SOLVING SYSTEMS OF EQUATIONS, DIRECT METHODS

Size: px
Start display at page:

Download "SOLVING SYSTEMS OF EQUATIONS, DIRECT METHODS"

Transcription

1 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

2 ELM Numecl Alyss D Muhem Mecmek Tody s lectue Commo dect techques fo le equto systems Guss Elmto Guss Elmto wth ow pvotg Guss Elmto fo tdgol systems

3 ELM Numecl Alyss D Muhem Mecmek Defto of the Polem We wt to solve: A = whee s the vecto of the ukows whle A d e gve. Codtos:. The ume of equtos s equl to the ume of ukows (tht s A s sque mt). The coeffcets of A d e el. The soluto of the system ests d t s uque

4 ELM Numecl Alyss D Muhem Mecmek Defto of the Polem The soluto of the system ests d t s uque A ests A s ot sgul A's colums e lely depedet A's les e lely depedet det(a) s ozeo k(a) s equl to A = oly f s ull vecto

5 ELM Numecl Alyss D Muhem Mecmek 5 Defto of the Polem Cme s ule The soluto of system of equtos: det det A A A eplces the th colum

6 ELM Numecl Alyss D Muhem Mecmek 6 Clculto of the detemt How to compute the detemt of sque mt whee C j s the cofcto of elemet j. det( A) N C j The cofcto C j s the detemt of the sumt oted y emovg the th ow d the j th colum of the mt multpled y () +j : j det M C j j No th ow M j j j j j j j j j No j th colum

7 ELM Numecl Alyss D Muhem Mecmek 7 Guss Elmto Emple : Solvg thee equtos thee ukows y 6y y z z 8z 8 y z y z y z y z 8y 9z z z y ( ) () ( ) z y

8 8 Usg Mt Notto: mtvecto fom : A = Fom ugmeted mt Guss elmto pocedue j j m Guss Elmto A 8 ELM Numecl Alyss D Muhem Mecmek

9 Pvot At the kth stge of Guss elmto pocedue the ppopte multple of the kth ow s used to educe ech of the etes the kth colum elow the kth ow to zeo elemet kk : pvot elemet Guss Elmto 9 ELM Numecl Alyss D Muhem Mecmek

10 ELM Numecl Alyss D Muhem Mecmek Guss Elmto Emple : V left loop R Ω R 5Ω ( ) ( ) V R Ω Ω uppe 5 ght loop ( ) ( ) R Ω R 5 Ω lowe ght loop ( ) ( ) V

11 ELM Numecl Alyss D Muhem Mecmek Guss Elmto Step 55 5 The pvot s = A 5 Multply the fst ow y / d dd t to the secod ow Multply the fst ow y / d dd t to the thd ow 55 A 5/ 5 / 5 / /

12 ELM Numecl Alyss D Muhem Mecmek Guss Elmto Step The pvot s = 5/ Multply the secod ow y /5 d dd t to the thd ow to get A 5/ By ck susttuto 5 /. / / 5 ( 5 / )(5) / 5/ ( )() ( )(5) /

13 ) ( ) ( ) )( ( k k k k k k k k k k k k Mesug computtol effot Mesue the ume of multplcto d dvsos The totl ume of multplcto d dvsos Guss Elmto Dscusso ELM Numecl Alyss D Muhem Mecmek

14 Emple : A IllCodtoed mt 6 5 Guss Elmto Dscusso ELM Numecl Alyss D Muhem Mecmek

15 ELM Numecl Alyss D Muhem Mecmek 5 Guss Elmto wth Row Pvotg I cet cses Reducg the ccuces Moe ccute th Guss Elmto Avodg (f possle) the flue Dvde y zeo Pvotg s eeded

16 ELM Numecl Alyss D Muhem Mecmek 6 Guss Elmto wth Row Pvotg Emple : Roudg to two sgfct dgts Wthout ow pvotg Wth ow Pvotg

17 8 8 6 z z z y y y ) ( 6() 8 ) ( 5 Guss Elmto wth Row Pvotg Emple 5: 7 ELM Numecl Alyss D Muhem Mecmek

18 ELM Numecl Alyss D Muhem Mecmek 8 Guss Elmto fo Tdgol System Specl Le System Asg Applcto A geel tdgol mt s mt whose ozeo elemets e foud oly o the dgol sudgol d supedgol of the mt.

