Analytic continuation and Riemann surface determination of algebraic functions by computer

Japan J. lndust. Appl. Math., 13 (1996), 107-116

Analytic Continuation and Riemann Surface Determination of Algebraic Functions by Computer

Kosuke SHIIHARA and Ta t eak i SASAKI

Institute of Mathematics, University of Tsukuba, Tsu~zuba-shi, Ibaraki 305, Japan

Received March 29, 1994

Let y = f(x) be an algebraic function which is defined a s a root of bivariate polynomial P(x, y). Given P(x, y), we presenta practical computation method for analytic continu- ation and R[emann surface determination of f(x). First, we determine the branch points of f{x) by calculating the roots of a univariate polynomial numerica[]y. Next, we expand f(ac) into the Puiseux series at each branch point by the extended Hensei construction method proposed by Sasaki and Kako recently. Then, using we[l-known Smith's theorem, we determine which series ate connected the each other for each pair of branch points, by numerically est[mating the values of the series at some middle point between the branch points. This allows u-~ to determine the l~iemann surface of f(x) compIetely. Analytic con- tinuation is performed sin-dlarly. We employ floating-point numbex arithmetic to perform these calculations.

Key words: algebraic geometry, approximate algebra, computer aigebra, Puiseux series, Riemann sur faee

1. I n t r o d u c t i o n

Ana ly t i c con t inua t ion and R iemann surface d e t e r m i n a t i o n ate very i m p o r t a n t coneepts in ma thema t i c s . To the au thors ~ knowledge, n o b o d y has carr ied out t h e ~ ope ra t i ons using c o m p u t e r a so lar. Our m e t h o d to de t e rmine the R i e m a n n surface of t he a lgebra ic funct ion consis ts of the following two opera t ions .

1. We expand the given f lmct ion into t he Pn i seux series a t each b r aneh point .

2. We de t e rmine whieh series a re connec ted the each o ther for each pa i r of b ranch points .

As for the Pu i seux series expansion, we are fami l ia r wi th t he Newton po lygon m e t h o d ; see [1], [9], [10]. T h e Newton po lygon m e t h o d has convent iona l ly been perforrned by in t roduc ing a lgebra ic numbers successively, which means t h u t the c o m p u t a t i o n is qu i te heavy a l though ir is exact . On the o the r hand , wha t we ac tua l ly need in R i e m a n n surface d e t e r m i n a t i o n is i n fo rma t ion on branch pohl ts , b ranches a t e a d l b ranch po in t and connect ions of b ranches be tween each pa i r of b ranch points . Then , the braa~ch po in t s are much b e t t e r given as a p p r o x i m a t e number s t h a n as a lgebra ic numbers . Using a p p r o x i m a t e numer ica l a r i thmet ic , we are able to pe r fo rm the e o m p u t a t i o n very efficiently a l though we nms t consider the a p p e a r a a c e of smal l error te rms. Therefore , we employ f loa t ing-poin t nurnber a r i t hm e t i c for t he numer ica l coet¡ Th is m e t h o d will be app roved only if the numer ica l errors

108 K. SHIIHARA and T. SASAKI

relating to our computation are small, and [2], [7] assures this. As for the second operation mentioned above, we use Smith's theorem [8] which is well-known in numerical analysis to determine an upper bound of error of the numerical root of the algebraic equation. We must emphasize that, although we employ approximate numerical arithmetic, Smith's theorem allows us to determine the cormections of branches rigorously.

We assume that the given po]ynomial P(x, y) is irreducible over C and monic w.r.t, variable y. These assulnptions ate reasonable for the following reasons. If P(x, y) is not irreducible, we factorize P(x, y) over C and treat each factor sep- arately; note tha t we have an approximate factorization algorithm which is ap- plicable for multivariate polynomials with tioating-point number coefficients [5]. If P(x, y) = p,,(x)y '~ + . . . + po(x) is not monic, we define

Then, q y) is monic w.r.t, variable y, and the root f (x ) of P(x, y) and the root f (x ) of P(x, y) are related with each other as f(x) = f(x)/p,~(x).

