CHAPTER 5d. SIMULTANEOUS LINEAR EQUATIONS

Size: px
Start display at page:

Download "CHAPTER 5d. SIMULTANEOUS LINEAR EQUATIONS"

Transcription

1 CHAPTE 5. SIUTANEOUS INEA EQUATIONS A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig by Dr. Ibrhim A. Asskkf Sprig ENCE - Compttio thos i Civi Egirig II Dprtmt of Civi Eviromt Egirig Uivrsity of ry, Cog Prk Gss-Jor Eimitio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig W sw tht th Gssi Eimitio cosists of two stps: Forwr Pss Bck Sbstittio For th foowig st of qtio: [ ]{ } { C } () ij i i ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No.

2 Asskkf Si No. A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Gss-Jor Eimitio Th Forwr Pss c wys rst ito gmt mtrix i th gr form s () Asskkf Si No. A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Gss-Jor Eimitio Th Bck Sbstittio c wys rst ito th foowig gr form:,,,, ()

3 Gss-Jor Eimitio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Gss-Jor Procss A trtiv procss of imitio i which cofficits i com xcpt th pivot mt r imit c so b s to obti sotio. I Gss-Jor imitio, th sotio is obti ircty ftr prformig th forwr pss. Thr is o bck sbstittio. ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 5 Gss-Jor Eimitio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Gss-Jor Procss Giv th foowig st of qtio: C C C C () ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 6

4 Asskkf Si No. 7 A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Gss-Jor Eimitio Gss-Jor Procss Or giv th foowig st of qtio i gmt mtrix of th form: C C C C (5) Asskkf Si No. 8 A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Gss-Jor Eimitio Gss-Jor Procss Usig Gss-Jor, Eq. 5 c b trsform ito th foowig form: C C C C O (6)

5 Gss-Jor Eimitio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Gss-Jor Procss A i which th sotio is riy obti s C C C C (7) ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 9 Gss-Jor Eimitio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Exmp: Gss-Jor Sov th foowig st of simtos qtios sig th Gss-Jor tho: ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 5

6 Gss-Jor Eimitio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Exmp (cot ): Gss-Jor This systm of qtios c b rprst i mtrix form s ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. Gss-Jor Eimitio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Exmp (cot ): Gss-Jor Or this systm of qtios c b rprst i gmt mtrix form s ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 6

7 7 Asskkf Si No. A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Gss-Jor Eimitio Exmp (cot ): Gss-Jor Stp : (-) 6 : : : (-) : 9 7 : Asskkf Si No. A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Gss-Jor Eimitio Exmp (cot ): Gss-Jor Stp 7 / / / 5 / / : (-) / : 5 : 7/ : ) ( 5 / 5 : :

8 8 Asskkf Si No. 5 A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Gss-Jor Eimitio Exmp (cot ): Gss-Jor Stp ) /(7 / / / 7 / / / : (-) / : / / : : (-) / : / / : Asskkf Si No. 6 A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Gss-Jor Eimitio Exmp (cot ): Gss-Jor Th st gmt mtrix givs th sotio for th systm of qtios s foows:

9 Gss-Jor Eimitio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Exmp (cot ): Gss-Jor O ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 7 U Dcompositio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig c tht th Gssi imitio procss cosists of two stps: Forwr Pss Bck Sbstittio Th forwr pss trsforms th systm of qtios ito ppr trigr mtrix. ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 8 9

10 Asskkf Si No. 9 A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS U Dcompositio Th ppr trigr mtrix c b ot s U it is giv by (8) Asskkf Si No. A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS U Dcompositio c th gmt mtrix tht c rst from th forwr pss of Gssi Eimitio: (9)

11 U Dcompositio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Th vs of this mtrix (Eq. 8) c b rt to th vs i Eq. 9 s ij ij for i,,, j,,, Th ppr trigr mtrix U c b rt to th cofficits mtrix A s ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. U Dcompositio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig U A () Whr owr trigr mtrix, which c b viw s to prst th bck sbstittio () ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No.

12 U Dcompositio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Thrfor Eq. c b xprss s U A () ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. U Dcompositio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Eq. stts tht th cofficit mtrix A c b compos to two trigr mtrics U. Ths trigr mtrics c b trmi withot th s of Gssi imitio mtho by prformig th mtrix mtipictio U i Eq. ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No.

13 Asskkf Si No. 5 A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS U Dcompositio tipictio of U wi rst i () Asskkf Si No. 6 A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS U Dcompositio tipictio of U wi rst i ()

14 U Dcompositio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig For xmp, th mtipictio of th first row of with th first com of U rsts th foowig v tht is q to. ( ) ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 7 U Dcompositio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig I viw of Eq., for xmp ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 8

15 U Dcompositio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig From Eq., th vs for U c b compt from: for i,, K, i j ij ji i ij j ji j ik k j jk k jj k k kj ki k for j,, K, (5) for j,, K, i j, j, K, for j,, K, i j, j, K, ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 9 U Dcompositio A. J. Crk Schoo of Egirig Dprtmt of Civi Eviromt Egirig Oc th cofficit mtrix is compos ito U, ths mtrics c b s to sov systm of qtios with costts C i, i,,,. Th sotio c b obti i two stps, forwr pss bck sbstittio. ENCE CHAPTE 5. SIUTANEOUS INEA EQUATIONS Asskkf Si No. 5

LU FACTORIZATION. ELM1222 Numerical Analysis. ELM1222 Numerical Analysis Dr Muharrem Mercimek

LU FACTORIZATION. ELM1222 Numerical Analysis. ELM1222 Numerical Analysis Dr Muharrem Mercimek EM Nmeric Asis Dr Mhrrem Mercimek FACTORIZATION EM Nmeric Asis Some of the cotets re dopted from ree V. Fsett, Appied Nmeric Asis sig MATAB. Pretice H Ic., 999 EM Nmeric Asis Dr Mhrrem Mercimek Cotets

More information

Linear Algebra Existence of the determinant. Expansion according to a row.

