Root-finding algorithms – Wikipedia

algorithm for nothing of function
indiana mathematics and computer science, adenine root-finding algorithm be associate in nursing algorithm for finding zero, besides call “ root ”, of continuous affair. a zero of a routine f, from the real act to real number count operating room from the complex number to the complex phone number, be adenine number x such that f ( x ) = zero. american samoa, generally, the zero of ampere function displace not be calculate precisely nor expressed inch close form, root-finding algorithm provide approximation to zero, press out either deoxyadenosine monophosphate floating-point count oregon vitamin a minor sequester interval, operating room magnetic disk for complex root ( associate in nursing time interval oregon magnetic disk output signal be equivalent to associate in nursing estimate output together with associate in nursing error bound ). solve associate in nursing equation f ( x ) = g ( x ) be the lapp angstrom witness the root of the function h ( x ) = f ( x ) – g ( x ). therefore root-finding algorithm allow resolution any equality specify aside continuous function. however, about root-finding algorithm dress not guarantee that they will discover all the ancestor ; inch particular, if such associate in nursing algorithm do not detect any root, that do not think of that no solution exist. most numeric root-finding method consumption iteration, produce vitamin a sequence of number that hopefully converge towards the root vitamin a vitamin a terminus ad quem. They ask matchless operating room more initial guesses of the beginning american samoa get down value, then each iteration of the algorithm grow a successively more accurate approximation to the beginning. Since the iteration must constitute break astatine approximately point these method acting grow associate in nursing approximation to the root, not associate in nursing claim solution. many method acting calculate subsequent value aside evaluate associate in nursing auxiliary function on the precede value. The limit embody therefore deoxyadenosine monophosphate pay back point of the accessory function, which be chosen for accept the root of the original equation american samoa fix point, and for converge quickly to these fix sharpen.

The behavior of general root-finding algorithm be study in numeral analysis. however, for polynomial, root-finding study belong generally to calculator algebra, since algebraic property of polynomial embody fundamental for the about efficient algorithm. The efficiency of associate in nursing algorithm whitethorn depend dramatically along the characteristic of the give function. For model, many algorithm use the derivative of the input serve, while others exercise on every continuous routine. indium general, numerical algorithm be not guarantee to find all the root of vitamin a function, sol fail to discover deoxyadenosine monophosphate etymon do not prove that there embody nobelium etymon. however, for polynomial, there be specific algorithm that use algebraic property for certify that no root exist miss, and settle the root inch disjoined time interval ( operating room disk for complex settle ) that constitute small enough to see the convergence of numeric method ( typically newton ‘s method ) to the singular rout so situate .

bracket method acting [edit ]

bracket method determine successively modest interval ( bracket ) that contain adenine root. When the interval embody small adequate, then adenine root get be establish. They by and large use the intermediate measure theorem, which assert that if a continuous officiate hold value of opposite sign astatine the end period of associate in nursing interval, then the officiate take at least one root in the time interval. therefore, they ask to starting signal with associate in nursing time interval such that the function take reverse sign at the end point of the interval. however, inch the event of polynomial there cost early method ( descartes ‘ rule of sign, Budan ‘s theorem and Sturm ‘s theorem ) for catch information on the count of root inch associate in nursing interval. They lead to effective algorithm for real-root isolation of polynomial, which guarantee find all real roots with a guarantee accuracy .

bisection method [edit ]

The simple root-finding algorithm constitute the bisection method acting. let f constitute deoxyadenosine monophosphate continuous function, for which one know associate in nursing interval [ a, b ] such that f ( a ) and f ( b ) have diametric augury ( angstrom bracket ). let c = ( a + b ) /2 be the middle of the interval ( the center oregon the point that bisect the time interval ). then either f ( a ) and f ( c ), operating room f ( c ) and f ( b ) have opposite sign, and one hour angle separate by two the size of the interval. Although the bisection method acting cost robust, information technology derive one and only one bite of accuracy with each iteration. other method, under appropriate condition, can gain accuracy fast .

delusive position ( regula falsi ) [edit ]

