9.5 Roots of Polynomials 369 9.5 Roots of Polynomials Here we present a few methods for finding roots of polynomials.These will serve for most practical problems involving polynomials of low-to-moderate degree or for well-conditioned polynomials of higher degree.Not as well appreciated as it ought to be is the fact that some polynomials are exceedingly ill-conditioned.The tiniest changes in a polynomial's coefficients can,in the worst case,send its roots sprawling all over the complex plane.(An infamous example due to Wilkinson is detailed by Acton [1].) Recall that a polynomial of degree n will have n roots.The roots can be real 81 or complex,and they might not be distinct.If the coefficients of the polynomial are real,then complex roots will occur in pairs that are conjugate,i.e.,if=a+bi 虽2 is a root then z2 =a-bi will also be a root.When the coefficients are complex, the complex roots need not be related. Multiple roots,or closely spaced roots,produce the most difficulty for numerical algorithms (see Figure 9.5.1).For example,P(x)=(x-a)2 has a double real root at z=a.However,we cannot bracket the root by the usual technique of identifying neighborhoods where the function changes sign,nor will slope-following methods 9 such as Newton-Raphson work well,because both the function and its derivative vanish at a multiple root.Newton-Raphson may work,but slowly,since large roundoff errors can occur.When a root is known in advance to be multiple,then special methods of attack are readily devised.Problems arise when(as is generally the case)we do not know in advance what pathology a root will display. 超% 9 Deflation of Polynomials When seeking several or all roots of a polynomial,the total effort can be 6 significantly reduced by the use of deftation.As each root r is found,the polynomial is factored into a product involving the root and a reduced polynomial of degree one less than the original,i.e.,P(x)=(x-r)Q(z).Since the roots of are exactly the remaining roots of P,the effort of finding additional roots decreases, because we work with polynomials of lower and lower degree as we find successive 10621 roots.Even more important,with deflation we can avoid the blunder of having our iterative method converge twice to the same(nonmultiple)root instead of separately "hg2 Numerical Recipes 43106 to two different roots. Deflation,which amounts to synthetic division,is a simple operation that acts (outside on the array of polynomial coefficients.The concise code for synthetic division by a monomial factor was given in 85.3 above.You can deflate complex roots either by North converting that code to complex data type,or else-in the case of a polynomial with real coefficients but possibly complex roots-by deflating by a quadratic factor, [x-(a+b)][x-(a-b)]=x2-2ax+(a2+b2) (9.5.1) The routine poldiv in 85.3 can be used to divide the polynomial by this factor. Deflation must,however,be utilized with care.Because each new root is known with only finite accuracy,errors creep into the determination of the coefficients of the successively deflated polynomial.Consequently,the roots can become more and more inaccurate.It matters a lot whether the inaccuracy creeps in stably (plus or
9.5 Roots of Polynomials 369 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). 9.5 Roots of Polynomials Here we present a few methods for finding roots of polynomials. These will serve for most practical problems involving polynomials of low-to-moderate degree or for well-conditioned polynomials of higher degree. Not as well appreciated as it ought to be is the fact that some polynomials are exceedingly ill-conditioned. The tiniest changes in a polynomial’s coefficients can, in the worst case, send its roots sprawling all over the complex plane. (An infamous example due to Wilkinson is detailed by Acton [1].) Recall that a polynomial of degree n will have n roots. The roots can be real or complex, and they might not be distinct. If the coefficients of the polynomial are real, then complex roots will occur in pairs that are conjugate, i.e., if x1 = a + bi is a root then x2 = a − bi will also be a root. When the coefficients are complex, the complex roots need not be related. Multiple roots, or closely spaced roots, produce the most difficulty for numerical algorithms (see Figure 9.5.1). For example, P(x)=(x − a) 2 has a double real root at x = a. However, we cannot bracket the root by the usual technique of identifying neighborhoods where the function changes sign, nor will slope-following methods such as Newton-Raphson work well, because both the function and its derivative vanish at a multiple root. Newton-Raphson may work, but slowly, since large roundoff errors can occur. When a root is