Lecture 3-4 Solutions of System of Linear Equations
|
|
|
- Timothy Pope
- 5 years ago
- Views:
Transcription
1 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)
2 VECTORS,,, colum vector row vector Emples: umers rry of oe dmesol : Vector e e e e vectors Idetty
3 MATRICES Trdgol, dgol detty mtr zero mtr Emples: umers rry of two dmesol Mtr :
4 MATRICES upper trgulr, symmetrc Emples:
5 Determt of MATRICES Defed for squre mtrces oly Emples: - det - - ( ) ( ) ( ) 8 -
6 Addg d Multplyg Mtrces The ddto of two mtrces A d B * Defed oly f they hve the sme sze * C A B c j j j, j Multplcto of two mtrces A( m) d B(pq) * The product C AB s defed oly f m p * C A B c j m k k kj, j
7 7 Systems of Ler Equtos Mtr form Stdrd form dfferet forms ler equtos c e preseted A system of
8 8 Solutos of Ler Equtos ; s soluto to the followg equtos : A set of equtos s cosstetf there ests o soluto to the system of equtos: These equtos re cosstet ;
9 9 Solutos of Ler Equtos Some systems of equtos my hve fte umer of solutos soluto for ll s ).( solutos hve fte umer of Both re sme equtos!!! lerly depedet equto I mtr form: ; A A Sgulr mtr
10 Grphcl Soluto of Systems of Ler Equtos Soluto =, =
11 Crmer s Rule s Not Prctcl Crmer' s Rule c e used to solve the system, Crmer' s To solve N y N system requres (N )(N -)N! multplctos. To solve y system,.8 It c e used f Rule s ot prctcl for lrge systems. multplctos re eeded. the determts re computed effcet wy
12 Nve Guss Elmto The method cossts of two steps: Forwrd Elmto: the system s reduced to upper trgulr form. A sequece of elemetry opertoss used. Bckwrd Susttuto: Solve the system strtg from the lst vrle. ' ' ' ' '
13 Elemetry Row Opertos Addg multple of oe row to other Multply y row y o-zero costt Bsclly To elmte ukow EX:...(); Add - () to () yelds...()
14 Emple Forwrd Elmto 8 7 8,, from equtos Elmte Step: Forwrd Elmto : Prt Row Row Row Row (/) Row Row (-) Row Row Row Row (-/)
15 Emple Forwrd Elmto 9 fromequto Elmte Step: 9, from equtos Elmte Step : Row Row
16 Emple Forwrd Elmto Elmto : the Forwrd Summry of
17 7 Emple Bckwrd Susttuto () ) ( (), () ) ( 9,,...solve for for solve the, for Solve 9
18 8 Forwrd Elmto j j j j j j j j ) ( To elmte ) ( To elmte
19 9 Forwrd Elmto s elmted. Cotue utl ) ( To elmte k kk k kj kk k j j k k j k
20 Bckwrd Susttuto j j j,,,,,,,,
21 Emple ) ( 8, from equtos Elmte Elmto Step: Forwrd : Prt Guss Elmto : Nve usg Solve eq eq eq eq eq eq equto pvot uchged eq
22 Emple ) ( 8 from equto Elmte Step : Forwrd Elmto : Prt eq eq eq equto pvot uchged eq uchged eq
23 Emple Bckwrd Susttuto The soluto s 8,,,,,,,
24 Determt The elemetry opertos do ot ffect the determt Emple: Elemetry opertos A A' det( A) det( A' )
25 How My Solutos Does System of Equtos A=Hve? Uque det( A) reduced mtr hs o zero rows No soluto det( A) reduced mtr hs oe or more zero rows correspodg B elemets Ifte det( A) reduced mtr hs oe or more zero rows correspodg B elemets
26 Emples.!. # : solutos fte # of No soluto Uque X mpossle X solutos Ifte No soluto soluto X X X X X X
27 Pseudocode: Forwrd Elmto Do k = to - Do = k+ to fctor =,k / k,k Do j = k+ to Ed Do,j =,j fctor * k,j = fctor * k Ed Do Ed Do 7
28 Pseudocode: Bck Susttuto = /, Do = - dowto sum = Do j = + to sum = sum,j * j Ed Do = sum /, Ed Do 8
29 9 Prolems wth Nve Guss Elmto o The Nve Guss Elmto my fl for very smple cses. (The pvotg elemet (dgol elemet; pvot ) s zero). o Also, very smll pvotg elemet my result serous computto errors.