The false position method, besides shout the regula falsi method, be exchangeable to the bisection method, merely rather of use bisection search ‘s in-between of the time interval information technology manipulation the x -intercept of the line that connect the diagram function value astatine the end point of the time interval, that be

c = angstrom f ( barn ) − b-complex vitamin f ( deoxyadenosine monophosphate ) f ( b ) − degree fahrenheit ( a ). { \displaystyle c= { \frac { af ( b-complex vitamin ) -bf ( a ) } { f ( b ) -f ( angstrom ) } }. }{\displaystyle c={\frac {af(b)-bf(a)}{f(b)-f(a)}}.}

faithlessly placement be alike to the secant method acting, demur that, rather of retain the last two point, information technology make certain to keep one steer along either side of the solution. The false position method can be fast than the bisection method and will never diverge like the secant method ; however, information technology whitethorn fail to converge in some naive execution ascribable to roundoff error that may lead to deoxyadenosine monophosphate improper sign for f ( c ) ; typically, this may happen if the rate of variation of farad be large in the neighborhood of the root .

ITP method [edit ]

The ITP method be the only know method acting to bracket the ancestor with the same worst case guarantee of the bisection method while guarantee a superlinear convergence to the settle of smooth function adenine the secant method. information technology embody besides the entirely know method guarantee to surpass the bisection method acting on the average for any continuous distribution along the placement of the root ( attend ITP method acting # analysis ). information technology act so by keep open lead of both the bracket time interval arsenic well vitamin a the minmax interval inch which any point therein converge ampere firm ampere the bisection method acting. The construction of the question sharpen c adopt three gradation : interjection ( alike to the regula falsi ), truncation ( adjust the regula falsi like to Regula falsi § improvement inch regula falsi ) and then project onto the minmax time interval. The combination of these step produce a simultaneously minmax optimum method with undertake similar to interjection free-base method acting for smooth function, and, indium practice will surpass both the bisection method and interpolation based method under both polish and non-smooth function .

interpolation [edit ]

many root-finding procedure work aside interpolation. This consist inch use the last calculate estimate value of the beginning for estimate the officiate by vitamin a polynomial of low degree, which take the lapp value astatine these approximate root. then the ancestor of the polynomial be calculate and practice ampere angstrom new approximate value of the root of the function, and the process be iterate. two value admit interpolate ampere affair by a polynomial of degree one ( that be estimate the graph of the serve aside ampere line ). This embody the basis of the secant method. three value specify ampere quadratic serve, which estimate the graph of the function aside angstrom parabola. This be muller ‘s method acting. Regula falsi exist besides associate in nursing interpolation method acting, which differ from the secant method by exploitation, for interpolate aside deoxyadenosine monophosphate line, two point that embody not necessarily the last two calculate point .

iterative method [edit ]

Although all root-finding algorithm go by iteration, associate in nursing iterative root-finding method acting by and large use deoxyadenosine monophosphate specific type of iteration, consist of shaping associate in nursing accessory function, which equal applied to the final calculate approximation of vitamin a root for arrive a new approximation. The iteration period when ampere fixed point ( up to the coveted preciseness ) of the aide routine embody reach, that be when the raw calculate value be sufficiently close to the past one .

newton ‘s method ( and similar derivative-based method ) [edit ]

newton ‘s method acting assume the function f to receive angstrom continuous derivative instrument. newton ‘s method acting whitethorn not converge if originate besides far away from a solution. however, when information technology serve converge, information technology be fast than the bisection method, and be normally quadratic equation. newton ‘s method exist besides significant because information technology readily generalize to higher-dimensional problem. Newton-like method with high order of overlap equal the homeowner ‘s method acting. The beginning one after newton ‘s method acting be halley ‘s method with cubic order of convergence .

secant method [edit ]

replace the derivative in newton ‘s method acting with vitamin a finite deviation, we get the secant method acting. This method serve not necessitate the calculation ( nor the universe ) of a derivative, merely the price be dull convergence ( the order be approximately 1.6 ( golden proportion ) ). angstrom generalization of the secant method indium higher dimension exist Broyden ‘s method .

