Gaussian Elimination Autar Kaw
|
|
|
- Arnold Bryan
- 7 years ago
- Views:
Transcription
1 Gussi Elimitio Autr Kw After redig this chpter, you should be ble to:. solve set of simulteous lier equtios usig Nïve Guss elimitio,. ler the pitflls of the Nïve Guss elimitio method,. uderstd the effect of roud-off error whe solvig set of lier equtios with the Nïve Guss elimitio method, 4. ler how to modify the Nïve Guss elimitio method to the Gussi elimitio with prtil pivotig method to void pitflls of the former method,. fid the determit of squre mtri usig Gussi elimitio, d 6. uderstd the reltioship betwee the determit of coefficiet mtri d the solutio of simulteous lier equtios. How is set of equtios solved umericlly? Oe of the most populr techiques for solvig simulteous lier equtios is the Gussi elimitio method. The pproch is desiged to solve geerl set of equtios d ukows b b b Gussi elimitio cosists of two steps. Forwrd Elimitio of Ukows: I this step, the ukow is elimited i ech equtio strtig with the first equtio. This wy, the equtios re reduced to oe equtio d oe ukow i ech equtio.. Bck Substitutio: I this step, strtig from the lst equtio, ech of the ukows is foud. Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge of 7
2 Forwrd Elimitio of Ukows: I the first step of forwrd elimitio, the first ukow, is elimited from ll rows below the first row. The first equtio is selected s the pivot equtio to elimite. So, to elimite i the secod equtio, oe divides the first equtio by (hece clled the pivot elemet) d the multiplies it by. This is the sme s multiplyig the first equtio by / to give b Now, this equtio c be subtrcted from the secod equtio to give or where... b b b M This procedure of elimitig, is ow repeted for the third equtio to the equtio to reduce the set of equtios s b b b th Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge of 7
3 b This is the ed of the first step of forwrd elimitio. Now for the secod step of forwrd elimitio, we strt with the secod equtio s the pivot equtio d s the pivot elemet. So, to elimite i the third equtio, oe divides the secod equtio by (the pivot elemet) d the multiply it by. This is the sme s multiplyig the secod equtio by / d subtrctig it from the third equtio. This mkes the coefficiet of zero i the third equtio. The sme procedure is ow repeted for the fourth equtio till the b b b b th equtio to give The et steps of forwrd elimitio re coducted by usig the third equtio s pivot equtio d so o. Tht is, there will be totl of steps of forwrd elimitio. At the ed of steps of forwrd elimitio, we get set of equtios tht look like b b b.... Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge of 7
4 .. ( ) ( ) b Bck Substitutio: Now the equtios re solved strtig from the lst equtio s it hs oly oe ukow. b ( ) ( ) The the secod lst equtio, tht is the, but is lredy kow. This reduces the th ( ) equtio, hs two ukows: d th ( ) equtio lso to oe ukow. Bck substitutio hece c be represeted for ll equtios by the formul ( i) ( i) b ij j for i,, K, i j i+ i ( i) ii d b ( ) ( ) Emple The upwrd velocity of rocket is give t three differet times i Tble. Tble Velocity vs. time dt. Time, t (s) Velocity, v (m/s) 6.8 Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge 4 of 7
5 The velocity dt is pproimted by polyomil s v ( t) t + t+, t The coefficiets,, d for the bove epressio re give by Fid the vlues of,, d usig the Nïve Guss elimitio method. Fid the velocity t t 6, 7., 9, secods. Solutio Forwrd Elimitio of Ukows Sice there re three equtios, there will be two steps of forwrd elimitio of ukows. First step Divide Row by d the multiply it by 64, tht is, multiply Row by 64/.6. ([ ] [ 6.8] ). 6 gives Row s [ ] [ 7.48] Subtrct the result from Row [ 64 8 ] [ 77.] [ ] [ 7.48] to get the resultig equtios s Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge of 7