19 ELM Numecl Alyss D Muhem Mecmek 9 Guss Elmto fo Tdgol System * Emple of Tdgol Mt = + = + = + =. We c tke dvtge of the zeo elemets tht e ledy peset the coeffcet mt d vods uecessy thmetc opetos. Thus we eed to stoe oly the ew vectos d.

20 .... )() ( ; )() ( ; )() ( ; Guss Elmto fo Tdgol System... ELM Numecl Alyss D Muhem Mecmek Emple 6:

21 d d d d d... Step : Fo the fst equto Step : Fo ech of the equto Step : Fo the lst equto Step : y ck susttuto Thoms Method ELM Numecl Alyss D Muhem Mecmek

22 ELM Numecl Alyss D Muhem Mecmek Thoms Method Emple 7: (6) d (); ( );. ( ); ().

23 . d d. ) / )( ( ) )(/ ( ) / )( ( d d. )() / ( / )() / ( / ) / ( / ELM Numecl Alyss D Muhem Mecmek Thoms Method

Advanced Algorithmic Problem Solving Le 3 Arithmetic. Fredrik Heintz Dept of Computer and Information Science Linköping University

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

More information

Lecture 3-4 Solutions of System of Linear Equations

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 information

ITERATIVE METHODS FOR SOLVING SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS

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 information

5 - Determinants. r r. r r. r r. r s r = + det det det

5 - 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 information

ME 501A Seminar in Engineering Analysis Page 1

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

More information

Chapter Linear Regression

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

More information

( m is the length of columns of A ) spanned by the columns of A : . Select those columns of B that contain a pivot; say those are Bi

( m is the length of columns of A ) spanned by the columns of A : . Select those columns of B that contain a pivot; say those are Bi Assgmet /MATH 47/Wte Due: Thusday Jauay The poblems to solve ae umbeed [] to [] below Fst some explaatoy otes Fdg a bass of the colum-space of a max ad povg that the colum ak (dmeso of the colum space)

More information

Chapter 17. Least Square Regression

Chapter 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 information

Describes techniques to fit curves (curve fitting) to discrete data to obtain intermediate estimates.

Describes 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 information

= y and Normed Linear Spaces

= y and Normed Linear Spaces 304-50 LINER SYSTEMS Lectue 8: Solutos to = ad Nomed Lea Spaces 73 Fdg N To fd N, we eed to chaacteze all solutos to = 0 Recall that ow opeatos peseve N, so that = 0 = 0 We ca solve = 0 ecusvel backwads

More information

Chapter Gauss-Seidel Method

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

More information

Chapter Unary Matrix Operations

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

More information

CS537. Numerical Analysis

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

More information

2. Elementary Linear Algebra Problems

2. Elementary Linear Algebra Problems . Eleety e lge Pole. BS: B e lge Suoute (Pog pge wth PCK) Su of veto opoet:. Coputto y f- poe: () () () (3) N 3 4 5 3 6 4 7 8 Full y tee Depth te tep log()n Veto updte the f- poe wth N : ) ( ) ( ) ( )

More information

SEPTIC B-SPLINE COLLOCATION METHOD FOR SIXTH ORDER BOUNDARY VALUE PROBLEMS

SEPTIC B-SPLINE COLLOCATION METHOD FOR SIXTH ORDER BOUNDARY VALUE PROBLEMS VOL. 5 NO. JULY ISSN 89-8 RN Joul of Egeeg d ppled Sceces - s Resech ulshg Netok RN. ll ghts eseved..pouls.com SETIC -SLINE COLLOCTION METHOD FOR SIXTH ORDER OUNDRY VLUE ROLEMS K.N.S. Ks Vsdhm d. Mul Ksh

More information

7.5-Determinants in Two Variables

7.5-Determinants in Two Variables 7.-eteminnts in Two Vibles efinition of eteminnt The deteminnt of sque mti is el numbe ssocited with the mti. Eve sque mti hs deteminnt. The deteminnt of mti is the single ent of the mti. The deteminnt

More information

PubH 7405: REGRESSION ANALYSIS REGRESSION IN MATRIX TERMS

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

More information

Computer Programming

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

More information

ICS141: Discrete Mathematics for Computer Science I

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

More information

Level-2 BLAS. Matrix-Vector operations with O(n 2 ) operations (sequentially) BLAS-Notation: S --- single precision G E general matrix M V --- vector

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

More information

Chapter 2 Intro to Math Techniques for Quantum Mechanics

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...