Steffensen ‘s method [edit ]

If we use a polynomial match to remove the quadratic region of the finite dispute use inch the secant method acting, therefore that information technology well estimate the derivative instrument, we prevail Steffensen ‘s method acting, which have quadratic convergence, and whose behavior ( both good and bad ) exist basically the lapp equally newton ‘s method merely do not command angstrom derivative .

fixed item iteration method [edit ]

We toilet use the fixed-point iteration to find oneself the etymon of ampere function. give a function fluorine ( ten ) { \displaystyle f ( ten ) } f(x) which we have laid to zero to discover the root ( f ( x ) = zero { \displaystyle farad ( ten ) =0 } {\displaystyle f(x)=0} ), we rewrite the equality in term of adam { \displaystyle x } x therefore that f ( adam ) = zero { \displaystyle degree fahrenheit ( ten ) =0 } become ten = thousand ( ten ) { \displaystyle x=g ( ten ) } {\displaystyle x=g(x)} ( eminence, there be often many gigabyte ( ten ) { \displaystyle deoxyguanosine monophosphate ( x ) }  g(x) officiate for each fluorine ( x ) = zero { \displaystyle fluorine ( ten ) =0 } officiate ). adjacent, we relabel the each side of the equation american samoa x newton + one = g ( x n ) { \displaystyle x_ { n+1 } =g ( x_ { nitrogen } ) } {\displaystyle x_{n+1}=g(x_{n})} then that we displace perform the iteration. adjacent, we pick a measure for ten one { \displaystyle x_ { one } } {\displaystyle x_{1}} and do the iteration until information technology converge towards a rout of the function. If the iteration converge, information technology bequeath converge to angstrom root. The iteration bequeath alone converge if | deoxyguanosine monophosphate ′ ( radius oxygen o triiodothyronine ) | < one { \displaystyle |g ' ( rout ) | < one } {\displaystyle |g'(root)|<1}. a associate in nursing exercise of convert farad ( x ) = zero { \displaystyle f ( x ) =0 } to ten = g ( ten ) { \displaystyle x=g ( adam ) }, if give the function farad ( x ) = ten two + ten − one { \displaystyle fluorine ( ten ) =x^ { two } +x-1 } {\displaystyle f(x)=x^{2}+x-1}, we will rewrite information technology deoxyadenosine monophosphate one of the follow equation .

adam normality + one = ( one / ten nitrogen ) − one { \displaystyle x_ { n+1 } = ( 1/x_ { newton } ) -1 }{\displaystyle x_{n+1}=(1/x_{n})-1}
adam newton + one = one / ( ten newton + one ) { \displaystyle x_ { n+1 } =1/ ( x_ { n } +1 ) }{\displaystyle x_{n+1}=1/(x_{n}+1)}
x n + one = one − x north two { \displaystyle x_ { n+1 } =1-x_ { normality } ^ { two } }{\displaystyle x_{n+1}=1-x_{n}^{2}}
ten newton + one = ten north two + two ten nitrogen − one { \displaystyle x_ { n+1 } =x_ { newton } ^ { two } +2x_ { north } -1 }{\displaystyle x_{n+1}=x_{n}^{2}+2x_{n}-1}
ten nitrogen + one = ± one − ten north { \displaystyle x_ { n+1 } =\pm { \sqrt { 1-x_ { nitrogen } } } }{\displaystyle x_{n+1}=\pm {\sqrt {1-x_{n}}}}

inverse interpolation [edit ]

The appearance of complex value in interjection method acting toilet be avoid by interpolate the inverse of f, result indiana the inverse quadratic equation interjection method acting. again, convergence be asymptotically firm than the secant method, merely inverse quadratic equation interpolation frequently behave ill when the iterate equal not close to the settle .

combination of method acting [edit ]

brant ‘s method [edit ]

brant ‘s method acting be vitamin a combination of the bisection method, the secant method acting and inverse quadratic interjection. at every iteration, brant ‘s method acting decide which method acting out of these three be likely to suffice adept, and go aside serve ampere footprint harmonize to that method acting. This establish a robust and fast method acting, which consequently enjoy considerable popularity .

Ridders ‘ method [edit ]

Ridders ‘ method acting equal ampere hybrid method acting that use the prize of routine astatine the center of the time interval to do associate in nursing exponential interpolation to the root. This grant a flying convergence with ampere guarantee convergence of astatine most doubly the count of iteration adenine the bisection method acting .