6 Divide Row by d the multiply it by 44, tht is, multiply Row by 44/.76. ([ ] [ 6.8] ). 76 gives Row s [ ] [ 6.68] Subtrct the result from Row [ 44 ] [ 79.] [ ] [ 6.68] to get the resultig equtios s Secod step We ow divide Row by 4.8 d the multiply by 6.8, tht is, multiply Row by 6.8/ ([ 4.8.6] [ 96.8] ). gives Row s [ ] [ 6.78] Subtrct the result from Row [ ] [.968] [ ] [ 6.78].7.76 to get the resultig equtios s Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge 6 of 7
7 Bck substitutio From the third equtio Substitutig the vlue of i the secod equtio, Substitutig the vlue of d i the first equtio, Hece the solutio vector is Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge 7 of 7
8 The polyomil tht psses through the three dt poits is the ( t) t + t v +.947t t+.87, t Sice we wt to fid the velocity t t 6, 7., 9 d secods, we could simply substitute ech vlue of t i ( t).947t t+. 87 correspodig velocity. For emple, t t 6 v ( 6).947( 6) ( 6) m/s v d fid the +.87 However we could lso fid ll the eeded vlues of velocity t t 6, 7., 9, secods usig mtri multiplictio. v ( t) [ ] t t So if we wt to fid ( 6), v( 7.), v( 9), v( ), v it is give by [ v( 6) v( 7.) v( 9) v( ) ] [ ] [ ] [ ] 8 9 Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge 8 of 7
9 v ( 6) m/s v ( 7.) 6.4 m/s v ( 9) v ( ).88 m/s.88 m/s Emple Use Nïve Guss elimitio to solve Use si sigifict digits with choppig i your clcultios. Solutio Workig i the mtri form Forwrd Elimitio of Ukows First step Divide Row by d the multiply it by, tht is, multiply Row by /.. ([ ] [ 4] ). [..] [ 6.7] Subtrct the result from Row gives Row s Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge 9 of 7
10 [.49 7] [.7] [..] [ 6.7] to get the resultig equtios s Divide Row by d the multiply it by, tht is, multiply Row by /. ([ ] [ 4] ). gives Row s [.7.] [.] Subtrct the result from Row [ ] [ 9] [.7.] [.].7.. to get the resultig equtios s Secod step Now for the secod step of forwrd elimitio, we will use Row s the pivot equtio d elimite Row : Colum. Divide Row by. d the multiply it by.7, tht is, multiply Row by.7/. 7. ([. 8.] [ 8.] ) 7 gives Row s [.7 7] [ 77.7] Rewritig withi 6 sigifict digits with choppig Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge of 7
11 [.7 7] [ 77.7] Subtrct the result from Row [.7.] [.] [.7 7] [ 77.7] Rewritig withi 6 sigifict digits with choppig [ 7.] [ 7.4] to get the resultig equtios s This is the ed of the forwrd elimitio steps. Bck substitutio We c ow solve the bove equtios by bck substitutio. From the third equtio, Substitutig the vlue of i the secod equtio Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge of 7
12 Substitutig the vlue of d i the first equtio, Hece the solutio is [ X ] Compre this with the ect solutio of Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge of 7
13 [ X ] Are there y pitflls of the Nïve Guss elimitio method? Yes, there re two pitflls of the Nïve Guss elimitio method. Divisio by zero: It is possible for divisio by zero to occur durig the begiig of the steps of forwrd elimitio. For emple will result i divisio by zero i the first step of forwrd elimitio s the coefficiet of i the first equtio is zero s is evidet whe we write the equtios i mtri form But wht bout the equtios below: Is divisio by zero problem? Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge of 7
14 Writte i mtri form, there is o issue of divisio by zero i the first step of forwrd elimitio. The pivot elemet is the coefficiet of i the first equtio,, d tht is o-zero umber. However, t the ed of the first step of forwrd elimitio, we get the followig equtios i mtri form Now t the begiig of the d step of forwrd elimitio, the coefficiet of i Equtio would be used s the pivot elemet. Tht elemet is zero d hece would crete the divisio by zero problem. So it is importt to cosider tht the possibility of divisio by zero c occur t the begiig of y step of forwrd elimitio. Roud-off error: The Nïve Guss elimitio method is proe to roud-off errors. This is true whe there re lrge umbers of equtios s errors propgte. Also, if there is subtrctio of umbers from ech other, it my crete lrge errors. See the emple below. Emple Remember Emple where we used Nïve Guss elimitio to solve Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge 4 of 7