More information

CS321. Introduction to Numerical Methods

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

More information

Chapter 2 Solving Linear Equation

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

More information

INTERPOLATION(2) ELM1222 Numerical Analysis. ELM1222 Numerical Analysis Dr Muharrem Mercimek

INTERPOLATION(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 information

Sequences and summations

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

More information

ANOTHER INTEGER NUMBER ALGORITHM TO SOLVE LINEAR EQUATIONS (USING CONGRUENCY)

ANOTHER INTEGER NUMBER ALGORITHM TO SOLVE LINEAR EQUATIONS (USING CONGRUENCY) ANOTHER INTEGER NUMBER ALGORITHM TO SOLVE LINEAR EQUATIONS (USING CONGRUENCY) Floet Smdche, Ph D Aocte Pofeo Ch of Deptmet of Mth & Scece Uvety of New Mexco 2 College Rod Gllup, NM 873, USA E-ml: md@um.edu

More information

VECTOR MECHANICS FOR ENGINEERS: Vector Mechanics for Engineers: Dynamics. In the current chapter, you will study the motion of systems of particles.

VECTOR MECHANICS FOR ENGINEERS: Vector Mechanics for Engineers: Dynamics. In the current chapter, you will study the motion of systems of particles. Seeth Edto CHPTER 4 VECTOR MECHNICS FOR ENINEERS: DYNMICS Fedad P. ee E. Russell Johsto, J. Systems of Patcles Lectue Notes: J. Walt Ole Texas Tech Uesty 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed. Seeth

More information

Previously. Extensions to backstepping controller designs. Tracking using backstepping Suppose we consider the general system

Previously. Extensions to backstepping controller designs. Tracking using backstepping Suppose we consider the general system 436-459 Advnced contol nd utomtion Extensions to bckstepping contolle designs Tcking Obseves (nonline dmping) Peviously Lst lectue we looked t designing nonline contolles using the bckstepping technique

More information

SOME REMARKS ON HORIZONTAL, SLANT, PARABOLIC AND POLYNOMIAL ASYMPTOTE

SOME 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 information

A Technique for Constructing Odd-order Magic Squares Using Basic Latin Squares

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

More information

means the first term, a2 means the term, etc. Infinite Sequences: follow the same pattern forever.

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,

More information

DATA FITTING. Intensive Computation 2013/2014. Annalisa Massini

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

More information

CURVE FITTING LEAST SQUARES METHOD

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

More information

CHAPTER 4 RADICAL EXPRESSIONS

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

More information

GCE AS/A Level MATHEMATICS GCE AS/A Level FURTHER MATHEMATICS

GCE AS/A Level MATHEMATICS GCE AS/A Level FURTHER MATHEMATICS GCE AS/A Level MATHEMATICS GCE AS/A Level FURTHER MATHEMATICS FORMULA BOOKLET Fom Septembe 07 Issued 07 Mesuto Pue Mthemtcs Sufce e of sphee = 4 Ae of cuved sufce of coe = slt heght Athmetc Sees S l d

More information

Mu Sequences/Series Solutions National Convention 2014

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

More information

RECAPITULATION & CONDITIONAL PROBABILITY. Number of favourable events n E Total number of elementary events n S

RECAPITULATION & CONDITIONAL PROBABILITY. Number of favourable events n E Total number of elementary events n S Fomulae Fo u Pobablty By OP Gupta [Ida Awad We, +91-9650 350 480] Impotat Tems, Deftos & Fomulae 01 Bascs Of Pobablty: Let S ad E be the sample space ad a evet a expemet espectvely Numbe of favouable evets

More information

12 Iterative Methods. Linear Systems: Gauss-Seidel Nonlinear Systems Case Study: Chemical Reactions

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

More information

GCE AS and A Level MATHEMATICS FORMULA BOOKLET. From September Issued WJEC CBAC Ltd.