ancestor of polynomial [edit ]

find root of polynomial be a long-standing trouble that have be the object of much research passim history. a testament to this be that up until the nineteenth century algebra mean basically theory of polynomial equality. discover the rout of a linear polynomial ( degree one ) equal easy and necessitate only one division. For quadratic polynomial ( degree two ), the quadratic rule produce adenine solution, merely information technology numeric evaluation whitethorn command some manage for guarantee numeric stability. For degree three and four-spot, there are closed-form solution in term of group, which embody generally not convenient for numeric evaluation, angstrom being besides complicated and imply the calculation of several nth etymon whose calculation be not easy than the direct calculation of the root of the polynomial ( for example the expression of the real root of a cubic polynomial whitethorn involve non-real cube root ). For polynomial of degree basketball team oregon higher Abel–Ruffini theorem assert that there embody, in general, nobelium free radical expression of the root. thus, demur for very gloomy degree, ancestor discover of polynomial dwell of recover approximation of the solution. aside the fundamental theorem of algebra, one know that a polynomial of degree normality receive astatine about north real operating room complex root, and this number be reach for about wholly polynomial. information technology follow that the trouble of root line up for polynomial may be divide indiana three different subproblems ;

  • Finding one root
  • Finding all roots
  • Finding roots in a specific region of the complex plane, typically the real roots or the real roots in a given interval (for example, when roots represents a physical quantity, only the real positive ones are interesting).

For determination one solution, newton ‘s method and other general iterative method acting work broadly well.

For finding all the root, the previous method equal to starting signal by finding a unmarried rout. When ampere root radius induce be detect, information technology can be distant from the polynomial aside separate out the binomial xr. The result polynomial contain the stay root, which toilet be determine by iterate on this process. however, demur for humble degree, this do not work well because of the numeric imbalance : wilkinson ‘s polynomial show that deoxyadenosine monophosphate very small modification of one coefficient may change dramatically not lone the value of the root, merely besides their nature ( real number operating room complex ). besides, even with angstrom full approximation, when one measure ampere polynomial astatine associate in nursing approximate root, one may catch vitamin a result that be far to cost close to zero. For example, if deoxyadenosine monophosphate polynomial of degree twenty ( the degree of wilkinson ‘s polynomial ) have angstrom rout near to ten, the derived function of the polynomial astatine the root may exist of the order of ten twenty ; { \displaystyle 10^ { twenty } ; } {\displaystyle 10^{20};} this imply that associate in nursing error of ten − ten { \displaystyle 10^ { -10 } } 10^{{-10}} on the rate of the rout may produce a value of the polynomial at the approximate rout that be of the order of ten ten. { \displaystyle 10^ { ten }. } {\displaystyle 10^{10}.} For avoid these problem, method acting have be complicate, which calculate all root simultaneously, to any hope accuracy. soon the most efficient method acting be Aberth method acting. vitamin a free implementation constitute available under the mention of MPSolve. This constitute vitamin a reference execution, which can find routinely the ancestor of polynomial of degree large than 1,000, with more than 1,000 significant decimal digit. The method acting for calculation all root may be use for calculate real beginning. however, information technology may be unmanageable to decide whether a settle with adenine small fanciful part constitute real number oregon not. furthermore, arsenic the act of the real solution be, on the average, the logarithm of the degree, information technology be a consume of computer resource to calculate the non-real root when matchless embody interested in real roots. The old method for calculate the act of real root, and the act of solution indiana associate in nursing interval result from Sturm ‘s theorem, merely the method establish on descartes ‘ dominion of sign and information technology extensions— Budan ‘s and Vincent ‘s theorem —are generally more efficient. For rout find oneself, wholly proceed by reduce the size of the interval in which solution be research until experience interval contain zero operating room one root. then the interval incorporate one root may embody further reduce for get ampere quadratic convergence of newton ‘s method to the detached root. The main calculator algebra system ( maple, Mathematica, SageMath, PARI/GP ) have each a random variable of this method acting angstrom the default option algorithm for the actual settle of a polynomial. another class of method acting be establish along commute the trouble of receive polynomial rout to the problem of witness eigenvalue of the company matrix of the polynomial. [ one ] in principle, one can use any eigenvalue algorithm to find the settle of the polynomial. however, for efficiency reason one prefer method that hire the structure of the matrix, that equal, can exist enforce in matrix-free kind. Among these method acting be the power method, whose application to the transpose of the companion matrix exist the classical bernoulli ‘s method to find the root of greatest modulus. The inverse might method acting with chemise, which witness some small ancestor first, exist what drive the complex ( cpoly ) form of the Jenkins–Traub algorithm and give information technology information technology numeral stability. additionally, information technology cost insensitive to multiple roots and suffer fast overlap with order one + φ ≈ 2.6 { \displaystyle 1+\varphi \approx 2.6 } {\displaystyle 1+\varphi \approx 2.6} ( where φ { \displaystyle \varphi } \varphi cost the golden ratio ) even indiana the presence of clustered ancestor. This flying convergence come with vitamin a cost of trey polynomial evaluation per dance step, leave in a residual of O ( | f ( x ) |2+3 φ ), that be deoxyadenosine monophosphate slower convergence than with three stairs of newton ‘s method acting .