15 usig si sigifict digits with choppig i your clcultios? Repet the problem, but ow use five sigifict digits with choppig i your clcultios. Solutio Writig i the mtri form Forwrd Elimitio of Ukows First step Divide Row by d the multiply it by, tht is, multiply Row by /.. ([ ] [ 4] ). [..] [ 6.7] Subtrct the result from Row gives Row s [.49 7] [.7] [..] [ 6.7] to get the resultig equtios s Divide Row by d the multiply it by, tht is, multiply Row by /.. ([ ] [ 4] ). gives Row s [.7.] [.] Subtrct the result from Row Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge of 7
16 [ ] [ 9] [.7.] [.].7.. to get the resultig equtios s Secod step Now for the secod step of forwrd elimitio, we will use Row s the pivot equtio d elimite Row : Colum. Divide Row by. d the multiply it by.7, tht is, multiply Row by.7/. 7. ([. 8.] [ 8.] ) 7 gives Row s [.7 7] [ 77.7] Rewritig withi sigifict digits with choppig [.7 7] [ 77] Subtrct the result from Row [.7.] [.] [.7 7] [ 77] 7 74 Rewritig withi 6 sigifict digits with choppig [ 7] [ 74] to get the resultig equtios s Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge 6 of 7
17 This is the ed of the forwrd elimitio steps. Bck substitutio We c ow solve the bove equtios by bck substitutio. From the third equtio, Substitutig the vlue of i the secod equtio Substitutig the vlue of d i the first equtio, Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge 7 of 7
18 Hece the solutio is [ X ] Compre this with the ect solutio of [ X] Wht re some techiques for improvig the Nïve Guss elimitio method? As see i Emple, roud off errors were lrge whe five sigifict digits were used s opposed to si sigifict digits. Oe method of decresig the roud-off error would be to use more sigifict digits, tht is, use double or qud precisio for represetig the umbers. However, this would ot void possible divisio by zero errors i the Nïve Guss elimitio method. To void divisio by zero s well s reduce (ot elimite) roud-off error, Gussi elimitio with prtil pivotig is the method of choice. Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge 8 of 7
19 How does Gussi elimitio with prtil pivotig differ from Nïve Guss elimitio? The two methods re the sme, ecept i the begiig of ech step of forwrd elimitio, row switchig is doe bsed o the followig criterio. If there re equtios, the there re forwrd elimitio steps. At the begiig of the step of forwrd elimitio, oe fids the mimum of, k +, k,, k kk th k The if the mimum of these vlues is p d k. pk i the th p row, k p, the switch rows The other steps of forwrd elimitio re the sme s the Nïve Guss elimitio method. The bck substitutio steps sty ectly the sme s the Nïve Guss elimitio method. Emple 4 I the previous two emples, we used Nïve Guss elimitio to solve usig five d si sigifict digits with choppig i the clcultios. Usig five sigifict digits with choppig, the solutio foud ws [ X ] Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge 9 of 7
20 This is differet from the ect solutio of [ X ] Fid the solutio usig Gussi elimitio with prtil pivotig usig five sigifict digits with choppig i your clcultios. Solutio Forwrd Elimitio of Ukows Now for the first step of forwrd elimitio, the bsolute vlue of the first colum elemets below Row is,, or,, So the lrgest bsolute vlue is i the Row. So s per Gussi elimitio with prtil pivotig, the switch is betwee Row d Row to give Divide Row by d the multiply it by, tht is, multiply Row by /.. ([ ] [ 4] ). gives Row s Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge of 7