GCE AS and A Level MATHEMATICS FORMULA BOOKLET. From September Issued WJEC CBAC Ltd. GCE AS d A Level MATHEMATICS FORMULA BOOKLET Fom Septeme 07 Issued 07 Pue Mthemtcs Mesuto Suce e o sphee = 4 Ae o cuved suce o coe = heght slt Athmetc Sees S = + l = [ + d] Geometc Sees S = S = o < Summtos

More information

SOLVING SYSTEMS OF EQUATIONS, ITERATIVE METHODS

SOLVING SYSTEMS OF EQUATIONS, ITERATIVE METHODS ELM Numericl Anlysis Dr Muhrrem Mercimek SOLVING SYSTEMS OF EQUATIONS, ITERATIVE METHODS ELM Numericl Anlysis Some of the contents re dopted from Lurene V. Fusett, Applied Numericl Anlysis using MATLAB.

More information

PROGRESSION AND SERIES

PROGRESSION AND SERIES INTRODUCTION PROGRESSION AND SERIES A gemet of umbes {,,,,, } ccodig to some well defied ule o set of ules is clled sequece Moe pecisely, we my defie sequece s fuctio whose domi is some subset of set of

More information

BINOMIAL THEOREM SOLUTION. 1. (D) n. = (C 0 + C 1 x +C 2 x C n x n ) (1+ x+ x 2 +.)

BINOMIAL THEOREM SOLUTION. 1. (D) n. = (C 0 + C 1 x +C 2 x C n x n ) (1+ x+ x 2 +.) BINOMIAL THEOREM SOLUTION. (D) ( + + +... + ) (+ + +.) The coefficiet of + + + +... + fo. Moeove coefficiet of is + + + +... + if >. So. (B)... e!!!! The equied coefficiet coefficiet of i e -.!...!. (A),

More information

Kinematics. Redundancy. Task Redundancy. Operational Coordinates. Generalized Coordinates. m task. Manipulator. Operational point

Kinematics. Redundancy. Task Redundancy. Operational Coordinates. Generalized Coordinates. m task. Manipulator. Operational point Mapulato smatc Jot Revolute Jot Kematcs Base Lks: movg lk fed lk Ed-Effecto Jots: Revolute ( DOF) smatc ( DOF) Geealzed Coodates Opeatoal Coodates O : Opeatoal pot 5 costats 6 paametes { postos oetatos

More information

Autar Kaw Benjamin Rigsby. Transforming Numerical Methods Education for STEM Undergraduates

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

More information

Chapter I Vector Analysis

Chapter I Vector Analysis . Chpte I Vecto nlss . Vecto lgeb j It s well-nown tht n vecto cn be wtten s Vectos obe the followng lgebc ules: scl s ) ( j v v cos ) ( e Commuttv ) ( ssoctve C C ) ( ) ( v j ) ( ) ( ) ( ) ( (v) he lw

More information

INTRODUCTION ( ) 1. Errors

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

More information

CS473-Algorithms I. Lecture 3. Solving Recurrences. Cevdet Aykanat - Bilkent University Computer Engineering Department

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

More information

The formulae in this booklet have been arranged according to the unit in which they are first

The formulae in this booklet have been arranged according to the unit in which they are first Fomule Booklet Fomule Booklet The fomule ths ooklet hve ee ge ccog to the ut whch the e fst touce. Thus cte sttg ut m e eque to use the fomule tht wee touce peceg ut e.g. ctes sttg C mght e epecte to use

More information

Parallel Newtonian Optimization without Hessian Approximation. Khalil K. Abbo College of Computer sciences and Mathematics University of Mosul

Parallel Newtonian Optimization without Hessian Approximation. Khalil K. Abbo College of Computer sciences and Mathematics University of Mosul f. J. of Comp. & Mth s., Vol., No., 6 Pllel Newto Optmzto wthout ess Appomto Collee of Compute sceces d Mthemtcs Uvesty of Mosul eceved o: /9/5 Accepted o: 6//5 الملخص الغرض من هذا البحث هو اقتراح خوارزمية