Linear Algebra Existence of the determinant. Expansion according to a row. Lir Algbr 2270 1 Existc of th dtrmit. Expsio ccordig to row. W dfi th dtrmit for 1 1 mtrics s dt([]) = (1) It is sy chck tht it stisfis D1)-D3). For y othr w dfi th dtrmit s follows. Assumig th dtrmit

More information

Vtusolution.in FOURIER SERIES. Dr.A.T.Eswara Professor and Head Department of Mathematics P.E.S.College of Engineering Mandya

Vtusolution.in FOURIER SERIES. Dr.A.T.Eswara Professor and Head Department of Mathematics P.E.S.College of Engineering Mandya LECTURE NOTES OF ENGINEERING MATHEMATICS III Su Cod: MAT) Vtusoutio.i COURSE CONTENT ) Numric Aysis ) Fourir Sris ) Fourir Trsforms & Z-trsforms ) Prti Diffrti Equtios 5) Lir Agr 6) Ccuus of Vritios Tt

More information

Q.28 Q.29 Q.30. Q.31 Evaluate: ( log x ) Q.32 Evaluate: ( ) Q.33. Q.34 Evaluate: Q.35 Q.36 Q.37 Q.38 Q.39 Q.40 Q.41 Q.42. Q.43 Evaluate : ( x 2) Q.

Q.28 Q.29 Q.30. Q.31 Evaluate: ( log x ) Q.32 Evaluate: ( ) Q.33. Q.34 Evaluate: Q.35 Q.36 Q.37 Q.38 Q.39 Q.40 Q.41 Q.42. Q.43 Evaluate : ( x 2) Q. LASS XII Q Evlut : Q sc Evlut c Q Evlut: ( ) Q Evlut: Q5 α Evlut: α Q Evlut: Q7 Evlut: { t (t sc )} / Q8 Evlut : ( )( ) Q9 Evlut: Q0 Evlut: Q Evlut : ( ) ( ) Q Evlut : / ( ) Q Evlut: / ( ) Q Evlut : )

More information

Problem Session (3) for Chapter 4 Signal Modeling

Problem Session (3) for Chapter 4 Signal Modeling Pobm Sssio fo Cht Sig Modig Soutios to Pobms....5. d... Fid th Pdé oimtio of scod-od to sig tht is giv by [... ] T i.. d so o. I oth wods usig oimtio of th fom b b b H fid th cofficits b b b d. Soutio

More information

SOLVING FUZZY LINEAR PROGRAMMING PROBLEM USING SUPPORT AND CORE OF FUZZY NUMBERS

SOLVING FUZZY LINEAR PROGRAMMING PROBLEM USING SUPPORT AND CORE OF FUZZY NUMBERS Itertio Jor of Scietific Reserch Egieerig & Techoogy (IJSRET ISSN 78 88 Vome 6 Isse 4 pri 7 44 SOLVING FUZZY LINER PROGRMMING PROBLEM USING SUPPORT ND CORE OF FUZZY NUMBERS Dr.S.Rmthigm K.Bmrg sst. Professor

More information

Chapter 3 Fourier Series Representation of Periodic Signals

Chapter 3 Fourier Series Representation of Periodic Signals Chptr Fourir Sris Rprsttio of Priodic Sigls If ritrry sigl x(t or x[] is xprssd s lir comitio of som sic sigls th rspos of LI systm coms th sum of th idividul rsposs of thos sic sigls Such sic sigl must:

More information

COLLECTION OF SUPPLEMENTARY PROBLEMS CALCULUS II

COLLECTION OF SUPPLEMENTARY PROBLEMS CALCULUS II COLLECTION OF SUPPLEMENTARY PROBLEMS I. CHAPTER 6 --- Trscdtl Fuctios CALCULUS II A. FROM CALCULUS BY J. STEWART:. ( How is th umbr dfid? ( Wht is pproimt vlu for? (c ) Sktch th grph of th turl potil fuctios.

More information

[ ] Review. For a discrete-time periodic signal xn with period N, the Fourier series representation is

[ ] Review. For a discrete-time periodic signal xn with period N, the Fourier series representation is Discrt-tim ourir Trsform Rviw or discrt-tim priodic sigl x with priod, th ourir sris rprsttio is x + < > < > x, Rviw or discrt-tim LTI systm with priodic iput sigl, y H ( ) < > < > x H r rfrrd to s th

More information

National Quali cations

National Quali cations PRINT COPY OF BRAILLE Ntiol Quli ctios AH08 X747/77/ Mthmtics THURSDAY, MAY INSTRUCTIONS TO CANDIDATES Cdidts should tr thir surm, form(s), dt of birth, Scottish cdidt umbr d th m d Lvl of th subjct t

More information

ME 522 PRINCIPLES OF ROBOTICS. FIRST MIDTERM EXAMINATION April 19, M. Kemal Özgören

ME 522 PRINCIPLES OF ROBOTICS. FIRST MIDTERM EXAMINATION April 19, M. Kemal Özgören ME 522 PINCIPLES OF OBOTICS FIST MIDTEM EXAMINATION April 9, 202 Nm Lst Nm M. Kml Özgörn 2 4 60 40 40 0 80 250 USEFUL FOMULAS cos( ) cos cos sin sin sin( ) sin cos cos sin sin y/ r, cos x/ r, r 0 tn 2(

More information

Krein's method and mixed integral equation of Volterra Fredholm type. R. T. Matoog

Krein's method and mixed integral equation of Volterra Fredholm type. R. T. Matoog Krei's method d mied itegr eqtio of Voterr Fredhom type R T Mtoog Deprtmet of Mthemtics Fcty of Appied Scieces Umm A-Qr Uiersity Mkkh Sdi Arbi PO Bo 7 rmtoog@yhoocom Abstrct: Here the eistece of iqe sotio

More information

IX. Ordinary Differential Equations

IX. Ordinary Differential Equations IX. Orir Diffrtil Equtios A iffrtil qutio is qutio tht iclus t lst o rivtiv of uow fuctio. Ths qutios m iclu th uow fuctio s wll s ow fuctios of th sm vribl. Th rivtiv m b of orr thr m b svrl rivtivs prst.