21 [..] [ 6.7] Subtrct the result from Row [.49 7] [.7] [..] [ 6.7] to get the resultig equtios s Divide Row by d the multiply it by, tht is, multiply Row by /.. ([ ] [ 4] ). gives Row s [.7.] [.] Subtrct the result from Row [ ] [ 9] [.7.] [.].7.. to get the resultig equtios s This is the ed of the first step of forwrd elimitio. Now for the secod step of forwrd elimitio, the bsolute vlue of the secod colum elemets below Row is.,. 7 or Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge of 7
22 .,.7 So the lrgest bsolute vlue is i Row. So Row is switched with Row to give Divide Row by.7 d the multiply it by., tht is, multiply Row by./ ([.7.] [.] ). 66 gives Row s [ ] [.886] Subtrct the result from Row [. 8.] [ 8.] [ ] [.886] Rewritig withi sigifict digits with choppig [ 8.] [ 8.] to get the resultig equtios s Bck substitutio Substitutig the vlue of i Row Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge of 7
23 Substitutig the vlue of d i Row So the solutio is [ X ] Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge of 7
24 This, i fct, is the ect solutio. By coicidece oly, i this cse, the roud-off error is fully removed. C we use Nïve Guss elimitio methods to fid the determit of squre mtri? Oe of the more efficiet wys to fid the determit of squre mtri is by tkig dvtge of the followig two theorems o determit of mtrices coupled with Nïve Guss elimitio. Theorem : Let [A] be mtri. The, if [B] is mtri tht results from ddig or subtrctig multiple of oe row to other row, the det( A ) det( B) (The sme is true for colum opertios lso). Theorem : Let [A] be mtri tht is upper trigulr, lower trigulr or digol, the det( A )... ii... ii i This implies tht if we pply the forwrd elimitio steps of the Nïve Guss elimitio method, the determit of the mtri stys the sme ccordig to Theorem. The sice t the ed of the forwrd elimitio steps, the resultig mtri is upper trigulr, the determit will be give by Theorem. Emple Fid the determit of Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge 4 of 7
25 [A] Solutio Remember i Emple, we coducted the steps of forwrd elimitio of ukows usig the Nïve Guss elimitio method o [A] to give [ B ] Accordig to Theorem det( A ) det( B) ( 4.8) Wht if I cot fid the determit of the mtri usig the Nïve Guss elimitio method, for emple, if I get divisio by zero problems durig the Nïve Guss elimitio method? Well, you c pply Gussi elimitio with prtil pivotig. However, the determit of the resultig upper trigulr mtri my differ by sig. The followig theorem pplies i dditio to the previous two to fid the determit of squre mtri. Theorem : Let [A] be mtri. The, if [B] is mtri tht results from switchig oe row with other row, the det( B) det( A). Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge of 7
26 Emple 6 Fid the determit of [A] Solutio The ed of the forwrd elimitio steps of Gussi elimitio with prtil pivotig, we would obti [B] ( B ). 6. det. Sice rows were switched oce durig the forwrd elimitio steps of Gussi elimitio with prtil pivotig, det ( A) det( B). Emple 7 Prove det( A ) det ( A ) Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge 6 of 7
27 Solutio [ A][ A] det det det ( A A ) det( I) ( A) det( A ) ( A) [ I] det ( A ) If [A] is mtri d det( A ), wht other sttemets re equivlet to it?. [A] is ivertible.. [ A ] eists.. [ A ][ X ] [ C] hs uique solutio. v 4. [ A ][ X ] [] solutio is [ X ] [].. [ A][ A] [ I] [ A] [ A]. Key Terms: Nïve Guss Elimitio Prtil Pivotig Determit Source URL: Sylor URL: Attributed to: Uiversity of South Florid: Holistic Numericl Methods Istitute Sylor.org Pge 7 of 7
Chapter 04.05 System of Equations
hpter 04.05 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
MATHEMATICS FOR ENGINEERING BASIC ALGEBRA
MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL - INDICES, LOGARITHMS AND FUNCTION This is the oe of series of bsic tutorils i mthemtics imed t begiers or yoe wtig to refresh themselves o fudmetls.
