GUARANTEED REAL ROOTS CONVERGENCE OF FITH ORDER POLYNOMIAL AND HIGHER by Fard A. Chouery, P.E. US Copyrght, 7 Itroducto: I dg the roots o the th order polyomal, we d the aalable terate algorthm do ot ge a guaratee o dg the roots. For eample, Newto s or Halley s terato ormula rus to dcultes whe ( ad ( or ( ad ( are smultaeously ear zero. Ths problem happes ot oly whe the root makes (, ( ad ( zero but what durg the terato process oe o the alues also ges ( = Newto s terato or both ( = ( = Halley s. Ths codto throws the mproed + to ty. Thus shg aroud or a ew + s requred wth o guarates, causg addtoal teratos that stll may stumble to a ew alues that ges ( = Newto s terato or both ( = ( = Halley s. Smlarly, ths problem also happes Lageree s Method. We also ote the coergece algorthm o L- Bastow the coecet the deomator s zero t throws the mproed + to ty. Aga they all do ot ge a guarateed algorthm o coergece. Aother problem s the tal alue t s at the programmer ow rsk t may cause dergece. Ths alue has to be sutable to cause coergece. So how do we pck the tal alue? We seek a guarateed algorthm or the th order polyomal that has a prescrbed tal alue. Ths ew algorthm s at least o ourth order coergece ad most o the tme s o th order coergece. So t s epected to coerge most o the tme e teratos depedg o the polyomal coecets. Settg up the soluto: We kow wth hgher order Taylor epaso t wll represets a better appromato o the ucto ( tha Newto or Halley s method. Newto ollows the slope o a le to d + alue. We wll ollow a th order polyomal cure stead ad s obtaed rom Taylor epaso, thus: ( ( ( ( ( = ( + ( ( + ( + ( + (.( Whch wll ge th ( ( order coergece. We ote that =.! ( ( ad =! or. So ( ( (. ( ( ( = ( + ( ( + ( + ( + ( + ( Thus Eq. lacks oe more term to brg us back to the orgal th order polyomal o Eq. where: Electrcal/Structural/Foudato Proessoal Egeer, BSEE, MSEE, MSE (Structural rom the Uersty o Washgto, WA. Presdet/CEO o FAC SYSTEMS INC. Seattle, WA (www.acsystems.com
( = + a + b + c+ d =. ( Assumg we are tryg to d the roots o the equato: y + py + qy + ry + sy+ t =. ( ad we hae substtuted y= p / to ge: a = + q b = q + r c = + q r + s d = q + r s + t the coecets Eq.. We do ot eed to proe that Eq. s a appromato to Eq. or Eq.. It s geerally uderstood that we ollow the polyomal cure o Eq. t would ge a closer alue to the actual root; more closer tha Newto terato because or ay t s a better appromato o the ucto tha Newto s appromato (see Fgure. These practcal assumptos ca be arly accepted by most mathematcas. All would agree that Taylor seres wth mssg term or two ges a closer appromato o ay ucto ( proded that Taylor Seres Remader term s small eough ad so the root + must be close to the root ad o proo s eeded ( ths s stll a questo the just compare alues wth arous appromated ucto. Whe solg the Quartc equato o Eq., the terato we pck the closest root o Eq., * = (, that s closest to zero so ( = Eq. ad the al becomes a root o (. ( We start our algorthm by rst checkg ( at ay root o ( otherwse we wll ru to ( problems dg the roots o Eq. whe ddg by ( proded s the eghborhood o * (. ( = = at =. Ths mea we ca eer hae a =. The oly way ths ca happes s whe d =. Thus, the rst check s dg out d =, so the root s = ad we d the remader o the roots by ddg Eq. by ad proceed to d the root o the Quartc equato. I d s ery close to zero but ot zero ad the selected tal alue, or some other reaso t ( causes ( = the treat Eq. to a lower order polyomal or oly that alue to d usg the equato bellow ( ( = ( + ( ( + ( + (. A (
I ( s also zero or the same alue the treat Eq. A to a lower order polyomal or oly that alue to d usg the equato bellow ( ( = ( + ( ( + (.. B I ( s also zero or the same alue the treat Eq. B to a lower order polyomal or oly that alue to d usg the equato bellow = ( + ( (.. C ( ( s also zero the use Eq. or a better appromato we replace by ad sole or. * [ ( ] = + =. D Obously s ( s also zero the = s the root. Ths guaratees coerso or the selected tal alue speced the et secto. Aother assumpto we hae s that the Taylor th order polyomal, Eq., has a real root ad ca eer ge magary roots. Ths assumpto ca be cotradcted whe ( has o roots or all the roots o ( ges ( >. To show the assumpto that whe ( has o roots the Taylor th order polyomal Eq. has magary roots. Dde Eq. by ( ad let u /( = yelds: ( ( ( ( h ( u = ( + u ( u + u + u+ =. ( Smlarly dde Eq. ( yelds: ( ( ( ( g( u = ( + u ( u + u + u + u+ =.. ( I we