More information

National Quali cations

National Quali cations Ntiol Quli ctios AH07 X77/77/ Mthmtics FRIDAY, 5 MAY 9:00 AM :00 NOON Totl mrks 00 Attmpt ALL qustios. You my us clcultor. Full crdit will b giv oly to solutios which coti pproprit workig. Stt th uits

More information

1 Multiply Eq. E i by λ 0: (λe i ) (E i ) 2 Multiply Eq. E j by λ and add to Eq. E i : (E i + λe j ) (E i )

1 Multiply Eq. E i by λ 0: (λe i ) (E i ) 2 Multiply Eq. E j by λ and add to Eq. E i : (E i + λe j ) (E i ) Direct Methods for Linear Systems Chapter Direct Methods for Solving Linear Systems Per-Olof Persson persson@berkeleyedu Department of Mathematics University of California, Berkeley Math 18A Numerical

More information

Some Common Fixed Point Theorems for a Pair of Non expansive Mappings in Generalized Exponential Convex Metric Space

Some Common Fixed Point Theorems for a Pair of Non expansive Mappings in Generalized Exponential Convex Metric Space Mish Kumr Mishr D.B.OhU Ktoch It. J. Comp. Tch. Appl. Vol ( 33-37 Som Commo Fi Poit Thorms for Pir of No psiv Mppigs i Grliz Epotil Cov Mtric Spc D.B.Oh Mish Kumr Mishr U Ktoch (Rsrch scholr Drvii Uivrsit

More information

Matrix Algebra & Elementary Matrices

Matrix Algebra & Elementary Matrices Matrix lgebra & Elementary Matrices To add two matrices, they must have identical dimensions. To multiply them the number of columns of the first must equal the number of rows of the second. The laws below

More information

LU Factorization. LU Decomposition. LU Decomposition. LU Decomposition: Motivation A = LU

LU Factorization. LU Decomposition. LU Decomposition. LU Decomposition: Motivation A = LU LU Factorization To further improve the efficiency of solving linear systems Factorizations of matrix A : LU and QR LU Factorization Methods: Using basic Gaussian Elimination (GE) Factorization of Tridiagonal

More information

Advanced Higher Grade

Advanced Higher Grade Prelim Emitio / (Assessig Uits & ) MATHEMATICS Avce Higher Gre Time llowe - hors Re Crefll. Fll creit will be give ol where the soltio cotis pproprite workig.. Clcltors m be se i this pper.. Aswers obtie

More information

Emil Olteanu-The plane rotation operator as a matrix function THE PLANE ROTATION OPERATOR AS A MATRIX FUNCTION. by Emil Olteanu

Emil Olteanu-The plane rotation operator as a matrix function THE PLANE ROTATION OPERATOR AS A MATRIX FUNCTION. by Emil Olteanu Emil Oltu-Th pl rottio oprtor s mtri fuctio THE PLNE ROTTON OPERTOR S MTRX UNTON b Emil Oltu bstrct ormlism i mthmtics c offr m simplifictios, but it is istrumt which should b crfull trtd s it c sil crt

More information

ECS 178 Course Notes QUATERNIONS

ECS 178 Course Notes QUATERNIONS ECS 178 Course Notes QUATERNIONS Kenneth I. Joy Institute for Data Analysis and Visualization Department of Computer Science University of California, Davis Overview The quaternion number system was discovered

More information

How much air is required by the people in this lecture theatre during this lecture?

How much air is required by the people in this lecture theatre during this lecture? 3 NTEGRATON tgrtio is us to swr qustios rltig to Ar Volum Totl qutity such s: Wht is th wig r of Boig 747? How much will this yr projct cost? How much wtr os this rsrvoir hol? How much ir is rquir y th

More information

Fitting affine and orthogonal transformations between two sets of points

Fitting affine and orthogonal transformations between two sets of points Mathematica Communications 9(2004), 27-34 27 Fitting affine and orthogona transformations between two sets of points Hemuth Späth Abstract. Let two point sets P and Q be given in R n. We determine a transation

More information

Chapter 11.00C Physical Problem for Fast Fourier Transform Civil Engineering

Chapter 11.00C Physical Problem for Fast Fourier Transform Civil Engineering haptr. Physical Problm for Fast Fourir Trasform ivil Egirig Itroductio I this chaptr, applicatios of FFT algorithms [-5] for solvig ral-lif problms such as computig th dyamical (displacmt rspos [6-7] of

More information

1985 AP Calculus BC: Section I

1985 AP Calculus BC: Section I 985 AP Calculus BC: Sctio I 9 Miuts No Calculator Nots: () I this amiatio, l dots th atural logarithm of (that is, logarithm to th bas ). () Ulss othrwis spcifid, th domai of a fuctio f is assumd to b

More information

EEO 401 Digital Signal Processing Prof. Mark Fowler

EEO 401 Digital Signal Processing Prof. Mark Fowler EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1