Let y = f(x) be the root of P(x, y) w.r . t .y . Algebraic functions may contain poles as well as branch points as singularities. The function f(x) has, however, no pole due to the assumption of monicness. Furthermore, irreducibility of P(x, y) means that P(x, y) is square-free, i.e., P(x, y) has no multiple factor.

This paper is organized as follows. In 2, we review the extended Hensel con- struction briefly. In 37 we explain how to determine the branch points. In 4, we describe a method of Riemann surface determination and gire an illustrative ex- ample. In 5, we describe a practical way to perform analytic continuation.

2. E x t e n d e d H e n s e l C o n s t r u c t i o n

In this section, we review the extended Hensel construction proposed recently in [61. For proofs, see [6].

DEFINITION 1 (Newton's line, Newton's polynomial). For each non-zero term c. y*x~ of P(x, y), we plot a d o t a t the point ( i , j ) in the two-dimensional Cartesian coordinate system. Let L be a straight line such that it passes the point (n, 0) as weU as another dot plotted and that no dot is plotted below L. The line L i s called Newton's Iine for P. The sum of all the terms of P(x, y), which are plotted on Newton's line is called Newton's polynomial for P. We denote Newton's polynomia] by p(0)(x, y). �9

REMARK 1. Newton's line is uniquely determined by P(x, y). Let Newton's line L in (%, e~)-plane be ey/n + e~/�91 ---- 1, where (~ is the intersection of L and ex-axis. Then, Newton's polynomial consists of some of the terms

yn yn-lx~/n ' yn-2x26/n,... ' xn~/n.

Analytic Continuation by Computer 109

Let 5 and ¡ be positive integers such that

~/~ = £ g o d ( ~ , ~ ) = ~.

Suppose that P(~ y) is factorized over Ca s

p(o)(1, y ) = (y _ ~,)m, . . . (y _ r (~)

~ I , . . . , r r 1 6 2 f o r a n y i # j .

Since P(~ is a homogeneous polynomial in y and x £ we can express P(~ y) in the following forro.

p(0)(x, y) = a~ ~ y) . . . G(2 )(~, y), G} ~ y) = (y - r

We define an ideal Ik as

Ik = (Y nx(k+~241 Y '~-lx(k+$)/¡ Yn-2x(k+2$)/¡ . . . , Y~241 �9 (2)

Then, we have the following lemma.

LEMMA 1 (Lagrange's interpolation polynomials). For each value of l = O , . . . , n - 1, there exists only one set o] polynomials {W(~)(x ,y) [ i = 1 , . . . , r } satisfying

W(O rE(o) . G(f) /G~o) l + . . . + W(~)rG(0) . . G(O)/G(f)] = yZ x(~/a)(,-z), t ~ l " " r [ 1 *

degv(W~O(x,y)) < degv(G~~ (i = 1 , . . . , r ) .

The next theorem follows from the above lemma.

T H E O R E M 1.

fying For all k E N, we can construct G~k)(x,y), i = 1,. . . ,r , satis-

P ( z , y ) ~_~ G~k)(z,y) . . .G(rk)(x,y) ( m o d Ik+l),

G}k)(x,y) =-- G}~ (mod Y1), i--- 1 , . . . ,r. m

G~k),. . . ,G @) ate calculated by the following procedure. Suppose that G i ~ - ' , , G(2 -1) have bee. calculated We expres~ P - G ? - I ) G t k-l) a~

p __ Gik-1) ..G(rk-1) ~ p(nk), . yn-axS/¡ .~ . . . ~_p(o k) . yO~nS/¡ (mod Ik+l).

