. Written in factored form it is easy to see that the roots are 2, 2, i,

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1 CMPS A Itroductio to Programmig Programmig Assigmet 4 I this assigmet you will write a java program that determies the real roots of a polyomial that lie withi a specified rage. Recall that the roots (or zeros) of a th degree polyomial c c x c x c x c x are those umbers x for which. I geeral a th degree polyomial has at most roots, some of which may be real ad some complex. Your program, which will be called Roots.java, will fid oly the real roots. Program Operatio Your program will prompt the user to eter the degree of the polyomial, the coefficiets, ad the left ad right edpoits of the iterval i which to search. For example, to fid the roots of the cubic polyomial x 6x x 6 i the iterval [-, ], your program would ru as follows. java Roots Eter the degree: Eter 4 coefficiets: -6-6 Eter the left ad right edpoits: - Root foud at. Root foud at. Root foud at. Notice that the user eters the coefficiets from lowest power of x to highest power. The degree ca be ay o-egative iteger, while the coefficiets ad rage limits are doubles. No error checig o these iputs is required for this project. You may therefore assume that your program will be tested oly o iputs that coform to these requiremets. The root values will be rouded to 5 decimal places of accuracy. 4 Cosider the fourth degree polyomial x x ( x )( x ). Writte i factored form it is easy to see that the roots are,, i, ad i. (See if you are ufamiliar with the umber i.) As we ca see the program does ot the fid complex roots. java Roots Eter the degree: 4 Eter 5 coefficiets: - - Eter the left ad right edpoits: -5 5 Root foud at -.44 Root foud at.44 Your program will fid oly those roots that are cotaied i the iterval specified by the user. If o roots are foud i that iterval, a message to that effect is prited to stdout. We illustrate this with two more 4 searches o the same polyomial: x x ( x )( x ).

2 java Roots Eter the degree: 4 Eter 5 coefficiets: - - Eter the left ad right edpoits: Root foud at.44 java Roots Eter the degree: 4 Eter 5 coefficiets: - - Eter the left ad right edpoits: No roots were foud i the specified rage. Notice that after typig the degree, your program will as the user for the correct umber of coefficiets. A th degree polyomial has the user. Whe testig your program you may fid it useful to store the iputs i a file ad redirect stdi to read from that file rather tha the eyboard. I that case the program operatio will appear as follows. coefficiets, icludig the zero terms, all of which must be etered by more ifile java Roots < ifile Eter the degree: Eter 6 coefficiets: Eter the left ad right edpoits: Root foud at -.75 Root foud at.75 Root foud at.544 The prompts all appear o a sigle lie because there is o oe at the eyboard hittig retur to eter the iputs. All of these examples have served oly to demostrate program operatio for this project. So far we've said othig about how your program will accomplish these calculatios. To start, read sectio 4. i the text, especially the example FidRoot.java at the ed of that sectio which uses the Bisectio Method to compute the square root of. This example is also posted o the class website. The Bisectio Method If is a cotiuous fuctio o a iterval [ a,, ad if f (a) ad have differet sigs, the must have a zero i [ a,. This fact is ow as the Itermediate Value Theorem, ad is clear from the figure below. f (b) a b

3 The bisectio method is a iterative techique that traps the zero i smaller ad smaller subitervals, util the size of the subiterval cotaiig the zero is sufficietly small. The locatio of the root is the ow with a error o more tha the width of that fial subiterval. To begi let m ( a b) / be the midpoit of [ a,. This midpoit serves as the curret approximatio to the root. Loo at the two subitervals ad. If ad have differet sigs, the the root is cotaied i. If ad have differet sigs, the the root is cotaied i. If the origial iterval cotaied oe root, the exactly oe of these two alteratives must hold. Let us assume that it is the first subiterval that cotais the root. The other subiterval is discarded ad the search cotiues o the ext iteratio i. Its midpoit will serve as the ext approximatio to the root. Notice that the search space has bee halved o oe iteratio. This meas that each iteratio icreases the accuracy of our root estimate by oe biary digit, ad about 4 iteratios gais oe decimal digit of accuracy. The algorithm cotiues to loop util the iterval i which the root is ow to exist has width less tha some predefied tolerace. f (b) [ m, f (a) f (m) [ m, f (m) If the iitial iterval cotais more tha oe root of the this method will oly fid oe of them. If we wish to fid all roots withi some rage L to R we must partitio [ L, R] ito a sequece of sufficietly small subitervals of equal width, the ru the Bisectio Method o each such subiterval for which f (a) ad f (b) have differet sigs. The width of these smaller subitervals therefore costitutes a limit o the possible resolutio with which roots ca be detected. I other words, roots i [ L, R] that are closer together tha the resolutio b a caot be distiguished. Whe ru o the example pictured below, this procedure will fid all three roots. b a a b L [ ] [ ] [ ] R The procedure will brea dow however whe has two roots that are "ifiitely close". For istace the polyomial ( x )( x ) x 5x 8x 4 has a so-called double root at the poit x ad a sigle root at x. odd root eve root y x 5x 8x 4 [ ] a b

