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finite componentsExamples include trusses,reactors,and electric circuits The three jumpersat the beginning Chap8 area lumpedsystem bungee of Conversely,distributed variable problems attempt to describethe spatial detail of systerns a continuous semicontinuous on or basis The distribution chemicals of alongthe lengthof an elonqated, reactor(Fig PT3lb) is an exantple a continuous rectangular of variablemodel Diffbrential equations derivedfiom conservation laws specify the distribution of the dependentvariable fbr such systemsThese differentiai equationscan be solved numericallyby convertingthem to an equivalentsystemof simultaneous algebraic equations The solutionof suchsetsof equations leplesents major applicationareatbr the metha cldsin the fbllowing chapters TheseeqLrations coupledbecause variables one locaare the at tion are dependent the variables acljoining on in regionsFor example,the concentration at the middle of the reactor in Fig PT311,is a function of the concentration adjoining in regionsSimilar examplescould be developedfor the spatialdistributionof temperature, m o n r e n l u m r e l e c t r i c i t l o Aside fiom phvsicalsystems simultiineous linearal-uebraic equations also arisein a varietyof mathernatical problemcontexts Theseresultwhen mathematical functions are required satisfyseveral sirnultaneously condition Each to conditions results an equation in that contains known coefflcients uuknownvariables and discussed this in The techniques part can be used to solve fbr the unknownswhen the equationsare linear and algebraic Somervidely usednunrerical techniques that employ simultaneous equations legresare sion analysis and splineintel'polation
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Due to its importancein fbrmulating and solving linear alsebraicequations Chap 8 provides a brief overviewof tnotrir algebruAside from coveling the rudimentsof matrix representation anclrnanipulation, clrapter the also describeshow man'icesare handledin MATLAB Chttpter 9 is devotedto the most fundamentaltechniquefor solving linear algebraic systems: launching into a detailed GausseLimittolionBefore discussion this technique, of a pleliminary sectiondealswith simple methodsfor solving small systerns These approaches presented provideyou witlr visualinsightandbecause oftlre rrrethodsare one to the elimination unknowns-represents basis Gauss of lbr the elimination After this prelirrrinary material,"naive" Gausseliminationis discussed starl with We it this "stripped-down"versionbecause allows the fundarnental technique be elaborated to potential probon without complicating Then,in subsequent sections, discuss details we lems of the naive appr-oirch presenta number of mocliflcations mjnirlize and cirto and cunlventtheseproblenrs The fbcus of this discussion will be the process switching of rows,orpdrlial pit'otingThe chapter endswith a brief description eflicientmethods of fbr solving tridiagonal ntatrrces how Gauss eliminationcanbe lbrmulatedas an LI) factoriZation 10 illustrates Such solutiontechniques valuable1br caseswhere many right-hand-side are vectorsneed The chaoterends witlr a brief outline of how MATLAB solveslinear to be evaluated s\/stet)-ls
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-1-1 startswith a descriptionof how LU factorizationcan be employedto efficiently calculatethe matir inverse,which has tremendous utility in analyzingstimulusrelationships physicalsystems response of The remainderof the chapteris devotedto the importantconceptof matrix conditionThe conditionnumberis introducedas a measure of the roundoff errorsthat can result when solving ill-conditionedmatrices I2 dealswith iterative solutiontechniques which are similar in spirit to the for in approximate methods rootsof equations discussed Chap6 That is they involve guessThe emphasis on the Gaussis ing a solutionand then iteratingto obtaina refinedestimate Seidel method, although a description is provided of an alternative approach, the Jacobi methodThe chapterendswith a brief description how nonlinearsimultaneous of equations canbe solved
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CHAPTER OBJECTIVES
The primary objectiveof this chapteris to acquaintyou with linear algebraicequations and their relationshipto matricesand matrix algebraSpecificobjectivesand topics coveredare e ' U n d e r s t a n d i n ga t r i xn o t a t i o n m Being able to identify the following typesof matrices:identity,diagonal, symmetric,triangular,and tridiagonal Knowing how to perlbrm matrix multiplication and being able to assess when it is feasible Knowing how to represent systemof linear algebraicequations matrix form a in Knowing how to solve linear algebraicequations with left division and matrix inversion MNLAII in
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by upposerhat three jumpers are connected bungeecordsFigure 8ln shows them It We but so Uing held in placeverticerlly that eachcord is fully extended unstretched \ dovnv'ard from eachof their t'1 crn define threedistances ,t'2and \ as measured 9/ grar po:itions ,ltierrhel are reJeased it1 takeshold and thejumpers s ill evenunstretched positions tually cometo the equilibriun-r shownin Fig 81b Suppose that you are askedto computethe displacernent eachof thejumpersIf we of as assumethat eachcord behaves a linear spring and follows Hooke's law, free-bodydiagramscan be developed eachjumper as depicted Fig 82 for in forcebalance be writtenforeachjumper: law, a steady-state can Using Newton'ssecond : m 1 gI k z ( x z- 1 1 )- f t 1 r 1 Q - x ) - k z ( x z- r r ) : 0 t1tlg kr(,r: * tntS - k: (r: - irz) : 0
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