multply Eq. by u we d the Eq. ad Eq. are detcal uctos ecept Eq. s traslated the y-as by. I ( or ( has o real root the Eq. has oe real root, so s Eq. t would hae oe real root. Thus, Eq. whe multpled by u to become uh(u would hae oe real root. Ths was doe to match Eq. ad t would oly be traslated by. Sce, uh(u = has oe real root amely u= the uh(u would hae o other real root. Thus, h(u has o real roots or Eq. has o real roots ad they are all magary. All ths because ( has o real root. Ths The problem ( = Newto s terato or ( = ( = Halley s terato or ay polyomal ca be aoded by addg a hgher term or a better appromato o Eq. ad d ad the term the bracket s greater the zero m s ee. m! ( m + = + ( m where (m ( s selected to be ot zero (
problem ca also happe ( has oly oe root. A sucet test s to determe k( = ( ( has a real root, where s a approprate tal alue used Eq. y Cure Eq. has o real roots Newto + ( Eq. root * Taylor Eq. a th order polyomal (-(-^ FIGURE ad k( matches Eq. (ths ca be dered by usg Eq. or ( ad subtractg ( to become Eq.. I so the all o ( ( has at least oe real roots Ths ca be easly see graphcally as gure whe traslatg the same ucto by the subtractg rom (. We sayg ( tersect oce the t s shted t wll tersect at least oce more. (Note: ( = k (. Thus we ca costruct the cure Fgure. Aother preerred alterate tha dog the test or the th order polyomals s to go ahead ad d the roots o ( ad (, (Note: we eed to do that ayway dg the tal codto descrbed orgog secto ad see ( roots are magary. I the test or the roots ( shows at least Note: ths test s useul because or eample ( s a odd ucto ad has a root ad s a hgher order polyomal or ee t s ot a polyomal that has bee appromated by a Taylor polyomal (For eample ( = ( + ( /! + ( /! + ( /! + + ( ( /!, where s odd ad ( ( s appromated umercally wth hgh accuracy wth a acceptable remader. I ths case (, (, (, ( (, does ot eed to be appromated wth hgh accuracy smlar to the Secat Method wth more tal alues. So the test becomes: k( = ( = ( + = ( (! ( ( ( + ( ( ( + ( + ( ( ( I k( has a real root usg the approprate tal alue the all o k( has a real root whe replacg by ad tells us the Quartc Eq. has o magary roots or all Ths ca be cocluded sce we are subtractg rom ( a traslated ucto by. I the case o a ee ucto ( t s saer to use Eq.7. I the tal codto all cases are guest at stead wthout usg the aboe guarateed coersos or dg the roots o ( the guessg the tal alue s permtted ad legally ad ca be deeded a court o law, the case o a ueducated perso the jury questos the egeer s result or guessg a umber. The reaso s t ca be permtted because t ca be ered wthout guessg wth our guarateed coersos o roots ad compared wth other roots - please see commetary the ed o the paper.
oe real root the Eq. wll always hae at least oe real root. I the selected test has o real root wthout checkg ( the t s ( ( y ( ( - Typcal root best to aod ths stuato ad use a cubc Taylor appromato stead o usg Eq. or we use the ollowg equato: ( ( Where, the order o coergece drops to th order ad the terato we pck the closest root ( = ( + ( ( + ( + ( (7 ( to zero the cubc equato.. I ths case we start our algorthm by rst checkg ( at ay root o ( where s the eghborhood o otherwse we wll ru to problems dg the roots o Eq.7 whe ddg by. ( = + a= at = ± a. I a> there s o eed to worry could eer be ( (
zero, other wse check ad see ( ± a =, true the root s oud ad we proceed usg sythetc dso to d the remader roots o the Quartc equato. I the root s ery close to ± a but ot that alue ad the selected tal alue or or some other reaso causes ( = the treat Eq. to a lower order polyomal or oly that alue to d by usg Eq. B. I ( s also zero use Eq. C. I ( s also zero the use Eq. or a better appromato wthout the orth order term by replacg by ad sole or. = [ ( ] = + (8 Obously ( s also zero the = s the root. Ths guaratees coerso or the selected tal alue speced the et secto. Fdg the tal alue : Ater makg the aboe check, we proceed by dg the roots o ( = ad d the zero slopes. Secod we d the roots o ( =, or the lecto pots. These roots ca be easly oud usg the Quartc ad Cubc polyomal solutos. Let m ad ma be the most egate ad the most poste root o ( ay ad let m ad ma be the most egate ad the most poste root o ( ay. By specto ( m > the our root * < m ad ( ma < the our root * > ma. Sce we kow the upper ad lower boud (-M, M: [ + ma( a, b, c, d, ma(, a + b + c d ] M = m + ad M < * < M The the tal alue ca be take as [ m( ', " M] m m = [m( m, m ] >, -M < * < m( m, m [ + ] " m m = [m( m, m ] < ad ( m >, m < * < m I m do ot est set m = m dg.. (9 or [ ma( ', " + M] ma ma = [ma( ma, ma ] <, M > * > ma( ma, ma [ + ] " ma ma = [ma( ma, ma ] > ad ( ma <, ma > * > ma I ma do ot est set ma = ma dg. (