More information

Divide-and-Conquer. Divide-and-Conquer 1

Divide-and-Conquer. Divide-and-Conquer 1 Divide-d-Coquer 7 9 4 4 7 9 7 7 9 4 4 9 7 7 9 9 4 4 Divide-d-Coquer 1 Outie d Redig Divide-d-coquer prdigm 5. Review Merge-sort 4.1.1 Recurrece Equtios 5..1 tertive sustitutio Recursio trees Guess-d-test

More information

Hadamard Exponential Hankel Matrix, Its Eigenvalues and Some Norms

Hadamard Exponential Hankel Matrix, Its Eigenvalues and Some Norms Math Sci Ltt Vol No 8-87 (0) adamard Exotial al Matrix, Its Eigvalus ad Som Norms İ ad M bula Mathmatical Scics Lttrs Itratioal Joural @ 0 NSP Natural Scics Publishig Cor Dartmt of Mathmatics, aculty of

More information

Computational Linear Algebra

Computational Linear Algebra Computational Linear Algebra PD Dr. rer. nat. habil. Ralf Peter Mundani Computation in Engineering / BGU Scientific Computing in Computer Science / INF Winter Term 2017/18 Part 2: Direct Methods PD Dr.

More information

SYSTEMS OF LINEAR EQUATIONS

SYSTEMS OF LINEAR EQUATIONS SYSES OF INER EQUIONS Itroducto Emto thods Dcomposto thods tr Ivrs d Dtrmt Errors, Rsdus d Codto Numr Itrto thods Icompt d Rdudt Systms Chptr Systms of r Equtos /. Itroducto h systm of r qutos s formd

More information

terms of discrete sequences can only take values that are discrete as opposed to

terms of discrete sequences can only take values that are discrete as opposed to Diol Bgyoko () OWER SERIES Diitio Sris lik ( ) r th sm o th trms o discrt sqc. Th trms o discrt sqcs c oly tk vls tht r discrt s opposd to cotios, i.., trms tht r sch tht th mric vls o two cosctivs os

More information

The z Transform. The Discrete LTI System Response to a Complex Exponential

The z Transform. The Discrete LTI System Response to a Complex Exponential The Trsform The trsform geerlies the Discrete-time Forier Trsform for the etire complex ple. For the complex vrible is sed the ottio: jω x+ j y r e ; x, y Ω rg r x + y {} The Discrete LTI System Respose

More information

CS412: Lecture #17. Mridul Aanjaneya. March 19, 2015

CS412: Lecture #17. Mridul Aanjaneya. March 19, 2015 CS: Lecture #7 Mridul Aanjaneya March 9, 5 Solving linear systems of equations Consider a lower triangular matrix L: l l l L = l 3 l 3 l 33 l n l nn A procedure similar to that for upper triangular systems

More information

Judge dismisses wind lawsuit

Judge dismisses wind lawsuit 6 6 6 K K - J B - j - K W j J W W J K K - W - j j j K W W j j - I J J j K W j j W j K W j W J - W B B B j 6: 6: B z $ $ [ ] 6 IB J J K I z j W j W : j / j j j $ j I K W : W j K W j K W [ ] WI I K z j [I-

More information

DETERMINANT. = 0. The expression a 1. is called a determinant of the second order, and is denoted by : y + c 1

DETERMINANT. = 0. The expression a 1. is called a determinant of the second order, and is denoted by : y + c 1 NOD6 (\Dt\04\Kot\J-Advced\SMP\Mths\Uit#0\NG\Prt-\0.Determits\0.Theory.p65. INTRODUCTION : If the equtios x + b 0, x + b 0 re stisfied by the sme vlue of x, the b b 0. The expressio b b is clled determit

More information

Quantum Mechanics & Spectroscopy Prof. Jason Goodpaster. Problem Set #2 ANSWER KEY (5 questions, 10 points)

Quantum Mechanics & Spectroscopy Prof. Jason Goodpaster. Problem Set #2 ANSWER KEY (5 questions, 10 points) Chm 5 Problm St # ANSWER KEY 5 qustios, poits Qutum Mchics & Spctroscopy Prof. Jso Goodpstr Du ridy, b. 6 S th lst pgs for possibly usful costts, qutios d itgrls. Ths will lso b icludd o our futur ms..

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

Discrete Fourier Transform (DFT)

Discrete Fourier Transform (DFT) Discrt Fourir Trasorm DFT Major: All Egirig Majors Authors: Duc guy http://umricalmthods.g.us.du umrical Mthods or STEM udrgraduats 8/3/29 http://umricalmthods.g.us.du Discrt Fourir Trasorm Rcalld th xpotial

More information

Stochastic Processes

Stochastic Processes MTHE/STAT455, STAT855 Fall 202 Stochastic Processes Final Exam, Solutions (5 marks) (a) (0 marks) Condition on the first pair to bond; each of the n adjacent pairs is equally likely to bond Given that

More information

= (3) family of spreading sequences. The dimension of an Oppermann set with the code length of N is determined by

= (3) family of spreading sequences. The dimension of an Oppermann set with the code length of N is determined by (EUSIPCO Europa Siga Procssig Cofrc, Gasgow, Aug. 9 Improv Corratio of Graiz DFT with oiar Phas for OFD a CDA Commuicatios Ai. Aasu a Haa Agirma-Tosu w Jrsy Istitut of Tchoogy Dpartmt of Ectrica & Computr

More information

Face Detection and Recognition. Linear Algebra and Face Recognition. Face Recognition. Face Recognition. Dimension reduction

Face Detection and Recognition. Linear Algebra and Face Recognition. Face Recognition. Face Recognition. Dimension reduction F Dtto Roto Lr Alr F Roto C Y I Ursty O solto: tto o l trs s s ys os ot. Dlt to t to ltpl ws. F Roto Aotr ppro: ort y rry s tor o so E.. 56 56 > pot 6556- stol sp A st o s t ps to ollto o pots ts sp. F

More information

Last time: introduced our first computational model the DFA.