known in advance to be multiple, then special methods of attack are readily devised. Problems arise when (as is generally the case) we do not know in advance what pathology a root will display. Deflation of Polynomials When seeking several or all roots of a polynomial, the total effort can be significantly reduced by the use of deflation. As each root r is found, the polynomial is factored into a product involving the root and a reduced polynomial of degree one less than the original, i.e., P(x)=(x − r)Q(x). Since the roots of Q are exactly the remaining roots of P, the effort of finding additional roots decreases, because we work with polynomials of lower and lower degree as we find successive roots. Even more important, with deflation we can avoid the blunder of having our iterative method converge twice to the same (nonmultiple) root instead of separately to two different roots. Deflation, which amounts to synthetic division, is a simple operation that acts on the array of polynomial coefficients. The concise code for synthetic division by a monomial factor was given in §5.3 above. You can deflate complex roots either by converting that code to complex data type, or else — in the case of a polynomial with real coefficients but possibly complex roots — by deflating by a quadratic factor, [x − (a + ib)] [x − (a − ib)] = x2 − 2ax + (a2 + b2) (9.5.1) The routine poldiv in §5.3 can be used to divide the polynomial by this factor. Deflation must, however, be utilized with care. Because each new root is known with only finite accuracy, errors creep into the determination of the coefficients of the successively deflated polynomial. Consequently, the roots can become more and more inaccurate. It matters a lot whether the inaccuracy creeps in stably (plus or
370 Chapter 9.Root Finding and Nonlinear Sets of Equations fx) 八x) e小N 83 from NUMERICAL RECIPESI 188891992 11800 (a) (b) Figure 9.5.1.(a)Linear,quadratic,and cubic behavior at the roots of polynomials.Only under high magnification (b)does it become apparent that the cubic has one,not three,roots,and that the quadratic has two roots rather than none. 9 minus a few multiples of the machine precision at each stage)or unstably (erosion of America computer, successive significant figures until the results become meaningless).Which behavior 9。 occurs depends on just how the root is divided out.Forward deflation,where the new polynomial coefficients are computed in the order from the highest power of x 9 down to the constant term,was illustrated in $5.3.This turns out to be stable if the root of smallest absolute value is divided out at each stage.Alternatively.one can do backward deflation,where new coefficients are computed in order from the constant term up to the coefficient of the highest power of z.This is stable if the remaining 可 root of largest absolute value is divided out at each stage. A polynomial whose coefficients are interchanged "end-to-end,"so that the constant becomes the highest coefficient,etc.,has its roots mapped into their reciprocals.(Proof:Divide the whole polynomial by its highest power x and rewrite it as a polynomial in 1/z.)The algorithm for backward deflation is therefore Fuurggoglrion Numerical Recipes 10621 virtually identical to that of forward deflation,except that the original coefficients are taken in reverse order and the reciprocal of the deflating root is used.Since we will 43108 use forward deflation below,we leave to you the exercise of writing a concise coding for backward deflation (as in 85.3).For more on the stability of deflation.consult [2]. To minimize the impact of increasing errors(even stable ones)when using (outside deflation.it is advisable to treat roots of the successively deflated polynomials as North Software. only tentative roots ofthe original polynomial.One then polishes these tentative roots by taking them as initial guesses that are to be re-solved for,using the nondeflated original polynomial P.Again you must beware lest two deflated roots are inaccurate enough that,under polishing,they both converge to the same undeflated root;in that case you gain a spurious root-multiplicity and lose a distinct root.This is detectable. since you can compare each polished root for equality to previous ones from distinct tentative roots.When it happens,you are advised to deflate the polynomial just once (and for this root only),then again polish the tentative root,or to use Maehly's procedure (see equation 9.5.29 below). Below we say more about techniques for polishing real and complex-conjugate