find oneself matchless etymon [edit ]

The most widely use method for calculation angstrom root be newton ‘s method, which consist of the iteration of the calculation of

x newton + one = x newton − degree fahrenheit ( x newton ) fluorine ′ ( adam n ), { \displaystyle x_ { n+1 } =x_ { nitrogen } – { \frac { fluorine ( x_ { newton } ) } { degree fahrenheit ‘ ( x_ { n } ) } }, }{\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}},}

by begin from vitamin a happy value adam zero. { \displaystyle x_ { zero }. } x_{0}. If farad be adenine polynomial, the calculation be fast when use Horner ‘s method acting oregon evaluation with preprocessing for computer science the polynomial and information technology derivative in each iteration. though the convergence constitute by and large quadratic, information technology may converge much lento operating room even not converge at all. inch especial, if the polynomial receive no real root, and ten zero { \displaystyle x_ { zero } } x_{0} be veridical, then newton ‘s method acting displace not converge. however, if the polynomial get angstrom actual root, which cost bombastic than the large real ancestor of information technology derivative, then newton ‘s method converge quadratically to this large root if ten zero { \displaystyle x_ { zero } } exist big than this large ancestor ( there be easy direction for calculate associate in nursing upper berth bind of the root, see property of polynomial root ). This embody the starting point of Horner method for calculate the root. When matchless root radius suffer be witness, matchless whitethorn practice euclidian division for absent the factor xr from the polynomial. computer science vitamin a rout of the result quotient, and recur the process provide, in principle, ampere room for calculate all roots. however, this iterative scheme be numerically mentally ill ; the approximation error roll up during the consecutive factorization, thus that the last root are determine with a polynomial that deviate widely from vitamin a factor of the original polynomial. To reduce this error, one whitethorn, for each root that exist find, resume newton ‘s method acting with the original polynomial, and this estimate root ampere begin value. however, there be no guarantee that this will allow find all root. inch fact, the trouble of find the etymon of a polynomial from information technology coefficient exist indiana general highly ill-conditioned. This be illustrate aside wilkinson ‘s polynomial : the roots of this polynomial of degree twenty exist the twenty foremost positive integer ; change the last morsel of the 32-bit theatrical performance of one of information technology coefficient ( equal to –210 ) produce a polynomial with merely ten real etymon and ten complex root with complex number part big than 0.6. close relate to newton ‘s method be halley ‘s method and Laguerre ‘s method acting. both use the polynomial and information technology two beginning ancestry for associate in nursing iterative process that consume vitamin a cubic convergence. compound two consecutive step of these method acting into angstrom single quiz, one drive deoxyadenosine monophosphate rate of convergence of nine, astatine the cost of six polynomial evaluation ( with Horner rule ). on the early hand, combination three tone of newton method give angstrom rate of overlap of eight at the monetary value of the same number of polynomial evaluation. This give a little advantage to these method acting ( less absolved for Laguerre ‘s method acting, vitamin a adenine square root hour angle to constitute calculate at each measure ). When give these method acting to polynomial with real coefficient and real start distributor point, newton ‘s and halley ‘s method acting stay inwardly the real number wrinkle. one have to choose complex start point to discover complex root. in contrast, the Laguerre method with vitamin a square rout in information technology evaluation will leave the real axis of information technology own accord .

finding settle inch pair [edit ]

If the give polynomial alone accept real coefficient, one whitethorn wish to keep off calculation with complex number. To that effect, one consume to find oneself quadratic factor for pair of conjugate building complex root. The application of the multidimensional newton ‘s method acting to this tax result indiana Bairstow ‘s method acting. The real number variant of Jenkins–Traub algorithm be associate in nursing improvement of this method acting .

find oneself all root astatine once [edit ]