Then, we construct G~ k) (x, y) by the following formula.

n--1 G~k)(x'Y) = G~k-I)(3~'Y) -~- E W(I)(x'y)p~k)(~)' i ~~- 1 ' ' " .,7". (3)

/=0

110 K. SHIIHARA and T. SASAZl

This construction is called extended Hensel construction. This method gives G~ k) (x, y) in fractional powers of x. If we apply the extended Hensel construction repeatedly, we finally obtain factors of P(x, y) which are linear w.r.t, y, i.e., Puiseux series expansion of y = f(x). For details, see [6].

3. D e t e r m i n a t i o n o f B r a n c h P o i n t s

The first step of analytic continuation azld Riemann surface determination is to find branch points. The branch points of algebraic function y = f(x) are specified by the following theorem.

THEOREM 2. The branch points of y = f(x) which is the root w.r.t, variable y, of monje polynomial P(x, y) ate contained in the roots of

( OP(x, y) ) R(x) = Resultanty P(x,y), Oy " (4)

Pro@ Put n = degy P(x, y). If P(xo, y), with x0 E C, is a square-free poly- nomial then ir can be factorized as follows,

P(xo,y) - - - (y - r l ) ( y - r 2 ) ' ' , (y - rn ) ,

r~CC, % r ( i , j = l , . . . , n , iT~j).

By Hensel construction wi~h modulus I = (x - x0), we obtain

P ( x , y ) - ( y - f l ( x ) ) ( y - f 2 ( x ) ) . . . ( y - f i ~ ( x ) ) ( m o d ( ( x - x 0 ) k + l ) ) , k = l , 2 , . . . .

Hence we find that the root y - f(x) of P(x, y) is expanded to the Taylor series around x = Xo. The contradiction of the above statement proves this theorem. �9

Note that not every root of R(x) is a branch point. Hence, for each root x0 of R(x), we must check whether x0 is a branch point or not. We perform this check by expanding y = f(x) at the point x = x0 : if the expansion yields a fractional power series then x0 is a branch point, otherwise x0 is n o t a branch point.

We calculate not only the roots of R(x) but also the roots of P(~ y) in (1) approximately by a numerical method.

As for the Puiseux series expansion, most textbooks on algebraic function or complex analysis introduce the so-called Newton polygon method [1]. Since we perform the computat ion approximately, we must handte small error terms in cal- culating power series. With the Newton polygon method, these small terms often behave as essential terms when the e• point is very close to a branch point. The branch point we calculate numerically is not exact but approximate; it is only a number which is very close to the true branch point. Therefore, the conventional Newton polygon method is very dangerotm in approximate numerical computa- tion. Instead, we emptoy Sa.saki-Kako's extended Hensel construction method. It

Analytic Continuation by Computer 111

determines a modulus I0 = ideal(Y ~, Y'~-lx;~,..., Y~ with )~ a non-negat ive ra- t ional nnmber, and performs the Puiseux expansion by selecting necessary terms by changing the modulus as I0 --*/1 ~ / 2 --~ "" ", where Ik = Io" (xk)'). Hence, the m e t h o d is completely safe to perform numerically.

The foliowing exaxaple shows how the branch points are determined and tha t not every root of R(x) is a branch point.

Ezample 1. Let P(x, y) be

P(x,y) = y6 + 3 ( - x + 1)y4 _ 2xy3 + 3(x2 _ 2x + 1)y2

+ 6 ( - x 2 + x)y + ( - x 3 + 4x 2 - 3x + 1).

Calculat ing the resul tant in (4), we obtain

R(x) = - 4 6 6 5 6 ( x - 1)3z4(64x3 - 165x 2 + 192x - 64) 2.

The roots of R(x) a r e a s follows.

x = 0, 1, 0 .516926 . . . , 1 . 0 3 0 6 . . . : k 0 . 9 3 4 0 1 1 . . . i.