370 Chapter 9. Root Finding and Nonlinear Sets of Equations Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). (a) x x (b) f(x) f(x) Figure 9.5.1. (a) Linear, quadratic, and cubic behavior at the roots of polynomials. Only under high magnification (b) does it become apparent that the cubic has one, not three, roots, and that the quadratic has two roots rather than none. minus a few multiples of the machine precision at each stage) or unstably (erosion of successive significant figures until the results become meaningless). Which behavior occurs depends on just how the root is divided out. Forward deflation, where the new polynomial coefficients are computed in the order from the highest power of x down to the constant term, was illustrated in §5.3. This turns out to be stable if the root of smallest absolute value is divided out at each stage. Alternatively, one can do backward deflation, where new coefficients are computed in order from the constant term up to the coefficient of the highest power of x. This is stable if the remaining root of largest absolute value is divided out at each stage. A polynomial whose coefficients are interchanged “end-to-end,” so that the constant becomes the highest coefficient, etc., has its roots mapped into their reciprocals. (Proof: Divide the whole polynomial by its highest power x n and rewrite it as a polynomial in 1/x.) The algorithm for backward deflation is therefore virtually identical to that of forward deflation, except that the original coefficients are taken in reverse order and the reciprocal of the deflating root is used. Since we will use forward deflation below, we leave to you the exercise of writing a concise coding for backward deflation (as in §5.3). For more on the stability of deflation, consult [2]. To minimize the impact of increasing errors (even stable ones) when using deflation, it is advisable to treat roots of the successively deflated polynomials as only tentative roots of the original polynomial. One then polishesthese tentative roots by taking them as initial guesses that are to be re-solved for, using the nondeflated original polynomial P. Again you must beware lest two deflated roots are inaccurate enough that, under polishing, they both converge to the same undeflated root; in that case you gain a spurious root-multiplicity and lose a distinct root. This is detectable, since you can compare each polished root for equality to previous ones from distinct tentative roots. When it happens, you are advised to deflate the polynomial just once (and for this root only), then again polish the tentative root, or to use Maehly’s procedure (see equation 9.5.29 below). Below we say more about techniques for polishing real and complex-conjugate
9.5 Roots of Polynomials 371 tentative roots.First,let's get back to overall strategy. There are two schools of thought about how to proceed when faced with a polynomial of real coefficients.One school says to go after the easiest quarry,the real,distinct roots,by the same kinds of methods that we have discussed in previous sections for general functions,i.e.,trial-and-error bracketing followed by a safe Newton-Raphson as in rtsafe.Sometimes you are only interested in real roots,in which case the strategy is complete.Otherwise,you then go after quadratic factors of the form(9.5.1)by any of a variety of methods.One such is Bairstow's method, which we will discuss below in the context of root polishing.Another is Muller's method,which we here briefly discuss. Muller's Method Muller's method generalizes the secant method,but uses quadratic interpolation 茶 among three points instead of linear interpolation between two.Solving for the zeros of the quadratic allows the method to find complex pairs of roots.Given three previous guesses for the rooti-2,-1,i,and the values of the polynomial P() RECIPES I at those points,the next approximation is produced by the following formulas, q三 Ti-Ti-1 University Press. 工i-1-xi-2 A=qP(xi)-q(1+q)P(i-1)+q2P(zi_2) (9.5.2) B=(2g+1)P(x)-(1+q)2P(x-1)+q2P(x-2) C≡(1+q)P(x) SCIENTIFIC followed by 2C xi+1=工2-(x-t-1) (9.5.3) B士VB2-4AC where the sign in the denominator is chosen to make its absolute value or modulus as large as possible.You can start the iterations with any three values of x that you Numerical Recipes 10.621 43106 like,e.g.,three equally spaced values on the real axis.Note that you must allow for the possibility of a complex denominator,and subsequent complex arithmetic, in implementing the method. (outside Muller's method is sometimes also used for finding complex zeros of analytic functions(not just polynomials)in the complex plane,for example in the IMSL routine ZANLY [3]. Laguerre's Method The second school regarding overall strategy happens to be the one to which we belong.That school advises you to use one of a very small number of methods that will converge(though with greater or lesser efficiency)to all types of roots:real, complex,single,or multiple.Use such a method to get tentative values for all n roots of your nth degree polynomial.Then go back and polish them as you desire