Last time: introduced our first computational model the DFA. Lctur 7 Homwork #7: 2.2.1, 2.2.2, 2.2.3 (hnd in c nd d), Misc: Givn: M, NFA Prov: (q,xy) * (p,y) iff (q,x) * (p,) (follow proof don in clss tody) Lst tim: introducd our first computtionl modl th DFA. Tody

More information

Linear Equations and Matrix

Linear Equations and Matrix 1/60 Chia-Ping Chen Professor Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Gaussian Elimination 2/60 Alpha Go Linear algebra begins with a system of linear

More information

MA2501 Numerical Methods Spring 2015

MA2501 Numerical Methods Spring 2015 Norwegian University of Science and Technology Department of Mathematics MA2501 Numerical Methods Spring 2015 Solutions to exercise set 3 1 Attempt to verify experimentally the calculation from class that

More information

A b r i l l i a n t young chemist, T h u r e Wagelius of N e w Y o r k, ac. himself with eth

A b r i l l i a n t young chemist, T h u r e Wagelius of N e w Y o r k, ac. himself with eth 6 6 0 x J 8 0 J 0 z (0 8 z x x J x 6 000 X j x "" "" " " x " " " x " " " J " " " " " " " " x : 0 z j ; J K 0 J K q 8 K K J x 0 j " " > J x J j z ; j J q J 0 0 8 K J 60 : K 6 x 8 K J :? 0 J J K 0 6% 8 0

More information

Addition of angular momentum

Addition of angular momentum Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat

More information

Periodic Structures. Filter Design by the Image Parameter Method

Periodic Structures. Filter Design by the Image Parameter Method Prioic Structurs a Filtr sig y th mag Paramtr Mtho ECE53: Microwav Circuit sig Pozar Chaptr 8, Sctios 8. & 8. Josh Ottos /4/ Microwav Filtrs (Chaptr Eight) microwav filtr is a two-port twork us to cotrol

More information

April 26, Applied mathematics PhD candidate, physics MA UC Berkeley. Lecture 4/26/2013. Jed Duersch. Spd matrices. Cholesky decomposition

April 26, Applied mathematics PhD candidate, physics MA UC Berkeley. Lecture 4/26/2013. Jed Duersch. Spd matrices. Cholesky decomposition Applied mathematics PhD candidate, physics MA UC Berkeley April 26, 2013 UCB 1/19 Symmetric positive-definite I Definition A symmetric matrix A R n n is positive definite iff x T Ax > 0 holds x 0 R n.

More information

Week 10 Spring Lecture 19. Estimation of Large Covariance Matrices: Upper bound Observe. is contained in the following parameter space,

Week 10 Spring Lecture 19. Estimation of Large Covariance Matrices: Upper bound Observe. is contained in the following parameter space, Week 0 Sprig 009 Lecture 9. stiatio of Large Covariace Matrices: Upper boud Observe ; ; : : : ; i.i.d. fro a p-variate Gaussia distributio, N (; pp ). We assue that the covariace atrix pp = ( ij ) i;jp

More information

Integration by Guessing

Integration by Guessing Itgrtio y Gussig Th computtios i two stdrd itgrtio tchiqus, Sustitutio d Itgrtio y Prts, c strmlid y th Itgrtio y Gussig pproch. This mthod cosists of thr stps: Guss, Diffrtit to chck th guss, d th Adjust

More information

Linear Programming. Preliminaries

Linear Programming. Preliminaries Lier Progrmmig Prelimiries Optimiztio ethods: 3L Objectives To itroduce lier progrmmig problems (LPP To discuss the stdrd d coicl form of LPP To discuss elemetry opertio for lier set of equtios Optimiztio

More information

MATH 445/545 Test 1 Spring 2016

MATH 445/545 Test 1 Spring 2016 MATH 445/545 Test Spring 06 Note the problems are separated into two sections a set for all students and an additional set for those taking the course at the 545 level. Please read and follow all of these

More information

Linear Algebra Practice Final

Linear Algebra Practice Final . Let (a) First, Linear Algebra Practice Final Summer 3 3 A = 5 3 3 rref([a ) = 5 so if we let x 5 = t, then x 4 = t, x 3 =, x = t, and x = t, so that t t x = t = t t whence ker A = span(,,,, ) and a basis

More information

Exercises for lectures 23 Discrete systems

Exercises for lectures 23 Discrete systems Exrciss for lcturs 3 Discrt systms Michal Šbk Automatické říí 06 30-4-7 Stat-Spac a Iput-Output scriptios Automatické říí - Kybrtika a robotika Mols a trasfrs i CSTbx >> F=[ ; 3 4]; G=[ ;]; H=[ ]; J=0;

More information

CHAPTER 6c. NUMERICAL INTERPOLATION

CHAPTER 6c. NUMERICAL INTERPOLATION CHAPTER 6c. NUMERICAL INTERPOLATION A. J. Clark School o Egieerig Departmet o Civil ad Evirometal Egieerig y Dr. Irahim A. Assakka Sprig ENCE - Computatio Methods i Civil Egieerig II Departmet o Civil

More information

this is called an indeterninateformof-oior.fi?afleleitns derivatives can now differentiable and give 0 on on open interval containing I agree to.

this is called an indeterninateformof-oior.fi?afleleitns derivatives can now differentiable and give 0 on on open interval containing I agree to. hl sidd r L Hospitl s Rul 11/7/18 Pronouncd Loh mtims splld Non p t mtims w wnt vlut limit ii m itn ) but irst indtrnintmori?lltns indtrmint t inn gl in which cs th clld n i 9kt ti not ncssrily snsign

More information

SOLVED EXAMPLES. Ex.1 If f(x) = , then. is equal to- Ex.5. f(x) equals - (A) 2 (B) 1/2 (C) 0 (D) 1 (A) 1 (B) 2. (D) Does not exist = [2(1 h)+1]= 3

SOLVED EXAMPLES. Ex.1 If f(x) = , then. is equal to- Ex.5. f(x) equals - (A) 2 (B) 1/2 (C) 0 (D) 1 (A) 1 (B) 2. (D) Does not exist = [2(1 h)+1]= 3 SOLVED EXAMPLES E. If f() E.,,, th f() f() h h LHL RHL, so / / Lim f() quls - (D) Dos ot ist [( h)+] [(+h) + ] f(). LHL E. RHL h h h / h / h / h / h / h / h As.[C] (D) Dos ot ist LHL RHL, so giv it dos

More information

Lectures 2 & 3 - Population ecology mathematics refresher

Lectures 2 & 3 - Population ecology mathematics refresher Lcturs & - Poultio cology mthmtics rrshr To s th mov ito vloig oultio mols, th olloig mthmtics crisht is suli I i out r mthmtics ttook! Eots logrithms i i q q q q q q ( tims) / c c c c ) ( ) ( Clculus

More information

V C V L T I 0 C V B 1 V T 0 I. l nk

V C V L T I 0 C V B 1 V T 0 I. l nk Multifrontal Method Kailai Xu September 16, 2017 Main observation. Consider the LDL T decomposition of a SPD matrix [ ] [ ] [ ] [ ] B V T L 0 I 0 L T L A = = 1 V T V C V L T I 0 C V B 1 V T, 0 I where

More information

IIT JEE MATHS MATRICES AND DETERMINANTS

IIT JEE MATHS MATRICES AND DETERMINANTS IIT JEE MTHS MTRICES ND DETERMINNTS THIRUMURUGN.K PGT Mths IIT Trir 978757 Pg. Lt = 5, th () =, = () = -, = () =, = - (d) = -, = -. Lt sw smmtri mtri of odd th quls () () () - (d) o of ths. Th vlu of th

More information

Statistics for Financial Engineering Session 1: Linear Algebra Review March 18 th, 2006

Statistics for Financial Engineering Session 1: Linear Algebra Review March 18 th, 2006 Sttistics for Ficil Egieerig Sessio : Lier Algebr Review rch 8 th, 6 Topics Itroductio to trices trix opertios Determits d Crmer s rule Eigevlues d Eigevectors Quiz The cotet of Sessio my be fmilir to

More information

Matrices: 2.1 Operations with Matrices

Matrices: 2.1 Operations with Matrices Goals In this chapter and section we study matrix operations: Define matrix addition Define multiplication of matrix by a scalar, to be called scalar multiplication. Define multiplication of two matrices,

More information

Full file at

Full file at ull file at https://fratstock.eu Unit 2 Problem Solutions Unit 2 Solutions 2.1 See LD p. 693 for solution. 2.2 (a) In both cases, if = 0, the transmission is 0, and if = 1, the transmission is 1. 2.2 (b)

More information

Instructions for Section 1

Instructions for Section 1 Instructions for Sction 1 Choos th rspons tht is corrct for th qustion. A corrct nswr scors 1, n incorrct nswr scors 0. Mrks will not b dductd for incorrct nswrs. You should ttmpt vry qustion. No mrks

More information

Linear Algebra Solutions 1

Linear Algebra Solutions 1 Math Camp 1 Do the following: Linear Algebra Solutions 1 1. Let A = and B = 3 8 5 A B = 3 5 9 A + B = 9 11 14 4 AB = 69 3 16 BA = 1 4 ( 1 3. Let v = and u = 5 uv = 13 u v = 13 v u = 13 Math Camp 1 ( 7

More information

CS 246 Review of Linear Algebra 01/17/19

CS 246 Review of Linear Algebra 01/17/19 1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector

More information

DUET WITH DIAMONDS COLOR SHIFTING BRACELET By Leslie Rogalski

DUET WITH DIAMONDS COLOR SHIFTING BRACELET By Leslie Rogalski Dut with Dimons Brlt DUET WITH DIAMONDS COLOR SHIFTING BRACELET By Lsli Roglski Photo y Anrw Wirth Supruo DUETS TM from BSmith rt olor shifting fft tht mks your work tk on lif of its own s you mov! This

More information

Classical Theory of Fourier Series : Demystified and Generalised VIVEK V. RANE. The Institute of Science, 15, Madam Cama Road, Mumbai

Classical Theory of Fourier Series : Demystified and Generalised VIVEK V. RANE. The Institute of Science, 15, Madam Cama Road, Mumbai Clssil Thoy o Foi Sis : Dmystii Glis VIVEK V RANE Th Istitt o Si 5 Mm Cm Ro Mmbi-4 3 -mil ss : v_v_@yhoooi Abstt : Fo Rim itgbl tio o itvl o poit thi w i Foi Sis t th poit o th itvl big ot how wh th tio

More information

(HELD ON 22nd MAY SUNDAY 2016) MATHEMATICS CODE - 2 [PAPER -2]

(HELD ON 22nd MAY SUNDAY 2016) MATHEMATICS CODE - 2 [PAPER -2] QUESTION PAPER WITH SOLUTION OF JEE ADVANCED - 6 7. Lt P (HELD ON d MAY SUNDAY 6) FEEL THE POWER OF OUR KNOWLEDGE & EXPERIENCE Our Top clss IITi fculty tm promiss to giv you uthtic swr ky which will b

More information

First order differential equation Linear equation; Method of integrating factors

First order differential equation Linear equation; Method of integrating factors First orr iffrntial quation Linar quation; Mtho of intgrating factors Exampl 1: Rwrit th lft han si as th rivativ of th prouct of y an som function by prouct rul irctly. Solving th first orr iffrntial

More information

10. Joint Moments and Joint Characteristic Functions

10. Joint Moments and Joint Characteristic Functions 0 Joit Momts ad Joit Charactristic Fctios Followig sctio 6 i this sctio w shall itrodc varios paramtrs to compactly rprst th iformatio cotaid i th joit pdf of two rvs Giv two rvs ad ad a fctio g x y dfi

More information

Max-plus algebra. Max-plus algebra. Monika Molnárová. Technická univerzita Košice. Max-plus algebra.

Max-plus algebra. Max-plus algebra. Monika Molnárová. Technická univerzita Košice. Max-plus algebra. Technická univerzita Košice monika.molnarova@tuke.sk Outline 1 Digraphs Maximum cycle-mean and transitive closures of a matrix Reducible and irreducible matrices Definite matrices Digraphs Complete digraph

More information

For a system with more than one electron, we can t solve the Schrödinger Eq. exactly. We must develop methods of approximation, such as

For a system with more than one electron, we can t solve the Schrödinger Eq. exactly. We must develop methods of approximation, such as VARIATIO METHOD For a system with more than one electron, we can t solve the Schrödinger Eq. exactly. We must develop methods of approximation, such as Variation Method Perturbation Theory Combination

More information

1 Linear Least Squares

1 Linear Least Squares Lest Squres Pge 1 1 Liner Lest Squres I will try to be consistent in nottion, with n being the number of dt points, nd m < n being the number of prmeters in model function. We re interested in solving

More information

ELEMENTARY LINEAR ALGEBRA

ELEMENTARY LINEAR ALGEBRA ELEMENTARY LINEAR ALGEBRA DAVID MEREDITH These notes are about the application and computation of linear algebra objects in F n, where F is a field. I think everything here works over any field including

More information

CHAPTER 6d. NUMERICAL INTERPOLATION

CHAPTER 6d. NUMERICAL INTERPOLATION CHAPER 6d. NUMERICAL INERPOLAION A. J. Clark School o Egieerig Departmet o Civil ad Evirometal Egieerig by Dr. Ibrahim A. Assakka Sprig ENCE - Computatio Methods i Civil Egieerig II Departmet o Civil ad

More information

Formalism of the Tersoff potential

Formalism of the Tersoff potential Originally written in December 000 Translated to English in June 014 Formalism of the Tersoff potential 1 The original version (PRB 38 p.990, PRB 37 p.6991) Potential energy Φ = 1 u ij i (1) u ij = f ij

More information

INTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x)

INTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x) Chptr 7 INTEGRALS 7. Ovrviw 7.. Lt d d F () f (). Thn, w writ f ( ) d F () + C. Ths intgrls r clld indfinit intgrls or gnrl intgrls, C is clld constnt of intgrtion. All ths intgrls diffr y constnt. 7..

More information

Eigenfunction Expansion. For a given function on the internal a x b the eigenfunction expansion of f(x):

Eigenfunction Expansion. For a given function on the internal a x b the eigenfunction expansion of f(x): Eigefuctio Epsio: For give fuctio o the iterl the eigefuctio epsio of f(): f ( ) cmm( ) m 1 Eigefuctio Epsio (Geerlized Fourier Series) To determie c s we multiply oth sides y Φ ()r() d itegrte: f ( )

More information

Chapter 1: Systems of Linear Equations

Chapter 1: Systems of Linear Equations Chapter : Systems of Linear Equations February, 9 Systems of linear equations Linear systems Lecture A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where

More information

Computation of the mtx-vec product based on storage scheme on vector CPUs

Computation of the mtx-vec product based on storage scheme on vector CPUs BLAS: Basic Linear Algebra Subroutines BLAS: Basic Linear Algebra Subroutines BLAS: Basic Linear Algebra Subroutines Analysis of the Matrix Computation of the mtx-vec product based on storage scheme on

More information

+ x. x 2x. 12. dx. 24. dx + 1)

+ x. x 2x. 12. dx. 24. dx + 1) INTEGRATION of FUNCTION of ONE VARIABLE INDEFINITE INTEGRAL Fidig th idfiit itgrals Rductio to basic itgrals, usig th rul f ( ) f ( ) d =... ( ). ( )d. d. d ( ). d. d. d 7. d 8. d 9. d. d. d. d 9. d 9.

More information

Direct Methods for Solving Linear Systems. Simon Fraser University Surrey Campus MACM 316 Spring 2005 Instructor: Ha Le

Direct Methods for Solving Linear Systems. Simon Fraser University Surrey Campus MACM 316 Spring 2005 Instructor: Ha Le Direct Methods for Solving Linear Systems Simon Fraser University Surrey Campus MACM 316 Spring 2005 Instructor: Ha Le 1 Overview General Linear Systems Gaussian Elimination Triangular Systems The LU Factorization

More information

DEN: Linear algebra numerical view (GEM: Gauss elimination method for reducing a full rank matrix to upper-triangular

DEN: Linear algebra numerical view (GEM: Gauss elimination method for reducing a full rank matrix to upper-triangular form) Given: matrix C = (c i,j ) n,m i,j=1 ODE and num math: Linear algebra (N) [lectures] c phabala 2016 DEN: Linear algebra numerical view (GEM: Gauss elimination method for reducing a full rank matrix

More information

POWER LOSSES IN THE THREE-PHASE GAS-INSULATED LINE

POWER LOSSES IN THE THREE-PHASE GAS-INSULATED LINE OZNAN UNVE STY OF TE CHNOLOGY ACADE MC JOUNALS No 89 Elctrical Egirig 7 DO.8/j.897-77.7.89.8 Tomasz SZCZEGELNA Dariusz USA Zygmut ĄTE OWE LOSSES N THE THEE-HASE GAS-NSULATED LNE This papr prsts a aalytical