A m o n g these, only x = 0 and x = 1 are branch points. In fact, at x = b = 0.516926, we can expand the roots of P(x, y) as follows.

yl(x) ~- -0 .401279 - 1.39007i + 0.107473 (z - b) 2 - 0.188506 (x - b) 3 + - - -

y2(x) ~- -0 .401279 + 1 . 3 9 0 0 7 / + 0.107473 (x - b) 2 - 0.188506 (x - b) 3 + - . .

y 3 ( Ÿ ~ 0.802557 - 0.695035 i + 0.214946 (x - 5 ) 2 - 0.377011 (x - 5) 3 + . . .

ya(x) ~-- 0.802557 + 0.695035 i § 0.214946 (x -- b) 2 - 0.377011 (x - b) 3 §

ys(x) ~-- -0 .401279 + 0.719388 i (x - b)

- (0 .322419 + 0.630897 i) (x - b) ~ + (0.565517 + 0.175175 i) (x - b)3+ . . .

y ~ ( ~ ) ~ - 0 . 4 0 1 ~ 7 9 - 0 . 7 1 9 3 8 8 i (~ - b)

- (0 .322419 - 0.630897 i) (x - b) 2 § (0.565517 - 0.175175 i) (x - b)3+ . . . .

The above calculat ion was performed by Sasaki-Kako's method. We also tested the Newton polygon me thod with floating-point number ar i thmetic for the coeffi- cient calculation, leading to an incorrect result. �9

4. R i e m a n n S u r f a c e D e t e r m i n a t i o n

Deter~Uning the R iemann surface of algebraic function f (x) is the same as determining how each branch is connected each other between every pair of branch points. Since the funct ion f (x) is single-valued on the Riemann surface, we can determine the connections by evaluating the funct ion at various points o ther th•n b ranch points.

Actually, de terminat ion of the connections ment ioned above is performed as follows. Let hi and b~ be two different branch points. First , consider the simple

112 K. SHIIHARA and T. SASAKI

c~,se in which the circles of convergence of Puiseux series a t x ---- bl mad x - bu overlap and the poin t x = b12 is inside the over lapping area. Pur thermore , asslmle tha t b12 is chosen so tha t P(b12, y) is square-free, i.e., P(bl2, y) has no mul t ip le root (since P(x, y) is irreducible by as sumpt ion , it is possible to choose bre satisfying this condit ion). Then, P(bl2,y) has n different roots which we pu t y = eXl, . . . , c~~, a r # c~8 for any r # s. For k = 1,2, let f}k)(x) (i = 1 , . . . , n ) be the roots of P(x, y) expanded into power series a t x = bk, then there is the following one- to-one cor respondence between e a ~ e lement of { a l , . . . , a,~} and each element of {Ÿ243 k = 1, 2.

(5) where { r l , . . . , r n } = { t , . . . , n t = { s l , . . . , s n } .

The f/(k)(x) (i = 1 , . . . , n) are iiffinite power series t, heoretically, but we can ha~~dle only t runca t ed power series practically. Fur thermore , the coefficients of f}k)(x) are ealeulated only approximate ly . Hence, we eannot calculate f(~k)(bl2) accurately. We can, however, de te rmine the above-ment ioned eorrespondence rigorously by using the following well-known theorem.

THEOREM 3 (Smith ' s t heorem [4], [8]). Let Q(y) be a polynomial over C, with deg(Q) = n, and Z l , . . . , z , be n different numbers in C. We define numbers q~(z l , . . . , z~) , i = 1 , . . . , n , as follows.

-Q(~~) Ÿ ' * ' , q i ( z ) = q i ( z 1 , Z 2 , . . . , Z n ) ---- Uy=i,7s Zj ) (i 1,2, n). (6)

Let D, be a dise u¡ the center at z, +q, (z ) and the radius (n - 1)]q,(z)l :

= { c �9 c I c - (~~ + q~(~))l -< ( ~ - 1)lq~(~)l }. (7) D,

Then, the n roots of Q(y) ave contained in n a U~=I Di in sueh way that ir m dises ave connected then the connected atea contains exaetIy the m roots. �9

Let C~l , . . . , ctn be the roots of Q(y) = P(b12, y). I f zi ~ (~~ (i = 1, 2 , . . . , n), then the radius of each dise D~ will be small. Therefore , so long as each zi is de te rmined to app rox ima te a , fairy welt, we can de te rmine the correspondence be tween each root a~ and i ts approx ima t ion zi uniquely.

