS 75/Mth 75 Brn T. Smth, UM, CS Dept. Sprng, 998 4/9/998 qud_ Intro to Gussn Qudrture s eore, the generl tretment chnges the ntegrton prolem to ndng the ntegrl w
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 informationApproximate 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:
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 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 informationSection 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
More informationRegression. 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
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 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 informationGRAPHING LINEAR EQUATIONS. Linear Equations. x l ( 3,1 ) _x-axis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1.
GRAPHING LINEAR EQUATIONS Qudrt II Qudrt I ORDERED PAIR: The first umer i the ordered pir is the -coordite d the secod umer i the ordered pir is the y-coordite. (, ) Origi ( 0, 0 ) _-is Lier Equtios Qudrt
More informationAvailable 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
More informationOn 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
More informationRank One Update And the Google Matrix by Al Bernstein Signal Science, LLC
Introducton Rnk One Updte And the Google Mtrx y Al Bernsten Sgnl Scence, LLC www.sgnlscence.net here re two dfferent wys to perform mtrx multplctons. he frst uses dot product formulton nd the second uses
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 informationAsymptotic 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:
More informationSolutions 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
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 informationEVALUATING COMPARISON BETWEEN CONSISTENCY IMPROVING METHOD AND RESURVEY IN AHP
ISAHP 00, Bere, Stzerld, August -4, 00 EVALUATING COMPARISON BETWEEN CONSISTENCY IMPROVING METHOD AND RESURVEY IN AHP J Rhro, S Hlm d Set Wto Petr Chrst Uversty, Surby, Idoes @peter.petr.c.d sh@peter.petr.c.d
More informationAnalytical 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
More informationNotes 17 Sturm-Liouville Theory
ECE 638 Fll 017 Dvid R. Jckso Notes 17 Sturm-Liouville Theory Notes re from D. R. Wilto, Dept. of ECE 1 Secod-Order Lier Differetil Equtios (SOLDE) A SOLDE hs the form d y dy 0 1 p ( x) + p ( x) + p (
More informationNumerical differentiation (Finite difference)
MA umercl Alyss Week 4 To tk d wrte portle progrms we eed strct model of ter computtol evromet. Ftful models do est ut tey revel tt evromet to e too dverse forcg portle progrmmers to lot eve te smplest
More informationHandout #2. Introduction to Matrix: Matrix operations & Geometric meaning
Hdout # Title: FAE Course: Eco 8/ Sprig/5 Istructor: Dr I-Mig Chiu Itroductio to Mtrix: Mtrix opertios & Geometric meig Mtrix: rectgulr rry of umers eclosed i pretheses or squre rckets It is covetiolly
More information1 Onto functions and bijections Applications to Counting
1 Oto fuctos ad bectos Applcatos to Coutg Now we move o to a ew topc. Defto 1.1 (Surecto. A fucto f : A B s sad to be surectve or oto f for each b B there s some a A so that f(a B. What are examples of
More informationQR Factorization and Singular Value Decomposition COS 323
QR Factorzato ad Sgular Value Decomposto COS 33 Why Yet Aother Method? How do we solve least-squares wthout currg codto-squarg effect of ormal equatos (A T A A T b) whe A s sgular, fat, or otherwse poorly-specfed?
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 informationScattering matrices of multiport radio devices
Gedj CAWKA Blystok Uversty of Techology do:599/4865 ctterg mtrces of multport rdo devces Abstrct Ths pper dels wth the mthemtcs models d propertes of three types the sctterg mtrces for rbtrry multport
More informationChapter 9 Jordan Block Matrices
Chapter 9 Jorda Block atrces I ths chapter we wll solve the followg problem. Gve a lear operator T fd a bass R of F such that the matrx R (T) s as smple as possble. f course smple s a matter of taste.
More informationFor combinatorial problems we might need to generate all permutations, combinations, or subsets of a set.
Addtoal Decrease ad Coquer Algorthms For combatoral problems we mght eed to geerate all permutatos, combatos, or subsets of a set. Geeratg Permutatos If we have a set f elemets: { a 1, a 2, a 3, a } the
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 information