9.5 Roots of Polynomials 371 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). tentative roots. First, let’s get back to overall strategy. There are two schools of thought about how to proceed when faced with a polynomial of real coefficients. One school says to go after the easiest quarry, the real, distinct roots, by the same kinds of methods that we have discussed in previous sections for general functions, i.e., trial-and-error bracketing followed by a safe Newton-Raphson as in rtsafe. Sometimes you are only interested in real roots, in which case the strategy is complete. Otherwise, you then go after quadratic factors of the form (9.5.1) by any of a variety of methods. One such is Bairstow’s method, which we will discuss below in the context of root polishing. Another is Muller’s method, which we here briefly discuss. Muller’s Method Muller’s method generalizes the secant method, but uses quadratic interpolation among three points instead of linear interpolation between two. Solving for the zeros of the quadratic allows the method to find complex pairs of roots. Given three previous guesses for the root xi−2, xi−1, xi, and the values of the polynomial P(x) at those points, the next approximation xi+1 is produced by the following formulas, q ≡ xi − xi−1 xi−1 − xi−2 A ≡ qP(xi) − q(1 + q)P(xi−1) + q2P(xi−2) B ≡ (2q + 1)P(xi) − (1 + q) 2P(xi−1) + q2P(xi−2) C ≡ (1 + q)P(xi) (9.5.2) followed by xi+1 = xi − (xi − xi−1) 2C B ± √ B2 − 4AC (9.5.3) where the sign in the denominator is chosen to make its absolute value or modulus as large as possible. You can start the iterations with any three values of x that you like, e.g., three equally spaced values on the real axis. Note that you must allow for the possibility of a complex denominator, and subsequent complex arithmetic, in implementing the method. Muller’s method is sometimes also used for finding complex zeros of analytic functions (not just polynomials) in the complex plane, for example in the IMSL routine ZANLY [3]. Laguerre’s Method The second school regarding overall strategy happens to be the one to which we belong. That school advises you to use one of a very small number of methods that will converge (though with greater or lesser efficiency) to all types of roots: real, complex, single, or multiple. Use such a method to get tentative values for all n roots of your nth degree polynomial. Then go back and polish them as you desire.
372 Chapter 9.Root Finding and Nonlinear Sets of Equations Laguerre's method is by far the most straightforward of these general,complex methods.It does require complex arithmetic,even while converging to real roots; however,for polynomials with all real roots,it is guaranteed to converge to a root from any starting point.For polynomials with some complex roots,little is theoretically proved about the method's convergence.Much empirical experience, however,suggests that nonconvergence is extremely unusual,and,further,can almost always be fixed by a simple scheme to break a nonconverging limit cycle.(This is implemented in our routine,below.)An example of a polynomial that requires this cycle-breaking scheme is one of high degree(20),with all its roots just outside of the complex unit circle,approximately equally spaced around it.When the method 81 converges on a simple complex zero,it is known that its convergence is third order. In some instances the complex arithmetic in the Laguerre method is no disadvantage,since the polynomial itself may have complex coefficients. To motivate(although not rigorously derive)the Laguerre formulas we can note 分 the following relations between the polynomial and its roots and derivatives Pn(x)=(x-x1)(x-x2)..(x-xn) (9.5.4) In Pn(x)=Inx-x1+Inz-z2+...+Inx-2n (9.5.5) 9 dx T-71 T-72 T-In P A2,9 d2In P() 1 1 1 dx2 三十1 -1产+(-2P++红-m $n∽ 9 (9.5.7) P Starting from these relations,the Laguerre formulas make what Acton [1]nicely calls "a rather drastic set of assumptions":The root xI that we seek is assumed to be located some distance a from our current guess x,while all other roots are assumed to be located at a distance b 9 x-x1=a x-zi=b i=2,3,...,n (9.5.8) Numerica 10.621 Then we can express (9.5.6),(9.5.7)as 43126 1,n-1 一十 -=G a b (9.5.9) 1n-1 a2+ 2=H (9.5.10) which yields as the solution for a Q= (9.5.11) G±√(n-1)(nH-G) where the sign should be taken to yield the largest magnitude for the denominator. Since the factor inside the square root can be negative,a can be complex.(A more rigorous justification of equation 9.5.11 is in [41.)