More information

3. REVIEW OF PROPERTIES OF EIGENVALUES AND EIGENVECTORS

3. 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 information

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

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

More information

Synthesis of Stable Takagi-Sugeno Fuzzy Systems

Synthesis of Stable Takagi-Sugeno Fuzzy Systems Sytess of Stle Tk-Sueo Fuzzy Systems ENATA PYTELKOVÁ AND PET HUŠEK Deptmet of Cotol Eee Fculty of Electcl Eee, Czec Teccl Uvesty Teccká, 66 7 P 6 CZECH EPUBLIC Astct: - Te ppe dels t te polem of sytess

More information

Objectives. Learning Outcome. 7.1 Centre of Gravity (C.G.) 7. Statics. Determine the C.G of a lamina (Experimental method)

Objectives. Learning Outcome. 7.1 Centre of Gravity (C.G.) 7. Statics. Determine the C.G of a lamina (Experimental method) Ojectves 7 Statcs 7. Cete of Gavty 7. Equlum of patcles 7.3 Equlum of g oes y Lew Sau oh Leag Outcome (a) efe cete of gavty () state the coto whch the cete of mass s the cete of gavty (c) state the coto

More information

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 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 information

2006 Jamie Trahan, Autar Kaw, Kevin Martin University of South Florida United States of America

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

More information

On EPr Bimatrices II. ON EP BIMATRICES A1 A Hence x. is said to be EP if it satisfies the condition ABx

On EPr Bimatrices II. ON EP BIMATRICES A1 A Hence x. is said to be EP if it satisfies the condition ABx Iteatoal Joual of Mathematcs ad Statstcs Iveto (IJMSI) E-ISSN: 3 4767 P-ISSN: 3-4759 www.jms.og Volume Issue 5 May. 4 PP-44-5 O EP matces.ramesh, N.baas ssocate Pofesso of Mathematcs, ovt. ts College(utoomous),Kumbakoam.

More information

Stats & Summary

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

More information

Numerical Methods for Eng [ENGR 391] [Lyes KADEM 2007] Direct Method; Newton s Divided Difference; Lagrangian Interpolation; Spline Interpolation.

Numerical Methods for Eng [ENGR 391] [Lyes KADEM 2007] Direct Method; Newton s Divided Difference; Lagrangian Interpolation; Spline Interpolation. Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 CHAPTER V Itepolto d Regesso Topcs Itepolto Regesso Dect Method; Newto s Dvded Deece; Lgg Itepolto; ple Itepolto Le d o-le Wht s tepolto? A ucto s ote gve ol t

More information

Summary: Binomial Expansion...! r. where

Summary: Binomial Expansion...! r. where Summy: Biomil Epsio 009 M Teo www.techmejcmth-sg.wes.com ) Re-cp of Additiol Mthemtics Biomil Theoem... whee )!!(! () The fomul is ville i MF so studets do ot eed to memoise it. () The fomul pplies oly

More information

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 [ ] 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 information

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.

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,

More information

MATRIX AND VECTOR NORMS

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

More information

such that for 1 From the definition of the k-fibonacci numbers, the firsts of them are presented in Table 1. Table 1: First k-fibonacci numbers F 1

such that for 1 From the definition of the k-fibonacci numbers, the firsts of them are presented in Table 1. Table 1: First k-fibonacci numbers F 1 Scholas Joual of Egeeg ad Techology (SJET) Sch. J. Eg. Tech. 0; (C):669-67 Scholas Academc ad Scetfc Publshe (A Iteatoal Publshe fo Academc ad Scetfc Resouces) www.saspublshe.com ISSN -X (Ole) ISSN 7-9

More information

Chapter 9 Jordan Block Matrices

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.

More information

On Several Inequalities Deduced Using a Power Series Approach

On Several Inequalities Deduced Using a Power Series Approach It J Cotemp Mth Sceces, Vol 8, 203, o 8, 855-864 HIKARI Ltd, wwwm-hrcom http://dxdoorg/02988/jcms2033896 O Severl Iequltes Deduced Usg Power Seres Approch Lored Curdru Deprtmet of Mthemtcs Poltehc Uversty

More information

CBSE , ˆj. cos CBSE_2015_SET-1. SECTION A 1. Given that a 2iˆ ˆj. We need to find. 3. Consider the vector equation of the plane.