I oe o the aboe s satsed the use or et cosecute hump at ad + where ( > ad ( + < wth < j < + " ' " ' j + j + + = ( j < ad = ( j >... ( Ths selected alue o wll guaratee coergece. I there are other termedate ad roots that maybe sutable ad do ot ole ±M the algorthm wll pck the aster alues M s ery large. The process ca be easly doe by sortg the combe data o ad ad estgate ( ad (. The proo o coersos ater selected tal codtos ca be see graphcally usg Newto s terato slope le or a lower cure appromato o Eq. or Eq. 7. Real Roots o Hgher Order Polyomals: I we go to the th order polyomal, we d usg the aboe th order polyomal soluto we ca d the slopes ad the lecto pots. I there s oe root or ( ' = ad ( ' > the all the roots o ( are all magary, other wse we ca proceed wth Eq.7 to d the real roots. Howeer, beore teratg we eed to make sure ( whe usg Eq.7 ths procedure or dg the ew + s smlar to the th order procedure. Thus, the real roots o the th order polyomal are guarateed to be etracted. I we go to the 7 th order polyomal aga usg the soluto o the th order we ca etract the slopes ad the lecto pots. I there s o zero slopes oud or all the roots o ( are magary, the there s oly oe root. The soluto s smlar to the th order polyomal ad the real roots are guarateed to be etracted. Fally, we go to the 8 th order we d smlartes to the th order kowg they are all the roots are magary. Wth ths we leae t to the reader to costruct a guarateed algorthm or hgher order polyomal. It s realzed that order to wrte the algorthm o polyomal all the preous algorthm o lower order has to be programmed. It s a ery ested challegg program or hgher order polyomals ad probably a epert programmer may be attracted to the problem. Commetary ad Hts to Guaratee Imagary Roots ad mult-dmeso uctos: For the th order polyomal that has magary roots substtute or = u + Eq. where u ad are real umbers ad caot be zero. Whe collectg the real ad magary Eq. t orces dg the root (u *, * o the ollowg two polyomal: ( u, = u g( u, = u + au + au + bu + cu+ d + bu+ c ( u ( u + au+ b + a + = + u =..... ( Sce or ay we ca guaraty a root u both equatos ad sce there s a u that makes both equatos o Eq. tersects (ths ca be see because oe equato s a th order u ad the other s a th order u the there s a soluto u that satses both equatos. 7
Now the tal alue ca be selected by takg =, = gg our sets o equato u related to as ollows: = = = = ( u + au+ b ( u + au+ b ( u + a or or g or g ( u + a or.. ( We see that yelds: < g ad < g or eery u. Also ± M ca be calculated rom Eq. M u + au+ b u + au + bu + cu+ d u + au+ b u + au + bu + cu+ d = m + ma,, ma,, u u u u ( I we use Eq. ad as we doe wth Eq. 9 ad to select as a ucto o u ad substtute Eq. ad d a u usg a guarateed coerso algorthm or each equato (such as usg the Author s or Improed Newto s or ( = problem or Halley s or ( = ( = problem. Now pck ay u the the tal codto (u, or guarateed root are oud. The we ca use a two dmesoal Newto s algorthm wth the Taylor seres epaso to d the roots. We kow ow how to aod the pt alls that was oe dmeso Newto s o ( = or Halley s or ( = ( = by usg hgher order terms the seres at that alue. Ths procedure ca be used or a two dmesoal Newto s algorthm. We ca coclude rom ths eample that or mult-dmeso ucto we ca dere a prescrbed tal codto t wll take some work but we hae a way to do t - that guaratees coersos usg a Taylor seres epaso algorthm or mult-dmeso that aods pt alls lke as ( = ( = oe dmeso the the soluto aoded guessg or the tal codto ad becomes a closed soluto. Wth that sad, guessg the tal alue should be permtted ad legally ad ca be deeded a court o law, the eet o a ueducated perso the jury questos the egeer s, scetsts etc., or hs or her result or guessg a umber. The reaso t ca be permtted because t ca be ered wthout guessg wth our guarateed coersos o roots ad compared wth other roots. The reaso ths problem was addressed was the author has soled seeral udametal usoled problems Structural ad Mechacs or Large delecto o Beams, Beam Bucklg ad Plates that requres dg zeros or a mult-dmeso uctos. Because ths problem s oles may areas egeerg the guessg questo had to be put rest or the beets o all. 8