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That is, the relative error of the norm of the computedsolutioncan be as large astherelative error of the norm of the coefficients [A] multiplied by the conditionnumber exof For 0n arnple,if the coefficientsof [A] are known to t-digit precision(ie,roundingerrorsare the order of 10-r) and Cond[Al : 10', the solution[X] may be valid to only r - cdigits (roundingelrorsry l0' -' ) EXAMPLE 13 I M o i r i x C o n d i t i o nE v o l u o t i o n ill-conditioned, bereprecan Problem Stotement The Hilbert matrix,which is notoriously sentedgenerallyas

t t ) , : y l : 5 , a n dL t : L t : 2 S e tt h ei n i t i a c o n d i t i o na s l s : where k: the springconstants,n n-rass, the lengthof "\t: Lt tutdx, : L, I u, + L2+ 6 Perfbrrn simulation L: the theunstretched springand w : the width of the massCom- from / : 0 to 20 Constructtime-series plots of both the disputethe positionsof the masses a functionof time usin-t placements as producea phaseand the velocitiesIn adclition, thefbllowing parameter values:t, : ft: : 5, mt : m):2, planeplot of r, versus r,

We will now use this function in exactly the same way we did when getting group information from a table Here is the query to put in the initialization block:

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The primary objectiveof this chapteris to introduceyou to solving boundary-value problemsfor ODEs Specific objectivesand topicscoveredare o problems the Understanding differencebetweeninitial-valueand boundary-value Knowin-qhow to expressan nth-orderODE as a systern n first-orderODEs of Knowing how to implementthe shootingmethod1brlinear ODEs by usinglinear ''shots" interpolationto generate accurate how derivativeboundaryconditionsare incorporated into the Understanding shootingmethod Knowing how to solve nonlinearODEs with the shootingmethodby usingroot "shots" locationto generate accurate Knowing how to implementthe finite-difference method Understanding how derivativeboundaryconditionsare incorporated into the finite-differencemethod Knowing how to solve nonlinearODEs with the finite-difference rnethod using by root locationmethodsfor systens of nonlinearalgebraicequations

SELECT 'GROUP' variable_name, SUBSTR(column_value,4) variable_value FROM TABLE(get_groups(':USER')) WHERE column_value LIKE 'BI-%' UNION ALL SELECT 'WEBGROUPS' variable_name, SUBSTR(column_value,7) variable_value FROM TABLE(get_groups(':USER')) WHERE column_value LIKE 'BIweb-%'

jumper o this point, we havebeencomputingthe velocity of a free-fallingbungee by integratinga singleODE:

tlu -; :8 atm c,t t --:u-

.

.

(222)

Thus, by solvingthe system two ODEs representedby of Eqs (221) and(222),we can simultaneously deterrnine both the velocityand the position However, because are now integratingtwo ODEs, we require two conditionsto we obtain the solutionWe are aheadytamiliar rvith one way to do this for the casewhere we for have varlues both positionand velocity at the initial time; x(r:0):r:, tr(l:0):ur Given such conditions,we can easily integratethe ODEs using the numericaltechniques in described Chaps20 and 2l This is refened to ils an initial-valueproblem But what if we do not know valueslbr both positionand velocity at / : 0 Let's say that we know the initial position but rather than having the initial velocity, we want the jumperto be at a specifiedpositionat a later time In other words: t(l : 0) :v; x(t:t1):ry variable,this is Because two conditionsare given at ditferentvaluesof the independent the problem called a boundary-value Such problemsrequire speciaisolution techniques Sonre theseare relatedto the of in Howmethodsfor initial value problemsthat were described the previoustwo chapters to ever, others emplov entirely different strategies obtain solutionsThis chapteris designedto introduceyou to the more comnlon of thesemethods

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