CBSE , ˆj. cos CBSE_2015_SET-1. SECTION A 1. Given that a 2iˆ ˆj. We need to find. 3. Consider the vector equation of the plane. CBSE CBSE SET- SECTION. Gv tht d W d to fd 7 7 Hc, 7 7 7. Lt,. W ow tht.. Thus,. Cosd th vcto quto of th pl.. z. - + z = - + z = Thus th Cts quto of th pl s - + z = Lt d th dstc tw th pot,, - to th pl.

More information

Class 13,14 June 17, 19, 2015

Class 13,14 June 17, 19, 2015 Class 3,4 Jue 7, 9, 05 Pla for Class3,4:. Samplg dstrbuto of sample mea. The Cetral Lmt Theorem (CLT). Cofdece terval for ukow mea.. Samplg Dstrbuto for Sample mea. Methods used are based o CLT ( Cetral

More information

Section 35 SHM and Circular Motion

Section 35 SHM and Circular Motion Section 35 SHM nd Cicul Motion Phsics 204A Clss Notes Wht do objects do? nd Wh do the do it? Objects sometimes oscillte in simple hmonic motion. In the lst section we looed t mss ibting t the end of sping.

More information

ECON 5360 Class Notes GMM

ECON 5360 Class Notes GMM ECON 560 Class Notes GMM Geeralzed Method of Momets (GMM) I beg by outlg the classcal method of momets techque (Fsher, 95) ad the proceed to geeralzed method of momets (Hase, 98).. radtoal Method of Momets

More information

Roberto s Notes on Integral Calculus Chapter 4: Definite integrals and the FTC Section 2. Riemann sums

Roberto s Notes on Integral Calculus Chapter 4: Definite integrals and the FTC Section 2. Riemann sums Roerto s Notes o Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 2 Rem sums Wht you eed to kow lredy: The defto of re for rectgle. Rememer tht our curret prolem s how to compute the re of ple rego

More information

under the curve in the first quadrant.

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 ( )

More information

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. 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 information

Section 7.3, Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors (the variable vector of the system) and

Section 7.3, Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors (the variable vector of the system) and Sec. 7., Boyce & DiPrim, p. Sectio 7., Systems of Lier Algeric Equtios; Lier Idepedece, Eigevlues, Eigevectors I. Systems of Lier Algeric Equtios.. We c represet the system...... usig mtrices d vectors

More information

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 ] 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 information

6.6 The Marquardt Algorithm

6.6 The Marquardt Algorithm 6.6 The Mqudt Algothm lmttons of the gdent nd Tylo expnson methods ecstng the Tylo expnson n tems of ch-sque devtves ecstng the gdent sech nto n tetve mtx fomlsm Mqudt's lgothm utomtclly combnes the gdent

More information

THE ROYAL STATISTICAL SOCIETY 2016 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5

THE ROYAL STATISTICAL SOCIETY 2016 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5 THE ROYAL STATISTICAL SOCIETY 06 EAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5 The Socety s provdg these solutos to assst cadtes preparg for the examatos 07. The solutos are teded as learg ads ad should

More information

Neville Robbins Mathematics Department, San Francisco State University, San Francisco, CA (Submitted August 2002-Final Revision December 2002)

Neville Robbins Mathematics Department, San Francisco State University, San Francisco, CA (Submitted August 2002-Final Revision December 2002) Nevlle Robbs Mathematcs Departmet, Sa Fracsco State Uversty, Sa Fracsco, CA 943 (Submtted August -Fal Revso December ) INTRODUCTION The Lucas tragle s a fte tragular array of atural umbers that s a varat

More information

XII. Addition of many identical spins

XII. Addition of many identical spins XII. Addto of may detcal sps XII.. ymmetc goup ymmetc goup s the goup of all possble pemutatos of obects. I total! elemets cludg detty opeato. Each pemutato s a poduct of a ceta fte umbe of pawse taspostos.

More information

Mathematically, integration is just finding the area under a curve from one point to another. It is b

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

More information

Chapter 2 Intro to Math Techniques for Quantum Mechanics

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...