30 Remedy : Pvotg & Sclg Pvotg Choose lrgest elemet ech colum for pvot Ths c e doe y swppg oth rows d colums complete pvotg. Or oly swppg rows Prtl pvotg Sclg Dvde ll elemets ech row y ts lrgest elemet
31 Emple 8 Pvotg : wth Prtl Solve the followg system usg Guss Elmto Prtl pvotg mes row swps usg pvots,.e., No colum swps.
32 Emple Itlzto step L Ide Vector 8
33 Why Ide Vector? Ide vectors re used ecuse t s much eser to echge sgle de elemet compred to echgg the vlues of complete row. I prctcl prolems wth very lrge N, echgg the cotets of rows my ot e prctcl,.e., c tke tme d memory.
34 Emple Forwrd Elmto-- Step : elmte ] [ the frst pvot equto Echge equto s to correspods m ] Frst colum [ ] [ 8 the pvot equto Selecto of L l d l l L T
35 Emple Forwrd Elmto-- Step : elmte Updte A d B Frst pvot equto
36 Emple Forwrd Elmto-- Step : elmte Selecto of L d [.. 8. Colum : the secod pvot equto ] L [ ]
37 7 Emple Forwrd Elmto-- Step : elmte ] [ L Thrd pvot equto
38 8 Emple Bckwrd Susttuto ,.. ] [ ,,,,,,,,,, L
39 9 Emple wth Sclg 8 Pvotg : wth Scled Prtl Solve the followg system usg Guss Elmto Scled Prtl pvotg mes row swps usg scled pvots.
40 Emple Itlzto step L Ide Vector 8 S Scle vector 8 Scle vector: dsregrd sg fd lrgest mgtude ech row
41 Emple Forwrd Elmto-- Step : elmte ] [ the frst pvot equto Echge equto s to correspods m, 8,,,,, ] [ ] 8 [ 8 the pvot equto Selecto of, L l d l l S Rtos L S l l
42 Emple Forwrd Elmto-- Step : elmte Updte A d B Frst pvot equto
43 Emple Forwrd Elmto-- Step : elmte Selecto of S. [.. 8 the secod pvot equto.7.7. ]..7.7 L [..7. ] Rtos : l, S l,,.. 8. L [ ]
44 Emple Forwrd Elmto-- Step : elmte ] [ L Thrd pvot equto
45 Emple Bckwrd Susttuto , 9 ] [ ,,,,,,,,,, L l
46 Emple : Scled Prtl Pvotg 8 Solve the followg system usg Guss elmto wth scled prtl pvotg
47 7 Emple Itlzto step L Ide Vector 8 S Scle vector 8
48 8 Emple Forwrd Elmto-- Step : elmte ] [ the frst pvot equto Echge equto s to correspods m, 8,,,,, ] [ ] 8 [ 8 the pvot equto Selecto of, L l d l l S Rtos L S l l
49 9 Emple Forwrd Elmto-- Step : elmte Updte A d B
50 ] [. 8..,, Rtos : ] [ ] 8 [ the secod pvot equto Selecto of, L S L S l l Emple Forwrd Elmto-- Step : elmte
51 Emple Forwrd Elmto-- Step : elmte ] [ Updtg A d B L
52 Emple Forwrd Elmto-- Step : elmte ] [ , Rtos : ] [ ] 8 [ the thrd pvot equto Selecto of, L S L S l l
53 Emple Forwrd Elmto-- Step : elmte ] [ L
54 Emple Bckwrd Susttuto ,.88.9 ] [ ,,,,,,,,,, L l l l l l l l l l l l l l l
55 How Do We Kow If Soluto s Good or Not Gve A= (A : mtr, : rghthd-sde (RHS) vector) s soluto f A-= Compute the resdul vector r= A- Due to roudg error, rmy ot e zero The soluto s cceptle f m r ε s usully clled tolerce.
