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0 Thusafter25 iterations searching, of It fzero finds a signchange thenuses interpoto lationandbisection until it getscloseenough theroot so that interpolation takes overand rapidlyconverges the root on Suppose we would like to usea lessstringent that We tolerance canusethe opt imset functionltl set a low maxinrumtolerance and a lessaccurate estimate ol'the rootresults:
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>- opl-ions = optim:,;e1,('l-olx', 1e-l); = tzcro(G (x) x^10 1,05,options) >) lx,lxl 1 0009 00090
Polynomialsare a specialtype of nonlinearalgebraicequationof the generalfbrnr t + | f r ( r ): d 1 , t "* a 2 x ' ' - t ' + o u t y 2 u , , ! a , , a 1
ertJ the MI tior foll
In where n is the order of the polynomial, and the a's are constantcoefficients many (but not all) cases, coefficientswill be realFor suchcases, rootscan be real and/orcomthe the plex In general,an nth order polynomial will haven roots Polynomialshave many applicationsin engineeringand scienceFor example,they are used extensivelyin curve fitting However,one of their most interestingand poweris linear systems ful applications in characterizing dynamicsystems-and,in particular, includereactors, mechanical devices, structures, electrical and circuits Examples 651 MATLAB Function: roots If you are dealing with a problem where you must determinea single real root of a polymethodcanhaveutility nomial,the techniques suchasbisection andthe Newton-Raphson desireto determine the roots,both real and complex all However,in many cases, engineers fbr like are simpletechniques bisection and Newton-Raphson not available Unfbrtunately, polynomials However,MATLAB has an exceldetermining the rootsof higher-order all the lent built-in capability, roots function,fbr this task The roots functionhasthe svntax
the wherex is a columnvectorcontaining rootsand c is a row vectorcontaining polythe nomial'scoefficients So hclwdoesthe roots functionwork MATLAB is very good at finding the eigenis the task as an valuesof a matrixConsequently, approach to recast root evaluation the problems problemBecause will be describing eigenvalue Iaterin the book, eigenvalue we we will merelyprovidean overviewhere Suppose havea polynomial we / t r 5 o 2 x 1+ , r , r t + a + x z* o 5 x+ a 6 : 0 + yields Dividing by rrsand rearranging
5 I : (Il -f dl I (l I -l' (ll I (14 At (l\ Ul (Irr Al
( 6 1) |
fiom the right-hand sideas A specialmatrix can be constructed usingthe coefflcients by 's the first row and with I and 0's writtenfbr the otherrows as shown:
-o2fu1 -u3fd1 -aafo1 -a5fu1 -aolar
10000 0t000 00r00 00010
r:ompanion matix It has the usefulpropEquation(6 l3) is calledthe polynomial's Thus, the algorithmunderlying are erty that its eigenvalues the rootsof the polynomial of the roots functionconsists merely settingup the companionmatrix and then using MATLAB's powerful eigenvalueevaluationfunction to determinethe roots Its applicapolynomial in manipulation functions, described the are tion, alongwith someotherrelated following example
ROOTS: OPEN METHODS
We shouldnote that roots has an inverse po1y, which whenpassed functionc-alled of the values the rootswill returnthe polynornial's coefficients syntaxis Its
'' '' l'1Y(r)
where r is a column vector containingthe roots and c is a row vectorcontainingthepoly nonliai'scclef ficients EXAMPLE 2 6 ' ; U s i n g M A T L A Bt o M o n i p u l o t eP o l y n o m i o l q n d D e t e r m i n e h e i r R o o t s s T Problem Siotement Use the following equation explorehow MATLAB canbeemto ployedto ntanipulate polynomials: 1 (E671) r / ! ( x ) : - r ' ' - 3 5 r '* 2 ' / 5 x t 1 2 1 2 5 , 1 2 3 u 7 5 + 1 2 - 5 Note that this polynomialhasthreereal rcots:05 - 10,and 2: and one pair ol complex I roots: +051 Solution Polynomials enteredinto MATLAI) by storingthe coefflcients a row are as \/ector example, Ftir entering following line stores coefficients the vector the the in a: -'' a = f I 35 215 2'l'2,\ i875 1251; We can then proceedto manipulate polynomialFor exarnplewe can evaluate at the it r : I, by ryping
:> p c )l y v a l (a, I )