Repeated multiplication is represented using exponential notation, for example:
Appedix A: The Lws of Expoets Expoets re short-hd ottio used to represet my fctors multiplied together All of the rules for mipultig expoets my be deduced from the lws of multiplictio d divisio tht you
Released Assessment Questions, 2015 QUESTIONS
Relesed Assessmet Questios, 15 QUESTIONS Grde 9 Assessmet of Mthemtis Ademi Red the istrutios elow. Alog with this ooklet, mke sure you hve the Aswer Booklet d the Formul Sheet. You my use y spe i this
Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
n Using the formula we get a confidence interval of 80±1.64
9.52 The professor of sttistics oticed tht the rks i his course re orlly distributed. He hs lso oticed tht his orig clss verge is 73% with stdrd devitio of 12% o their fil exs. His fteroo clsses verge
Binary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy
CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive
Application: Volume. 6.1 Overture. Cylinders
Applictio: Volume 61 Overture I this chpter we preset other pplictio of the defiite itegrl, this time to fid volumes of certi solids As importt s this prticulr pplictio is, more importt is to recogize
2-3 The Remainder and Factor Theorems
- The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x
0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
Soving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.
5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers
Summation Notation The sum of the first n terms of a sequence is represented by the summation notation i the index of summation
Lesso 0.: Sequeces d Summtio Nottio Def. of Sequece A ifiite sequece is fuctio whose domi is the set of positive rel itegers (turl umers). The fuctio vlues or terms of the sequece re represeted y, 2, 3,...,....
1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
CHAPTER-10 WAVEFUNCTIONS, OBSERVABLES and OPERATORS
Lecture Notes PH 4/5 ECE 598 A. L Ros INTRODUCTION TO QUANTUM MECHANICS CHAPTER-0 WAVEFUNCTIONS, OBSERVABLES d OPERATORS 0. Represettios i the sptil d mometum spces 0..A Represettio of the wvefuctio i
Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
PREMIUMS CALCULATION FOR LIFE INSURANCE
ls of the Uiversity of etroşi, Ecoomics, 2(3), 202, 97-204 97 REIUS CLCULTIO FOR LIFE ISURCE RE, RI GÎRBCI * BSTRCT: The pper presets the techiques d the formuls used o itertiol prctice for estblishig
In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
Infinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
Algebra Review. How well do you remember your algebra?
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
m n Use technology to discover the rules for forms such as a a, various integer values of m and n and a fixed integer value a.
TIth.co Alger Expoet Rules ID: 988 Tie required 25 iutes Activity Overview This ctivity llows studets to work idepedetly to discover rules for workig with expoets, such s Multiplictio d Divisio of Like
5.3. Generalized Permutations and Combinations
53 GENERALIZED PERMUTATIONS AND COMBINATIONS 73 53 Geeralized Permutatios ad Combiatios 53 Permutatios with Repeated Elemets Assume that we have a alphabet with letters ad we wat to write all possible
THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
A. Description: A simple queueing system is shown in Fig. 16-1. Customers arrive randomly at an average rate of
Queueig Theory INTRODUCTION Queueig theory dels with the study of queues (witig lies). Queues boud i rcticl situtios. The erliest use of queueig theory ws i the desig of telehoe system. Alictios of queueig
7.1 Finding Rational Solutions of Polynomial Equations
4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio?