In the general case, the circles of convergence a t x = bl and x = b2 m a y not overlap. In such a case, we choose severa/ points, say x = ao, a l , . . . ,a~+t , with a0 = bl and at+l = b2, such t h a t the circles of convergence at x = a~ and x = a i+l (i = 0, 1 . . . . . l) overlap and P(a~, y) is square-free. Then, the n roots of P(x, y) can be expanded into Taylor series at x = ai by Newton ' s me thod [3], aald let t he series obta ined be { g l ( x ) , . . . , g ~ ( x ) } . Then, there is a one-to- one cor respondenee between each e lement of { f ~ k ) ( x ) , . . . , f~(k)(x)} ana each el- ement of { g l ( x ) , . . . , g ~ ( x ) } . Therefore , apply ing the above~mentioned procedure

Analytic Continuation by Computer 113

to a l , a 2 , . . . , successively, we can de t e rmine the cor respondence be tween e lements of { f ~ l ) ( x ) , . . . , f ( ~ l ) ( x ) } and { f ~ 2 ) ( x ) , . . . , f ( 2 ) ( x ) } .

Example 2. Let P ( x , y) be

P ( x , y) = y3 _ x y + x. (8)

T h e cand ida t e s of b ranch po in t s are x -- 0 and x -- 6.75 and b o t h are reaUy b ranch poin ts . In fact, we can e x p a n d the roots of (8) to the following P u i s e u x series, respect ively, a t these points .

Yll (x) ~- - 1 . X 1 / 3 - - 0.33333 X 2 / 3 Ac 0.012346 X 4 / 3 - - 0.0041152 x 5/3 + . . .

Yl2(X) --~ (0.5 - 0.86603 i) x 1/3 + (0.16667 + 0 .28868i) x 2/3

+ ( -0 .0061728 + 0.010692 i) x 4/3 + (0.0020576 + 0.0035639 i) x 5/3 + . . .

y13(x) "" (0.5 + 0 . 8 6 6 0 3 i ) x W3 + (0.16667 - 0 .28868i) x 2/3

+ ( -0 .0061728 - 0.010692 i ) x 4/3 + (0.0020576 - 0.0035639 i ) x sI3 + . . -

Y21(x) --~ --3. -- 0.197531 (X -- 6.75) + 0.00758692 (x -- 6.75) 2 + . . .

Y22(x) --~ 1.5 + 0.333333 (x -- 6.75) 1/2 + 0.0987654 (x -- 6.75)

+ 0.00731596 (x - 6 . 7 5 ) 3 / 2 - 0.00379346 (x - 6.75) 2 + . . .

y23(x) " 1.5 - 0.333333 (x - 6.75) ~/2 + 0.0987654 (x - 6.75)

- 0.00731596 (x - 6.75) 3/2 - 0.00379346 (x - 6.75) 2 + . . . .

Le t us de t e rmine the corre8pondence be tween bra~tches at x -- 0 a n d b ranches a t x -- 6.75, by choosing b12 -- 3. The roo t s of P ( 3 , y) = 0, let t h e m be c~1, (~2, c~3, a te ca lcu la ted as follow8.

(~1 ~ -2 .1038 , c~2 ~ 1.0519 - 0.565236i , ~:~ ~ 1.0519 + 0.565236i

Cons ider ing only the lcad ing one of two t e r m s of expans ions f rom y~3(x) (i = 1, 2, j = 1, 2, 3), mld eva lua t ing t h e m at x ---- 3, we o b t a i n

( Yll (3) ~ -1 .44224 , Y12(3) ~ 0 .721124-1 .24902 i, Y13(3) ~ 0.721124+1.24902 i,

Y21(3) - 3 . 0 , Y22(3) 1 .5+0.645497i , y2~(3) ~ 1 .5 -0 .645497 i .