56 How Good s the Soluto?.... Resdues : R soluto
57 Remrks: We use de vector to vod the eed to move the rows whch my ot e prctcl for lrge prolems. If we order the equto s the lst vlue of the de vector, we hve trgulr form. Scle vector s formed y tkg mmum mgtude ech row. Scle vector does ot chge. The orgl mtr Ad vector re used 7 checkg the resduls.
58 8 Trdgol Systems: The o-zero elemets re the m dgol, super dgold sudgol. j = f -j > Trdgol Systems
59 Trdgol Systems Occur my pplctos Needs less storge(- compred to + for the geerl cses) Selecto of pvotg rows s uecessry (uder some codtos) Effcetlysolved y Guss elmto 9
60 Algorthm to Solve Trdgol Systems Bsed o Nve Guss elmto. As prevous Guss elmto lgorthms Forwrd elmto step Bckwrd susttuto step Elemets the super dgolre ot ffected. Elemets the m dgol, d eed updtg
61 Trdgol System ' ' ' ' ' ' elemets re ot updted The elemets d to updte the eed e zeros, wll elemets the All d c d c d c d d c d c d c d c d M M O O M M O O O
62 Dgol Domce A mtr A j, j j s dgolly domt f for ( ) The mgtude of ech dgol elemet s lrger th the sum of elemets the correspodg row.
63 Dgol Domce Emples : Dgolly domt Not Dgolly domt
64 Dgolly Domt Trdgol System A trdgol system s dgolly domt f d c ( ) Forwrd Elmto preserves dgol domce
65 Solvg Trdgol System,...,, for Susttuto Bckwrd Elmto Forwrd c d d d c d d d
66 Emple,, for, Susttuto Bckwrd, Elmto Forwrd 8 9,,, 8 9 Solve c d d d c d d d B C A D
67 7 Emple.9...9, ,.. 9., ForwrdElmto 8 9,,, d c d d d d c d d d d c d d d B C A D
68 8 Emple Bckwrd Susttuto After the Forwrd Elmto: Bckwrd Susttuto:...., ,.. d c d c d c d T T d
69 How does LU Decomposto work? If solvg set of ler equtos If A = LU the Multply y L - whch gves Rememer L - L = I whch leds to Now, sce IU = U the Now, let A = LU = L - LU = L - IU = L - U = L - L - = z Whch eds wth Lz = () Ad U = z () Thus, gve A =,. Decompose A = LU. Solve Lz= for z. Solve U= z for 9
70 Is LU Decomposto etter th Guss Elmto? Solve A = T = clock cycle tme d = sze of the mtr Forwrd Elmto Decomposto CT FE 8 T 8 CT DE 8 T Bck Susttuto CT BS T Forwrd Susttuto CT FS T Bck Susttuto CT BS T 7
71 Is LU Decomposto etter th Guss Elmto? Tme tke y methods To solve A = Guss Elmto LU Decomposto T 8 T = clock cycle tme d = sze of the mtr T 8 So oth methods re eqully effcet. 7
72 To fd verse of A Tme tke y Guss Elmto Tme tke y LU Decomposto CT 8 T FE CT BS CT T LU CT FS CT BS 7
73 To fd verse of A Tme tke y Guss Elmto 8 T Tme tke y LU Decomposto T Tle Comprg computtol tmes of fdg verse of mtr usg LU decomposto d Guss elmto. CT verse GE / CT verse LU
74 Method: A Decomposes to L d U A LU l l l u u u u u u U s the sme s the coeffcet mtr t the ed of the forwrd elmto step. L s oted usg the multplers tht were used the forwrd elmto process 7
75 Fdg the U mtr Usg the Forwrd Elmto Procedure of Guss Elmto 8 Step :.;.7; Row Row. Row Row
76 Fdg the UMtr Step : ;.8.8 Row Row.7..8 U Mtr fter Step : 7
77 Fdg the L mtr l l Usg the multplers used durg the Forwrd Elmto Procedure From the frst l. step of forwrd 8 elmto l.7 l 77