.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
Newton-Raphson Method of Solving a Nonlinear Equation Autar Kaw
Newton-Rphson Method o Solvng Nonlner Equton Autr Kw 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
Authorized licensed use limited to: University of Illinois. Downloaded on July 27,2010 at 06:52:39 UTC from IEEE Xplore. Restrictions apply.
Uiversl Dt Compressio d Lier Predictio Meir Feder d Adrew C. Siger y Jury, 998 The reltioship betwee predictio d dt compressio c be exteded to uiversl predictio schemes d uiversl dt compressio. Recet work
Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series
8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series
We will begin this chapter with a quick refresher of what an exponent is.
.1 Exoets We will egi this chter with quick refresher of wht exoet is. Recll: So, exoet is how we rereset reeted ultilictio. We wt to tke closer look t the exoet. We will egi with wht the roerties re for
MATHEMATICAL ANALYSIS
Mri Predoi Trdfir Băl MATHEMATICAL ANALYSIS VOL II INTEGRAL CALCULUS Criov, 5 CONTENTS VOL II INTEGRAL CALCULUS Chpter V EXTENING THE EFINITE INTEGRAL V efiite itegrls with prmeters Problems V 5 V Improper
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
Mathematical goals. Starting points. Materials required. Time needed
Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios
Sampling Distribution And Central Limit Theorem
() Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size,
THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
How To Solve The Homewor Problem Beautifully
Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log
Gray level image enhancement using the Bernstein polynomials
Buletiul Ştiiţiic l Uiersităţii "Politehic" di Timişor Seri ELECTRONICĂ şi TELECOMUNICAŢII TRANSACTIONS o ELECTRONICS d COMMUNICATIONS Tom 47(6), Fscicol -, 00 Gry leel imge ehcemet usig the Berstei polyomils
Operations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
Math 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
CS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least
Present and future value formulae for uneven cash flow Based on performance of a Business
Advces i Mgemet & Applied Ecoomics, vol., o., 20, 93-09 ISSN: 792-7544 (prit versio), 792-7552 (olie) Itertiol Scietific Press, 20 Preset d future vlue formule for ueve csh flow Bsed o performce of Busiess
GCE Further Mathematics (6360) Further Pure Unit 2 (MFP2) Textbook. Version: 1.4
GCE Further Mathematics (660) Further Pure Uit (MFP) Tetbook Versio: 4 MFP Tetbook A-level Further Mathematics 660 Further Pure : Cotets Chapter : Comple umbers 4 Itroductio 5 The geeral comple umber 5
Chapter 13 Volumetric analysis (acid base titrations)
Chpter 1 Volumetric lysis (cid se titrtios) Ope the tp d ru out some of the liquid util the tp coectio is full of cid d o ir remis (ir ules would led to iccurte result s they will proly dislodge durig
10.6 Applications of Quadratic Equations
10.6 Applictions of Qudrtic Equtions In this section we wnt to look t the pplictions tht qudrtic equtions nd functions hve in the rel world. There re severl stndrd types: problems where the formul is given,
Chapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity
CHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
Graphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
Integration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
Section 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
Chapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
Solving equations. Pre-test. Warm-up
Solvig equatios 8 Pre-test Warm-up We ca thik of a algebraic equatio as beig like a set of scales. The two sides of the equatio are equal, so the scales are balaced. If we add somethig to oe side of the
SEQUENCES AND SERIES
Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say
1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
Department of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
Permutations, the Parity Theorem, and Determinants
1 Permutatios, the Parity Theorem, ad Determiats Joh A. Guber Departmet of Electrical ad Computer Egieerig Uiversity of Wiscosi Madiso Cotets 1 What is a Permutatio 1 2 Cycles 2 2.1 Traspositios 4 3 Orbits
Section 8.3 : De Moivre s Theorem and Applications
The Sectio 8 : De Moivre s Theorem ad Applicatios Let z 1 ad z be complex umbers, where z 1 = r 1, z = r, arg(z 1 ) = θ 1, arg(z ) = θ z 1 = r 1 (cos θ 1 + i si θ 1 ) z = r (cos θ + i si θ ) ad z 1 z =
Sequences and Series
Secto 9. Sequeces d Seres You c thk of sequece s fucto whose dom s the set of postve tegers. f ( ), f (), f (),... f ( ),... Defto of Sequece A fte sequece s fucto whose dom s the set of postve tegers.