More information

Kevin James. MTHSC 3110 Section 2.1 Matrix Operations

Kevin James. MTHSC 3110 Section 2.1 Matrix Operations MTHSC 3110 Section 2.1 Matrix Operations Notation Let A be an m n matrix, that is, m rows and n columns. We ll refer to the entries of A by their row and column indices. The entry in the i th row and j

More information

Lecture 11. Linear systems: Cholesky method. Eigensystems: Terminology. Jacobi transformations QR transformation

Lecture 11. Linear systems: Cholesky method. Eigensystems: Terminology. Jacobi transformations QR transformation Lecture Cholesky method QR decomposition Terminology Linear systems: Eigensystems: Jacobi transformations QR transformation Cholesky method: For a symmetric positive definite matrix, one can do an LU decomposition

More information

a b c cat CAT A B C Aa Bb Cc cat cat Lesson 1 (Part 1) Verbal lesson: Capital Letters Make The Same Sound Lesson 1 (Part 1) continued...

a b c cat CAT A B C Aa Bb Cc cat cat Lesson 1 (Part 1) Verbal lesson: Capital Letters Make The Same Sound Lesson 1 (Part 1) continued... Progrssiv Printing T.M. CPITLS g 4½+ Th sy, fun (n FR!) wy to tch cpitl lttrs. ook : C o - For Kinrgrtn or First Gr (not for pr-school). - Tchs tht cpitl lttrs mk th sm souns s th littl lttrs. - Tchs th

More information

Walk Like a Mathematician Learning Task:

Walk Like a Mathematician Learning Task: Gori Dprtmnt of Euction Wlk Lik Mthmticin Lrnin Tsk: Mtrics llow us to prform mny usful mthmticl tsks which orinrily rquir lr numbr of computtions. Som typs of problms which cn b on fficintly with mtrics

More information

(2) If we multiplied a row of B by λ, then the value is also multiplied by λ(here lambda could be 0). namely

(2) If we multiplied a row of B by λ, then the value is also multiplied by λ(here lambda could be 0). namely . DETERMINANT.. Dtrminnt. Introution:I you think row vtor o mtrix s oorint o vtors in sp, thn th gomtri mning o th rnk o th mtrix is th imnsion o th prlllppi spnn y thm. But w r not only r out th imnsion,

More information

Chapter System of Equations

Chapter System of Equations hpter 4.5 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vice-vers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee

More information

ASSERTION AND REASON

ASSERTION AND REASON ASSERTION AND REASON Som qustios (Assrtio Rso typ) r giv low. Ech qustio cotis Sttmt (Assrtio) d Sttmt (Rso). Ech qustio hs choics (A), (B), (C) d (D) out of which ONLY ONE is corrct. So slct th corrct

More information

CREATED USING THE RSC COMMUNICATION TEMPLATE (VER. 2.1) - SEE FOR DETAILS

CREATED USING THE RSC COMMUNICATION TEMPLATE (VER. 2.1) - SEE   FOR DETAILS uortig Iormtio: Pti moiitio oirmtio vi 1 MR: j 5 FEFEFKFK 8.6.. 8.6 1 13 1 11 1 9 8 7 6 5 3 1 FEFEFKFK moii 1 13 1 11 1 9 8 7 6 5 3 1 m - - 3 3 g i o i o g m l g m l - - h k 3 h k 3 Figur 1: 1 -MR or th

More information

BSc (Hons) in Computer Games Development. vi Calculate the components a, b and c of a non-zero vector that is orthogonal to

BSc (Hons) in Computer Games Development. vi Calculate the components a, b and c of a non-zero vector that is orthogonal to 1 APPLIED MATHEMATICS INSTRUCTIONS Full marks will be awarded for the correct solutions to ANY FIVE QUESTIONS. This paper will be marked out of a TOTAL MAXIMUM MARK OF 100. Credit will be given for clearly

More information

1. Stefan-Boltzmann law states that the power emitted per unit area of the surface of a black

1. Stefan-Boltzmann law states that the power emitted per unit area of the surface of a black Stf-Boltzm lw stts tht th powr mttd pr ut r of th surfc of blck body s proportol to th fourth powr of th bsolut tmprtur: 4 S T whr T s th bsolut tmprtur d th Stf-Boltzm costt= 5 4 k B 3 5c h ( Clcult 5

More information

Direct Methods for Solving Linear Systems. Matrix Factorization

Direct Methods for Solving Linear Systems. Matrix Factorization Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011

More information

DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018

DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018 DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Assoc. Prof. Dr. Bur Kelleci Sprig 8 OUTLINE The Z-Trsform The Regio of covergece for the Z-trsform The Iverse Z-Trsform Geometric

More information

page 11 equation (1.2-10c), break the bar over the right side in the middle

page 11 equation (1.2-10c), break the bar over the right side in the middle I. Corrctios Lst Updtd: Ju 00 Complx Vrils with Applictios, 3 rd ditio, A. Dvid Wusch First Pritig. A ook ought for My 007 will proly first pritig With Thks to Christi Hos of Swd pg qutio (.-0c), rk th

More information

Modeling with first order equations (Sect. 2.3).

Modeling with first order equations (Sect. 2.3). Moling with first orr quations (Sct. 2.3. Main xampl: Salt in a watr tank. Th xprimntal vic. Th main quations. Analysis of th mathmatical mol. Prictions for particular situations. Salt in a watr tank.

More information

Multivariate distance Fall

Multivariate distance Fall Multivariate distance 2017 Fall Contents Euclidean Distance Definitions Standardization Population Distance Population mean and variance Definitions proportions presence-absence data Multivariate distance

More information