372 Chapter 9. Root Finding and Nonlinear Sets of Equations Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). Laguerre’s method is by far the most straightforward of these general, complex methods. It does require complex arithmetic, even while converging to real roots; however, for polynomials with all real roots, it is guaranteed to converge to a root from any starting point. For polynomials with some complex roots, little is theoretically proved about the method’s convergence. Much empirical experience, however, suggests that nonconvergenceis extremely unusual, and, further, can almost always be fixed by a simple scheme to break a nonconverging limit cycle. (This is implemented in our routine, below.) An example of a polynomial that requires this cycle-breaking scheme is one of high degree (>∼ 20), with all its roots just outside of the complex unit circle, approximately equally spaced around it. When the method converges on a simple complex zero, it is known that its convergence is third order. In some instances the complex arithmetic in the Laguerre method is no disadvantage, since the polynomial itself may have complex coefficients. To motivate (although not rigorously derive) the Laguerre formulas we can note the following relations between the polynomial and its roots and derivatives Pn(x)=(x − x1)(x − x2)...(x − xn) (9.5.4) ln |Pn(x)| = ln |x − x1| + ln |x − x2| + ... + ln |x − xn| (9.5.5) d ln |Pn(x)| dx = + 1 x − x1 + 1 x − x2 + ... + 1 x − xn = P n Pn ≡ G (9.5.6) −d2 ln |Pn(x)| dx2 = + 1 (x − x1)2 + 1 (x − x2)2 + ... + 1 (x − xn)2 = P n Pn 2 − P n Pn ≡ H (9.5.7) Starting from these relations, the Laguerre formulas make what Acton [1] nicely calls “a rather drastic set of assumptions”: The root x1 that we seek is assumed to be located some distance a from our current guess x, while all other roots are assumed to be located at a distance b x − x1 = a ; x − xi = b i = 2, 3,...,n (9.5.8) Then we can express (9.5.6), (9.5.7) as 1 a + n − 1 b = G (9.5.9) 1 a2 + n − 1 b2 = H (9.5.10) which yields as the solution for a a = n G ± (n − 1)(nH − G2) (9.5.11) where the sign should be taken to yield the largest magnitude for the denominator. Since the factor inside the square root can be negative, a can be complex. (A more rigorous justification of equation 9.5.11 is in [4].)
9.5 Roots of Polynomials 373 The method operates iteratively:For a trial value z.a is calculated by equation (9.5.11).Then x-a becomes the next trial value.This continues until a is sufficiently small. The following routine implements the Laguerre method to find one root of a given polynomial of degree m,whose coefficients can be complex.As usual,the first coefficient a[o]is the constant term,while a [m]is the coefficient of the highest power of x.The routine implements a simplified version of an elegant stopping criterion due to Adams [5].which neatly balances the desire to achieve full machine accuracy.on the one hand,with the danger of iterating forever in the presence of roundoff error,on the other. #include <math.h> 19881992 #include "complex.h" #include "nrutil.h" 1.800 #define EPSS 1.0e-7 #define MR 8 #define MT 10 from NUMERICAL RECIPESI #define MAXIT (MT*MR) Here EPSS is the estimated fractional roundoff error.We try to break (rare)limit cycles with MR different fractional values,once every MT steps,for MAXIT total allowed iterations. server (Nort void laguer(fcomplex a[],int m,fcomplex *x,int *its) THE Given the degreemand the m1complex coefficients a[.]of the polynomial America computer and given a complex value x,this routine improves x by Laguerre's method until it converges, ART within the achievable roundoff limit,to a root of the given polynomial.The number of iterations taken is returned as its. 9 Programs int iter,j; float abx,abp,abm,err; fcomplex dx,x1,b,d,f,g,h,sq,gp,gm,g2; stat1cf1 oat frac[R+1]={0.0,0.5,0.25,0.75,0.13,0.38,0.62,0.88,1.0J: Fractions used to break a limit cycle. for (iter=1;iter<=MAXIT;iter++){ Loop over iterations up to allowed maximum. 