Hence, for each yi j (3) (1 < i <: 2, disc Dij axe ca lcu la t ed as follows.

D~~

Dl2

D~3

D21

922

D2a

1 < j _< 3), t h e center and the r ad ius of Smi th ' s

Center

-2 .13561

1.06781 - 0.648553i

Rad ius

1.38675

1.38672

1.06781 + 0.648553i 1.38672

-2 .27419 1.45161

1.1371 + 0.697553i 0.733236

1.1371 - 0 .697553i 0.733236

I14 K. SHIIHARA and T. SASAKI

T h e S m i t h ' s c i rc les o b t a i n e d a re s h o w n in Fig. 1 (sol id c i rc les for Y11, Y12, Y13, a~ld d o t t e d ones for Y21, Y22, Y23)- A l t h o u g h t h e a b o v e va lues a r e c o n s i d e r a b l y dif- f e ren t f r o m p r o p e r roo t s a l , c~2, ~~, we c a n d e t e r m i n e t h e fo l lowing c o r r e s p o n d e n c e a m o n g e x p a n s i o n s f r o m F]g. 1.

{ Yll ~ Otl ~ ' Y21 Y12 ~ ' Of 2 r ) Y23 Y13 r ~ O~3 ~ ) Y22

-,~~ -~ , I

- - 1

Fig. I.

�9 : The roots of P(3 , y) = 0 - - : Smi th ' s circle generated by the first two terms of y11, y12 and Y~3 at x = 3 - - - : Smi th ' s circle generated by the first two terms of y21, y22 and y23 at x = 3.

REMARK 2. I n Fig . 1, two sol id c i rc les ove r l ap each o t h e r . However , we can d e t e r m i n e p r e c i s e l y t h e o n e - t o - o n e c o r r e s p o n d e n c e b e t w e e n each r o o t and each ex- pans ion , s ince ea~h c i rc le c o n t a i n s o n l y one p r o p e r r o o t of P ( 3 , y) = 0. �9

REMARK 3. I n c a l c u l a t i n g q u a n t i t i e s in (7), r o u n d - o f f e r ro r s m a y s o m e t i m e s b e c o m e ser ious . In such a case, we c a n use t h e in t e rva l a r i t h m e t i c to check ah u p p e r - b o u n d of t h e r o u n d - o f f er rors . �9

R i e m a n n su r f ace d e t e r m i n a t i o n also r equ i r e s t h e b e h a v i o r o f f i m c t i o n f(x) a t t h e in f in i ty p o i n t x = c~. We shat l c o n s i d e r t h e b e h a v i o r o f t h e r o o t of P(l/z, y) at z = 0 i n s t ead o f f(x) at x = c~ [1]. I n genera l , P(1/z,y) c a n b e w r i t t e n as

P(�88 =Q~Y), Q(z,y) ec[~,y], kez. Obvious ly , t h e r o o t y -= g(z) of P(1/z, y) sat isf ies

AnMytic Continuation by Computer 115

where y -- ~(z) is the root of Q ( z , y). So, it is sufficient to check only Q ( z , y) by the m e t h o d described so far. Note tha t Q ( z , y) may not be monic, a l though P ( x , y) is monic.

Exampte 3. Consider the P ( x , y) given in Example 2. We have

Q ( z , y ) = z y 3 - y + 1.

The branches of algebraic function defined by Q ( z , y~ are expanded at z --- 0 as follows.

Y31 (Z) ~ 1 + Z + 3 Z 2 + 12 Z 3 + "'"

~ 3 2 ( z ) _ ~ - - -

~ 3 3 ( z ) - ~

1 0.5 - 0.375 z 1/2 - 0.5 z - 0.820312 Z 3 / 2

z l /2

--1.5 Z 2 -- 2.93262 Z 5/2 -- 6 Z 3 "-~ "" �9

1 0.5 q- 0.375 z 1/2 - 0.5 z -t- 0.820312 Z 3 / 2

z l / 2

- 1 . 5 z 2 § 2.93262z 5/2 - 6 z 3 §

Hence, the branches of algebraic function defined by P ( x , y) are expanded at x -= c~ as follows.