4 No matter how fie a resolutio we pic, we caot distiguish betwee the "two" roots at the bisectio method caot fid this root at all sice it is ot the case that ad sigs, o matter how small b a is. f (a) f (b) x. I fact have differet We ca divide the roots of a polyomial ito two classes called the odd roots ad the eve roots, respectively. The poit is a odd root if ad oly if the graph of crosses the x-axis at. We call a eve root if ad oly if the graph touches the x-axis at. Sice it requires a sig chage, the bisectio method ca oly fid the odd roots of a polyomial. However the eve roots of are amog the odd roots of a related polyomial called its derivative. We illustrate this with the polyomial from the previous example. f (x) y f ( x) x x 8 ew odd root at As we ca see, ot all the odd roots of the derivative f (x) are eve roots of the origial polyomial f(x). It ca be show however that every eve root of is a odd root of f (x). Represetig Polyomials Derivatives are studied extesively i Calculus. It is ot ecessary however to ow how to differetiate geeral fuctios or eve to ow what a derivative is i order to use the above fact, so studets who have ot had Calculus eed ot be dismayed. The world of polyomials is much simpler tha the world of geeral cotiuous fuctios i Mathematics. To specify a th degree polyomial oe eed oly specify a list of umbers, amely its coefficiets. c c x c x c x c 4 x odd root at x x c x I this project these umbers c (for ) will be stored a array of legth polyomial is a array. The derivative f (x) is a polyomial of degree whose coefficiets are give by a simple rule. c c x c x ( c ) x c x. For us the, a I other words, if the coefficiet array for is c, c, c, c,, c, c ) the the coefficiet array for ( f (x) is ( c, c, c,, ( ) c, c ). To put it aother way, if c (for ) are the coefficiets of ad d (for ) are the coefficiets of f (x), the d ( ) c (for ). 4

5 Oce the coefficiets of are calculated, oe ca ru the Bisectio Method o to fid its odd roots. Each of the roots of f (x) are the plugged ito to see if. If so, we have foud a eve root of. We ru ito aother problem at this stage though. All of these calculatios will be doe usig the type double. Wheever two quatities u ad v are idepedetly computed ad stored usig a floatig poit data type, roud off ad approximatio errors are itroduced. Eve if some Mathematical theory tells us that u v, the stored floatig poit values may ot i geeral be equal. We ca expect however that is less tha some positive threshold value which is close to zero, but ot closer tha u v f (x) 5 f ( ) the accuracy that the calculatio allows. I particular we caot directly cofirm that is at best oly a approximatio to a root of a approximatio to its actual value. Istead, to chec whether f ( ) threshold. f (x) f ( ), ad for that matter our computed value for is also a root of f ( ) sice is oly, we chec that Program Specificatios Your program is required to implemet the followig static fuctios to carry out the operatios described above. Your fuctios must match the oes below i their ame, retur type, ad parameter list. Your iitial tas is to fill i the braces with appropriate Java code ad test each fuctio thoroughly. static double poly(double[] C, double x){...} A call to poly(c, x) will retur the value at x of the polyomial with coefficiet array C. You ca accomplish this by writig a loop that multiplies each coefficiet by a appropriate power of x, ad accumulates the sum of all such terms. Whe the loop termiates, retur the sum. static double[] diff(double[] C){...} The call diff(c) will retur a referece to a ewly allocated array D cotaiig the coefficiets of the polyomial that is the derivative of the polyomial with coefficiet array C. I other words the fuctio poly(d, x) will be the derivative of the fuctio poly(c, x). static double fidroot(double[] C, double a, double b, double tolerace){...} Assumig poly(c, a) ad poly(c, b) have differet sigs, a call to fidroot(c, a, b, tolerace) will retur a approximatio to a root of poly(c, x) i the iterval [a, whose error is o more tha tolerace. Implemet this fuctio by usig the Bisectio Method illustrated at the ed of sectio 4. ad i the examples FidRoot.java, FidRoot.java ad FidRoot.java o the class webpage. This fuctio has a precoditio that says the polyomial taes opposite sigs at the edpoits of the iterval. Therefore it should oly be called whe that precoditio is satisfied. Oly after these fuctios are worig, should you begi writig fuctio mai() for the project. The followig steps are offered as a rough outlie of program logic for fuctio mai().. Declare double variables resolutio, tolerace ad threshold, ad iitialize them to, 7 ad respectively. Declare ay other variables eeded by fuctio mai().. Get the degree of the polyomial ad its coefficiets, alog with the left ad right edpoits of the search iterval [L, R] from the user.. Calculate the coefficiets of the derivative polyomial by callig your fuctio diff(). 4. Begi a loop iteratig over all subitervals [a, of width resolutio formig a partitio of [L, R]. 5. For each such subiterval [a, 6. If the polyomial chages sigs across [a, 7. Fid a odd root of the polyomial i [a, accurate to withi tolerace 8. Prit the value of the root rouded to 5 decimal places

6 9. Else if the derivative polyomial chages sigs across [a,. Fid a odd root of the derivative polyomial i [a, accurate to withi tolerace. If the absolute value of the polyomial evaluated at this root is less tha threshold. Prit the value of the root rouded to 5 decimal places. If o eve or odd root was foud i [L, R] prit a message to that effect The above outlie is a example of what programmers call pseudo-code. Somethig closer to Eglish tha actual Java code, but structured to express algorithm steps. Note that idetatio is sigificat i pseudocode, ulie i Java. Loop bodies, as well as the true ad false braches of coditioals are idicated solely by idetatio. You may defie other fuctios as you see fit, but the oes described above are required for full credit. A example will be posted o the webpage illustratig how oe ca roud ad format a double value to a specific umber of decimal places. You may play with the parameters resolutio, tolerace ad threshold, but the specific values metioed above should result i your output matchig the examples exactly. More such examples will be posted o the webpage for testig purposes. What to tur i Your fial tas is to write a Maefile for this project alog the lies of the oe i lab assigmet 4. This Maefile should create a executable jar file called Roots allowig oe to ru the program without havig to type java at the commad lie. Iclude a clea target as i lab4. You may also write a submit target lie the oe i lab4. You may also try to write aother phoy target called chec that checs the files you submitted. Submit the two files Maefile ad Roots.java to the assigmet ame pa4. This project is cosiderably more ivolved tha ay of the earlier oes, so you will have more time to complete it. You should evertheless start early, wor o oe thig at a time, ad as questios of myself, the TAs ad o Piazza. 6