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Lake Havasu
FIGURE I II2 P TheLower River Colorodo
coefficientsof the fourth-orderpolynomialf(r) : p6" * pzx3t pzx2+ p$ + psthat passes throughthe following five (300,0616), (400,0525), points:(200,0746),(250,0675), into and (500, 0457)Each of thesepairs can be substituted (Pl 114) l p2x"-2+ + pn rx * pn Eq (Pl1l4) to yield a systemof five equations with five to thep's are constant coefficientsA straightforward unknowns(thep's) Use this approach solve for the coeffor computingthe coefficients is to generate n linear ficients In addition, determineand interpret the condition number equations that we can solve simultaneouslyfor that we want to determine the coefficients Suppose Polynomial interpolationconsistsof determiningthe (n - 1)th-orderpolynomial that fits n data points polynomials havethe generalform,
,','"
Iterotive Methods
CHAPTER OBJECTIVES
The primary objectiveof this chapteris to acquaintyou with iterativernethods for solving simuitaneous equations Specificobjectivesand topicscoveredare ' ' ' ' Understanding differencebetweenthe Gauss-Seidel Jacobimethods the and Knowing how to assess diagonaldominanceand knowing what it means Recognizinghow relaxationcan be usedto improve the convergence iterative of methods Understanding how to solve systems nonlinearequations of with successive substitutionand Newton-Raphson
EXAM
terative or approximatemethods provide an alternativeto the eliminationmethodi I describedto this point Such approaches similar to the techniques developed are we to I I obtain the roots of a single equationin Chaps5 and 6 Those approaches consistedof guessinga value and then using a systematic method to obtain a refined estimate the 0f root Because present part of the book dealswith a similar problem-obtaining the the valuesthat simultaneously satistya setol'equations-we might suspect that suchapproximate methodscould be useful in this context In this chapter,we will presentapproaches for solving both linear and nonlinearsimultaneous equations
l2l
IINEAR SYSTEMS: GAUSS-SEIDEL
The Gauss-Seidel methodis the most commonly used iterativemethod for solving lineu algebraicequations Assumethat we are given a setof il equations: lAl{"r} : {b} Suppose that for conciseness limit ourselves a 3 x 3 setof equations thedragonal we to If elementsare all nonzero,the first equationcan be solvedforr1, the second x2,andthe for
121 LINEAR SYSTEMS: GAUSS-SEIDEL
third for xr to yield
, Dr-4ttX
-clrr-{l ,-!
(t21a)
, i tl D) -AtqXi -d)tt; a22
(12llr)
bt-ayrl-a:txl
(12rc)
whereT and j - I are the presentand previousiterations To start the solution process,initial guesses must be made for the x's A simple approach is to assumethat they are all zeroThesezeroscan be substituted into Eq (121a) which can be used to calculatea new value for -r1 : bt latr Then we substitute this new valueof r1 alongwith the previous guess zerofor-rr into Eq (121b)to compute'a of new is value for x2 The process repeated Eq (121c)to calculatea new estimate x3 Then for for we return to the first equationand repeatthe entire procedureuntil our solutionconverges closely enoughto the true valuesConvergence be checkedusing the criterion that for can lli' Jri -r/'I e,,': l--l lr;l :XAMPLE l l2 Gouss-Seidel ethod M ProblemStqtemeni Use the Gauss-Seidel methodto obtainthe solutionfor 3xr-01r2-02x3: 185 0lxr* 7r2-03xj: -193 0 3 1 10 2 t 2 + l 0 r 3 714 Note that the solutionis ixlr : |3 -25 i ) < x 100% r, rll rr
Solution First,solveeachof the equations its unknownon the diagonal: for 785+01,r2f01yl
3 - 193 0lrr t 03r,r 1 1 14- 03x -l 02x2 r
( Er 2 r l ) (Er212) (E1213)
By assuming that;r2 x3 arezero,Eq and (E1211) be used compute can to
^t -
785+0r(0)+02(0)
:2616661
METHODS ITERATIVE
into valueof -rr : 0, can be substituted Eq'(El2'l This value,alongwith the assumed to calculate
- 1 9 3 0 1 ( 2 6 1 6 6 6 1 ) 3 ( 0 ) +0 -
1qL\)L
by The first iteration is completecl substitutingthe calculatedvaluesfor 'rt and'r: into to E q ( E 1 2 1 3 ) y i e l d
a 1 -
1 14- 03(2616661 02(-2191524) 7005610 )+ : l0
to is For the seconditeration,the sameprocess repeated compLlte
I 0) + 785+ 0r(-2194524) 02(1'0056 :2990551 3 - 1 9 3- 0 1 Q 9 9 0 s 5 7 0 3 ( 7 0 0 s 6 1 0 - 1 4 q q 6 r 5 + ) - ) 7 (2990s57 02(-2499625) 1 r4 - 03 )+ :1000291 l0
couldh The method is, therefore,convergingon the true solutionAdditional iterations th know However, in an actualproblem, we would not applied to improve the answers theenorFr to a Eq true answeraprioriConsequently, (12'2) provides means estitnate for x1: example, ', - 2'616661| ,00" : |2'990551 : 125(/'
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