78 Fdg the LMtr From the secod step of forwrd elmto l.8.8. L
79 Does LU= A? LU ? 79
80 Geerl Formul (Doolttle s method) 8 ;,,,, ;,,,, k k j u u j u j j k u u k u k s sk js jk kk jk j j j s sk js jk jk k k K l l K l K l K
81 Prevous Emple Revsted 8 A.7.).( /..8) ) /( / (.; /.8; / 8 ; / ; / ; ; ; u u u u u u l l l 8
82 Usg LU Decomposto to solve SLEs Solve the followg set of ler equtos usg LU Decomposto Usg the procedure for fdg the L d U mtrces A LU
83 Emple Set Lz = Solve for z z z z z z z z z z z z z z z z z z z 8
84 Emple Set U = z Solve for The equtos ecome
85 Emple From the rd equto Susttutg d usg the secod equto
86 Emple Susttutg d usg the frst equto Hece the Soluto Vector s:
87 Fdg the verse of squre mtr How c LU Decomposto e used to fd the verse? Let B = A - d ssume the frst colum of B to e [ ] T Usg ths d the defto of mtr multplcto Frst colum of B, Secod colum of B, M M A M M A The remg colums B c e foud the sme mer 87
88 Emple: Iverse of Mtr Fd the verse of squre mtr A A 8 Usg the decomposto procedure, the L d U mtrces re foud to e A LU
89 Emple: Iverse of Mtr Solvg for the ech colum of B requres two steps ) Solve Lz = for z ) Solve U = z for Step :..7. z z z Lz Ths geertes the equtos:..7. z z z z z z 89
90 Emple: Iverse of Mtr Solvg for z z. z. z.. z. z z.. z z z z 9
91 Emple: Iverse of Mtr Solvg U = z for
92 Emple: Iverse of Mtr Usg Bckwrd Susttuto So the frst colum of the verse of A s:
93 Emple: Iverse of Mtr Repetg for the secod d thrd colums of the verse Secod Colum Thrd Colum
94 Emple: Iverse of Mtr The verse of A s A To check your work do the followg operto AA - = I = A - A 9
95 Guss-Jord Method The method reduces the geerl system of equtos A=to I= where Is detty mtr. Oly Forwrd elmto s doe d o ckwrd susttuto s eeded. It hs the sme prolems s Nve Guss elmto d c e modfed to do scled prtl pvotg. It tkes % more tme th Nve Guss method. It s ofte used to fd verse mtrces. 9
96 9 Guss-Jord Method Emple 7 / from Elemte 7 eq eq eq eq eq eq eq eq d equtos Step
97 97 Guss-Jord Method Emple / from Elemte 7 eq eq eq eq eq eq eq eq d equtos Step
98 98 Guss-Jord Method Emple.8.7 / from Elemte eq eq eq eq eq eq eq eq d equtos Step
99 99 Guss-Jord Method Emple 7 s soluto to trsformed s
100 Guss-Jord Method Fdg verse mtr () A Fd the verse mtr of A I B Frst costruct the ugmeted mtr:
101 Guss-Jord Method Fdg verse mtr () Applyg the Guss-Jord method: / / / / / / / / / / / / / / / / / / / /
102 Guss-Jord Method Fdg verse mtr () / / / / / / / / / / / / / / / / A Therefore,
103 LU Decomposto Method For most o-sgulr mtr A tht oe could coduct Nïve Guss Elmto forwrd elmto steps, oe c lwys wrte t s A = LU where L = lower trgulr mtr U = upper trgulr mtr
104 Guss-Sedel Method Guss-Sedel s the smplest tertve method for solvg mtr equto. Bsc Procedure: - Algerclly solve ech ler equto for - Assume tl guess soluto rry - Solve for ech d repet - Use solute ppromte error fter ech terto to check f error s wth pre-specfed tolerce.
105 Advtges If oly few tertos re requred, computtol cost s lower compred to Guss elmto or LU decomposto. If the physcs of the prolem re uderstood, close tl guess c e mde, decresg the umer of tertos eeded. However, f the mtr s ll-codtoed, the soluto mght ot coverge.
106 Algorthm Iput : mtr A, vector, tl guess, tolerce. Updte j, j=,, usg ( m) j Ajj j j k A ( m) k. Check f the covergece crtero s met. m j Repet utl covergece. jk ( m ) ( m) j j k j A jk ( m) k
107 Emple Solve ; 7 st terto: =[. -.7.] d terto: =[.8 -..] rd terto: =[.7 -..] th terto: =[.7 -..] NOTE: Oly few tertos re requred for dgolly domtmtrces (well-codtoed, or smll codto umer). 7
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
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
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
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
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
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
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
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
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
Chapter 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
Computer 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
PubH 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
Chapter 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
MATRIX 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
CS321. 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
INTRODUCTION ( ) 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
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
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
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...
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
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
Methods 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
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
Matrix. 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
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
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
2006 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
Advanced Algorithmic Problem Solving Le 6 Math and Search
Advced Algorithmic Prolem Solvig Le Mth d Serch Fredrik Heitz Dept of Computer d Iformtio Sciece Liköpig Uiversity Outlie Arithmetic (l. d.) Solvig lier equtio systems (l. d.) Chiese remider theorem (l.5
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
CHAPTER 4 RADICAL EXPRESSIONS
6 CHAPTER RADICAL EXPRESSIONS. The th Root of a Real Number A real umber a s called the th root of a real umber b f Thus, for example: s a square root of sce. s also a square root of sce ( ). s a cube
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
CS473-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
Mu Sequences/Series Solutions National Convention 2014
Mu Sequeces/Seres Solutos Natoal Coveto 04 C 6 E A 6C A 6 B B 7 A D 7 D C 7 A B 8 A B 8 A C 8 E 4 B 9 B 4 E 9 B 4 C 9 E C 0 A A 0 D B 0 C C Usg basc propertes of arthmetc sequeces, we fd a ad bm m We eed
Numerical 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
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
1 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,
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
means 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,
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
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
The 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
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...
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.
Systems 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
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.
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
G 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
F. Inequalities. HKAL Pure Mathematics. 進佳數學團隊 Dr. Herbert Lam 林康榮博士. [Solution] Example Basic properties
![F. Inequalities. HKAL Pure Mathematics. 進佳數學團隊 Dr. Herbert Lam 林康榮博士. [Solution] Example Basic properties](/thumbs/82/86768390.jpg "F. Inequalities. HKAL Pure Mathematics. 進佳數學團隊 Dr. Herbert Lam 林康榮博士. [Solution] Example Basic properties")
進佳數學團隊 Dr. Herbert Lam 林康榮博士 HKAL Pure Mathematcs F. Ieualtes. Basc propertes Theorem Let a, b, c be real umbers. () If a b ad b c, the a c. () If a b ad c 0, the ac bc, but f a b ad c 0, the ac bc. Theorem
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
Chapter 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
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
CS321. 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
ESS Line Fitting
ESS 5 014 17. Le Fttg A very commo problem data aalyss s lookg for relatoshpetwee dfferet parameters ad fttg les or surfaces to data. The smplest example s fttg a straght le ad we wll dscuss that here
Autar 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
Chapter 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.
Chapter 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
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
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
PGE 310: Formulation and Solution in Geosystems Engineering. Dr. Balhoff. Interpolation
PGE 30: Formulato ad Soluto Geosystems Egeerg Dr. Balhoff Iterpolato Numercal Methods wth MATLAB, Recktewald, Chapter 0 ad Numercal Methods for Egeers, Chapra ad Caale, 5 th Ed., Part Fve, Chapter 8 ad
Notes 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 (
Mathematically, 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
Review 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
Asynchronous 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
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
The 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
Math 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
3/20/2013. 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
GRAPHING 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
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
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
Rank 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
CS321. 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
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,
Fibonacci 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
( ) 2 2. Multi-Layer Refraction Problem Rafael Espericueta, Bakersfield College, November, 2006
Mult-Layer Refracto Problem Rafael Espercueta, Bakersfeld College, November, 006 Lght travels at dfferet speeds through dfferet meda, but refracts at layer boudares order to traverse the least-tme path.
An 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 ** *
ENGI 4430 Numerical Integration Page 5-01
ENGI 443 Numercal Itegrato Page 5-5. Numercal Itegrato I some o our prevous work, (most otaly the evaluato o arc legth), t has ee dcult or mpossle to d the dete tegral. Varous symolc algera ad calculus
The 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
Limit of a function:
- Limit of fuctio: We sy tht f ( ) eists d is equl with (rel) umer L if f( ) gets s close s we wt to L if is close eough to (This defiitio c e geerlized for L y syig tht f( ) ecomes s lrge (or s lrge egtive
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
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 ( )
6.6 Moments and Centers of Mass
th 8 www.tetodre.co 6.6 oets d Ceters of ss Our ojectve here s to fd the pot P o whch th plte of gve shpe lces horzotll. Ths pot s clled the ceter of ss ( or ceter of grvt ) of the plte.. We frst cosder
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
GAUSS ELIMINATION. Consider the following system of algebraic linear equations
Numercl Anlyss for Engneers Germn Jordnn Unversty GAUSS ELIMINATION Consder the followng system of lgebrc lner equtons To solve the bove system usng clsscl methods, equton () s subtrcted from equton ()
PTAS for Bin-Packing
CS 663: Patter Matchg Algorthms Scrbe: Che Jag /9/00. Itroducto PTAS for B-Packg The B-Packg problem s NP-hard. If we use approxmato algorthms, the B-Packg problem could be solved polyomal tme. For example,
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
For 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
International 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
Chapter 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
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
ECE 102 Engineering Computation
ECE Egieerig Computtio Phillip Wog Mth Review Vetor Bsis Mtri Bsis System of Lier Equtios Summtio Symol is the symol for summtio. Emple: N k N... 9 k k k k k the, If e e e f e f k Vetor Bsis A vetor is
Linear 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)
Chapter 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
2/20/2013. Topics. Power Flow Part 1 Text: Power Transmission. Power Transmission. Power Transmission. Power Transmission
/0/0 Topcs Power Flow Part Text: 0-0. Power Trassso Revsted Power Flow Equatos Power Flow Proble Stateet ECEGR 45 Power Systes Power Trassso Power Trassso Recall that for a short trassso le, the power
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;
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
Lecture Notes 2. The ability to manipulate matrices is critical in economics.
Lecture Notes. Revew of Matrces he ablt to mapulate matrces s crtcal ecoomcs.. Matr a rectagular arra of umbers, parameters, or varables placed rows ad colums. Matrces are assocated wth lear equatos. lemets
Algorithms 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
QR 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?
Chapter 4: Distributions
Chpter 4: Dstrbutos Prerequste: Chpter 4. The Algebr of Expecttos d Vrces I ths secto we wll mke use of the followg symbols: s rdom vrble b s rdom vrble c s costt vector md s costt mtrx, d F m s costt