米its=1ter; OF SCIENTIFIC COMPUTING(ISBN b=a[m]; 19881982 err=Cabs(b); d=f=Complex(0.0,0.0) v@cam abx=Cabs(*x); 10-621 for（j=m-1;j>=0;j--)( Efficient computation of the polynomial and f=Cadd(Cmul(*x,f),d); its first two derivatives.f stores P/2. Numerical Recipes -43108 d=Cadd(Cmul(*x,d),b); b=Cadd(Cmul(*x,b),a[j]) err-Cabs(b)+abxerr; (outside err米=EPSS North Software. Estimate of roundoff error in evaluating polynomial. if (Cabs(b)<=err)return; We are on the root. g=Cdiv(d,b); The generic case:use Laguerre's formula Ame ying of g2=Cmul(g,g); h=Csub(g2,RCmul(2.0,Cdiv(f,b))); sq=Csqrt(RCmul((float)(m-1),Csub(RCmul((float)m,h),g2))); gp=Cadd(g,sq); gm=Csub(g,sq): abp=Cabs(gp); abm=Cabs(gm); if (abp abm)gp=gm; dx=((FMAX(abp,abm)>0.0 Cdiv(Complex((float)m,0.0),gp) RCmul(1+abx,Complex(cos((float)iter),sin((float)iter))))); x1=Csub(*x,dx);
9.5 Roots of Polynomials 373 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). The method operates iteratively: For a trial value x, a is calculated by equation (9.5.11). Then x − a becomes the next trial value. This continues until a is sufficiently small. The following routine implements the Laguerre method to find one root of a given polynomial of degree m, whose coefficients can be complex. As usual, the first coefficient a[0] is the constant term, while a[m] is the coefficient of the highest power of x. The routine implements a simplified version of an elegant stopping criterion due to Adams [5], which neatly balances the desire to achieve full machine accuracy, on the one hand, with the danger of iterating forever in the presence of roundoff error, on the other. #include <math.h> #include "complex.h" #include "nrutil.h" #define EPSS 1.0e-7 #define MR 8 #define MT 10 #define MAXIT (MT*MR) Here EPSS is the estimated fractional roundoff error. We try to break (rare) limit cycles with MR different fractional values, once every MT steps, for MAXIT total allowed iterations. void laguer(fcomplex a[], int m, fcomplex *x, int *its) Given the degree m and the m+1 complex coefficients a[0..m] of the polynomial m i=0 a[i]xi, and given a complex value x, this routine improves x by Laguerre’s method until it converges, within the achievable roundoff limit, to a root of the given polynomial. The number of iterations taken is returned as its. { int iter,j; float abx,abp,abm,err; fcomplex dx,x1,b,d,f,g,h,sq,gp,gm,g2; static float frac[MR+1] = {0.0,0.5,0.25,0.75,0.13,0.38,0.62,0.88,1.0}; Fractions used to break a limit cycle. for (iter=1;iter<=MAXIT;iter++) { Loop over iterations up to allowed maximum. *its=iter; b=a[m]; err=Cabs(b); d=f=Complex(0.0,0.0); abx=Cabs(*x); for (j=m-1;j>=0;j--) { Efficient computation of the polynomial and its first two derivatives. f stores P f=Cadd(Cmul(*x,f),d); /2. d=Cadd(Cmul(*x,d),b); b=Cadd(Cmul(*x,b),a[j]); err=Cabs(b)+abx*err; } err *= EPSS; Estimate of roundoff error in evaluating polynomial. if (Cabs(b) <= err) return; We are on the root. g=Cdiv(d,b); The generic case: use Laguerre’s formula. g2=Cmul(g,g); h=Csub(g2,RCmul(2.0,Cdiv(f,b))); sq=Csqrt(RCmul((float) (m-1),Csub(RCmul((float) m,h),g2))); gp=Cadd(g,sq); gm=Csub(g,sq); abp=Cabs(gp); abm=Cabs(gm); if (abp < abm) gp=gm; dx=((FMAX(abp,abm) > 0.0 ? Cdiv(Complex((float) m,0.0),gp) : RCmul(1+abx,Complex(cos((float)iter),sin((float)iter))))); x1=Csub(*x,dx);