1 3 12 y31(x) ~ 1 + - + + + . . .

0.375 0.5 0.820312 1.5 2.93262 6 y3~(x) - -~x 1/~ - 0.5 - X l / ~ - x x3/2 X 2 xS/2 x 3 + ' ' "

0.375 0.5 0.820312 1.5 2.93262 6 y33(x) --~ - x 1/2 - 0.5 + - - + - - +

X 1/2 X X 3/2 X 2 X 5/2 X 3 - ~ ' ' ' "

Considering only the leading two terms of expansions from y i j ( x ) (i ~- 2, 3,

j -- 1, 2, 3), and evaluat ing these equations at x -- 10, we obta in

| y21(10) ~ -3 .64198, y22(10) ~ 2.10092, Y23(10) ~ 0.899075,

ya1(10) 1.1, y32(10) ~ 2.66228, Y33(10) -3 .66228.

Then , for each yij (2 <: i < 3, 1 ___ j ~ 3), we can Mso calculate the center and the radius of Smith 's disc D~~ as follows.

Disc

D21

D22

D23

D31

D32

D33

Center Radius

-3 .5696 0.144753

2.35244 0.503044

1.21716 0.636174

1.14449 0.0889783

2.43489 -0 .454771

-3 .57938 0.165792

116 K. SHIIHARA and T. SASAKI

T h e r e f o r e , we o b t a i n t h e fo l lowing o n e - t o - o n e c o r r e s p o n d e n c e a m o n g b ranches y21, y22, y23 a n d b r a n c h e s y31, y32~ y33 by S m i t h ' s t h e o r e m .

{ Y23 ~ ~ 1.15347 , ~ Y31 Y22 ~ * 2 .42362 , ~ Y32 Y21 ~ ~ - 3 . 5 7 7 0 9 , * Y33

whe re 1.15347, 2 .42362 a n d - 3 . 5 7 7 0 9 a r e t h e r o o t s o f P ( 1 0 , y) = 0, r e spec t ive ly .

5. Analyt ic C o n t i n u a t i o n

S u p p o s e t h a t we w a n t t o c o n t i n u e t h e series, e i t h e r t h e T a y l o r ser ies o r t h e P u i s e u x series , e x p a n d e d a t x = a t o t h e p o i n t x = b. T h e m e t h o d of a n a l y t i c c o n t i n u a t i o n d e s c r i b e d in m o s t t e x t b o o k s is as follows. We first c a l c u l a t e t h e ser ies at x = a, t h e n we c o n s t r u c t t h e ser ies a t x ---- b. Th i s m e t h o d is v e r y ineff icient , be- cause we m u s t c a l c u l a t e qu i t e a la rge n u m b e r o f t e r m s t o o b t a i n a des i r ed a c c u r a c y in t h e coef f ic ien ts o f c o n s t r u c t e d series .

I n s t e a d , we p e r f o r m as follows. W e f i rs t c a l c u l a t e t h e r o o t s o f P(b, y). I f t h e roo t s a t e d i f fe ren t f r o m each o the r , t h e n we p e r f o r m t h e ser ies e x p a n s i o n a t x = b by N e w t o n ' s m e t h o d . I f P(b, y) is n o t squa re - f r ee , t h e n we e x p a n d t h e roo t s o f P(x, y) in to t h e P u i s e u x ser ies at x = b by S a s a k i - K a k o ' s m e t h o d . T h e n , we d e t e r m i n e t h e c o r r e s p o n d e n c e b e t w e e n one of these ser ies a n d t h e ser ies a t x - a b y t h e p r o c e d u r e d e s c r i b e d in 4.

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