More information

Algorithms Theory, Solution for Assignment 2

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

More information

Minimizing spherical aberrations Exploiting the existence of conjugate points in spherical lenses

Minimizing spherical aberrations Exploiting the existence of conjugate points in spherical lenses Mmzg sphecal abeatos Explotg the exstece of cojugate pots sphecal leses Let s ecall that whe usg asphecal leses, abeato fee magg occus oly fo a couple of, so called, cojugate pots ( ad the fgue below)

More information

1 Onto functions and bijections Applications to Counting

1 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 information

10.3 The Quadratic Formula

10.3 The Quadratic Formula . Te Qudti Fomul We mentioned in te lst setion tt ompleting te sque n e used to solve ny qudti eqution. So we n use it to solve 0. We poeed s follows 0 0 Te lst line of tis we ll te qudti fomul. Te Qudti

More information

Rendering Equation. Linear equation Spatial homogeneous Both ray tracing and radiosity can be considered special case of this general eq.

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

More information

PGE 310: Formulation and Solution in Geosystems Engineering. Dr. Balhoff. Interpolation

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

More information

Numerical Differentiation and Integration

Numerical 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 information

Dr. Shalabh. Indian Institute of Technology Kanpur

Dr. Shalabh. Indian Institute of Technology Kanpur Aalyss of Varace ad Desg of Expermets-I MODULE -I LECTURE - SOME RESULTS ON LINEAR ALGEBRA, MATRIX THEORY AND DISTRIBUTIONS Dr. Shalabh Departmet t of Mathematcs t ad Statstcs t t Ida Isttute of Techology

More information

Lattice planes. Lattice planes are usually specified by giving their Miller indices in parentheses: (h,k,l)

Lattice planes. Lattice planes are usually specified by giving their Miller indices in parentheses: (h,k,l) Ltte ples Se the epol ltte of smple u ltte s g smple u ltte d the Mlle des e the oodtes of eto oml to the ples, the use s ey smple lttes wth u symmety. Ltte ples e usully spefed y gg the Mlle des petheses:

More information

Professor Wei Zhu. 1. Sampling from the Normal Population

Professor Wei Zhu. 1. Sampling from the Normal Population AMS570 Pofesso We Zhu. Samplg fom the Nomal Populato *Example: We wsh to estmate the dstbuto of heghts of adult US male. It s beleved that the heght of adult US male follows a omal dstbuto N(, ) Def. Smple

More information

MATH 371 Homework assignment 1 August 29, 2013

MATH 371 Homework assignment 1 August 29, 2013 MATH 371 Homework assgmet 1 August 29, 2013 1. Prove that f a subset S Z has a smallest elemet the t s uque ( other words, f x s a smallest elemet of S ad y s also a smallest elemet of S the x y). We kow

More information

Chapter 3 Supplemental Text Material

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

More information

Numerical Analysis Topic 4: Least Squares Curve Fitting

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

More information

148 CIVIL ENGINEERING

148 CIVIL ENGINEERING STRUTUR NYSS fluee es fo Bems d Tusses fluee le sows te vto of effet (eto, se d momet ems, foe tuss) used movg ut lod oss te stutue. fluee le s used to deteme te posto of movele set of lods tt uses te

More information

In Calculus I you learned an approximation method using a Riemann sum. Recall that the Riemann sum is

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

More information

The formulae in this booklet have been arranged according to the unit in which they are first

The formulae in this booklet have been arranged according to the unit in which they are first Fomule Booklet Fomule Booklet The fomule ths ooklet hve ee ge og to the ut whh the e fst toue. Thus te sttg ut m e eque to use the fomule tht wee toue peeg ut e.g. tes sttg C mght e epete to use fomule

More information

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.

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

More information

Lesson 4 Linear Algebra

Lesson 4 Linear Algebra Lesso Lier Algebr A fmily of vectors is lierly idepedet if oe of them c be writte s lier combitio of fiitely my other vectors i the collectio. Cosider m lierly idepedet equtios i ukows:, +, +... +, +,

More information

Lecture Notes 2. The ability to manipulate matrices is critical in economics.

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

More information

Bond Additive Modeling 5. Mathematical Properties of the Variable Sum Exdeg Index

Bond 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 information