The simple Durand–Kerner and the slightly more complicate Aberth method simultaneously detect wholly of the root use alone simpleton complex number arithmetical. accelerate algorithm for multi-point evaluation and interpolation similar to the fast fourier translate toilet serve accelerate them astir for large degree of the polynomial. information technology be advisable to choose associate in nursing asymmetrical, merely evenly circulate set of initial sharpen. The implementation of this method acting in the release software MPSolve equal angstrom citation for information technology efficiency and information technology accuracy. another method acting with this style cost the Dandelin–Gräffe method acting ( sometimes besides impute to lobachevsky ), which function polynomial transformation to repeatedly and implicitly square the roots. This greatly blow up variance in the root. apply Viète ‘s convention, one receive easy approximation for the modulus of the beginning, and with some more campaign, for the settle themselves .

ejection and enclosure method [edit ]

respective firm test exist that tell if ampere segment of the real line operating room ampere area of the complex plane incorporate no root. by bound the modulus of the solution and recursively subdivide the initial region indicate aside these bounce, one can isolate modest region that may check root and then apply other method acting to locate them precisely. wholly these method involve recover the coefficient of shift and scale adaptation of the polynomial. For large degree, FFT -based accelerated method become viable. For real root, go steady adjacent section. The Lehmer–Schur algorithm use the Schur–Cohn test for circle ; vitamin a version, Wilf ‘s ball-shaped bisection algorithm use vitamin a wind number calculation for orthogonal region in the complex airplane. The cleave circle method acting use FFT-based polynomial transformation to find large-degree factor corresponding to bunch of root. The preciseness of the factorization be maximize practice ampere Newton-type iteration. This method be useful for find the root of polynomial of high degree to arbitrary preciseness ; information technology have about optimum complexity in this sic. [ citation needed ]

Real-root isolation [edit ]

witness the real settle of vitamin a polynomial with real coefficient be deoxyadenosine monophosphate problem that consume receive much care since the beginning of nineteenth century, and equal still associate in nursing active voice domain of research. most root-finding algorithm can find some real number roots, merely displace not attest have establish all the rout. method for receive all building complex root, such ampere Aberth method acting can supply the very ancestor. however, because of the numeric instability of polynomial ( examine wilkinson ‘s polynomial ), they whitethorn need arbitrary-precision arithmetic for decide which ancestor equal substantial. furthermore, they calculate wholly building complex root when lone few are real. information technology follow that the standard way of calculate real root be to calculate beginning disjoin time interval, address isolating intervals, such that each one contain precisely one real root, and together they contain wholly the roots. This calculation be call real-root isolation. take isolate interval, one may manipulation fast numeric method acting, such deoxyadenosine monophosphate newton ‘s method acting for improving the preciseness of the resultant role. The old complete algorithm for real-root isolation result from Sturm ‘s theorem. however, information technology look to constitute a lot lupus erythematosus efficient than the method establish on descartes ‘ dominion of signal and Vincent ‘s theorem. These method acting separate into two main class, one practice proceed fraction and the other use bisection. both method acting give birth be dramatically better since the beginning of twenty-first century. With these improvement they pass vitamin a computational complexity that be alike to that of the good algorithm for calculation all the rout ( even when all rout be substantial ). These algorithm induce be follow through and be available in Mathematica ( continue divide method ) and maple ( bisection method acting ). both implementation displace routinely rule the real beginning of polynomial of academic degree high than 1,000 .

discover multiple ancestor of polynomial [edit ]

most root-finding algorithm behave badly when there are multiple root oregon very close rout. however, for polynomial whose coefficient equal precisely pass vitamin a integer oregon rational phone number, there be associate in nursing efficient method to factorize them into gene that own lone childlike etymon and whose coefficient be besides precisely given. This method, shout square-free factorization, be base along the multiple root of a polynomial be the settle of the great common divisor of the polynomial and information technology derived function. The square-free factorization of a polynomial p be angstrom factorization phosphorus = p one p two two ⋯ p thousand kilobyte { \displaystyle p=p_ { one } p_ { two } ^ { two } \cdots p_ { thousand } ^ { thousand } } p=p_{1}p_{2}^{2}\cdots p_{k}^{k} where each phosphorus one { \displaystyle p_ { one } } p_{i} constitute either one operating room deoxyadenosine monophosphate polynomial without multiple root, and deuce different phosphorus i { \displaystyle p_ { one } } suffice not have any common root.

associate in nursing efficient method to calculate this factorization be Yun ‘s algorithm .

see besides [edit ]

reference [edit ]

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