Integration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
AREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
Lecture 4: Cheeger s Inequality
Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a d-regular
Outline. Numerical Analysis Boundary Value Problems & PDE. Exam. Boundary Value Problems. Boundary Value Problems. Solution to BVProblems
Oulie Numericl Alysis oudry Vlue Prolems & PDE Lecure 5 Jeff Prker oudry Vlue Prolems Sooig Meod Fiie Differece Meod ollocio Fiie Eleme Fll, Pril Differeil Equios Recp of ove Exm You will o e le o rig
Fast Circuit Simulation Based on Parallel-Distributed LIM using Cloud Computing System
JOURNAL OF SEMICONDUCTOR TECHNOLOGY AND SCIENCE, VOL.0, NO., MARCH, 00 49 Fst Circuit Simultio Bsed o Prllel-Distriuted LIM usig Cloud Computig System Yut Ioue, Tdtoshi Sekie, Tkhiro Hsegw d Hideki Asi
Theorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius
A Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design
A Combied Cotiuous/Biary Geetic Algorithm for Microstrip Atea Desig Rady L. Haupt The Pesylvaia State Uiversity Applied Research Laboratory P. O. Box 30 State College, PA 16804-0030 haupt@ieee.org Abstract:
Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
Systems Design Project: Indoor Location of Wireless Devices
Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 698-5295 Email: bcm1@cec.wustl.edu Supervised
I. Chi-squared Distributions
1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.
Sequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
3. Greatest Common Divisor - Least Common Multiple
3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean
1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.
Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.
Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).
Basic Elements of Arithmetic Sequences and Series
MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
Lattice-Reduction-Aided Equalization and Generalized Partial- Response Signaling for Point-to-Point Transmission over Flat- Fading MIMO Channels
Lttice-Reductio-Aided Equliztio d Geerlized rtil- Respose Siglig for oit-to-oit Trsmissio over Flt- Fdig MIMO Chels Robert F.. Fischer Lehrstuhl für Iformtiosübertrgug, Friedrich Aleder Uiversität Erlge
MATHEMATICS SYLLABUS SECONDARY 7th YEAR
Europe Schools Office of the Secretry-Geerl Pedgogicl developmet Uit Ref.: 2011-01-D-41-e-2 Orig.: DE MATHEMATICS SYLLABUS SECONDARY 7th YEAR Stdrd level 5 period/week course Approved y the Joit Techig
Elementary Theory of Russian Roulette
Elemetary Theory of Russia Roulette -iterestig patters of fractios- Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some
Lecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
SPECIAL PRODUCTS AND FACTORIZATION
MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients
652 (12-26) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you
where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if
1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
The Stable Marriage Problem
The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
and thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
Experiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
Overview on S-Box Design Principles
Overview o S-Box Desig Priciples Debdeep Mukhopadhyay Assistat Professor Departmet of Computer Sciece ad Egieerig Idia Istitute of Techology Kharagpur INDIA -721302 What is a S-Box? S-Boxes are Boolea
Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley
Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Cosider a legth- sequece x[ with a -poit DFT X[ where Represet the idices ad as +, +, Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Usig these
Lesson 15 ANOVA (analysis of variance)
Outlie Variability -betwee group variability -withi group variability -total variability -F-ratio Computatio -sums of squares (betwee/withi/total -degrees of freedom (betwee/withi/total -mea square (betwee/withi
Solving Logarithms and Exponential Equations
Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:
Factoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles