ASSOCIATED POLYNOMIALS AND UNIFORM M E T H O D S FOR THE SOLUTION OF LINEAR PROBLEMS J. F. TRAUB SIAM REVIEW Vol. 8, N o . 3, July, 1966 SIAM REVIEW Vol. 8, N o . 3, July, 1966 ASSOCIATED POLYNOMIALS A N D UNIFORM M E T H O D S FOR T H E SOLUTION OF LINEAR PROBLEMS* J. F. TRAUBf TABLE OF CONTENTS 1. 2. 3. 4. 5. 6. 7. 8. Introduction Brief survey of results The basic orthonormality relation The inverse of the Vandermonde matrix Interpolation Approximation of linear functionals The variation of the coefficients of a polynomial with the variation of its zeros Solution of inhomogeneous difference and differential equations with constant coefficients 9. Divided differences 10. Applications of divided difference calculations 11. Some interesting formulas 12. The inverse transformation and Wronski's aleph function 13. A result concerning powers of zeros 14. Historical survey Appendix. Notation References 277 279 279 281 285 287 289 290 291 292 293 295 297 298 299 300 1. Introduction. T o every polynomial P of degree n we associate a sequence of n + 1 polynomials of increasing degree which we call the associated polynomials of P. T h e associated polynomials depend in a particularly simple way on the coefficients of P. These polynomials have appeared in m a n y guises in the literature, usually related to some particular application and most often going unrecognized. T h e y have been called Horner polynomials and Laguerre polynomials. (For a historical discussion see §14.) Often w h a t occurs is not an associated polynomial itself b u t a number which is an associated polynomial evaluated at a zero of P. T h e properties of associated polynomials have never been investigated in themselves. W e shall t r y to demonstrate t h a t associated polynomials provide a useful unifying concept. Although m a n y of the results of this paper are new, we shall also present known results in our framework. Let P(t) be a monic polynomial of degree n, n (1.1) P(t) = Zfln-/, O0 = I- 3=0 T h e condition t h a t P be monic is convenient rather t h a n essential. T h e a - and t are in general complex. Let the zeros of P be labeled pj. I n some of our applica3 * Received by the editors June 10, 1965, and in final revised form December 20, 1965. t Mathematics Research Center, Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey. 277 278 J. F. TRAUB tions we refer to t h e set of py as t h e sample point set. W e assume here t h a t the Pj are distinct; t h e uniform extension of our results to confluent zeros will be reported elsewhere. Let u be a complex parameter and define polynomials Aj(u, P),j = 0 , 1 , • • • , n, by t (1.2) P { t \ U ~ P { U ) - t An-jiu, P)t\ t * u. j=0 T h e remainder theorem shows t h a t (1.2) is a valid definition. W e verify below t h a t Aj(u, P) is a polynomial in u of degree j . W e call the set Aj(u, P),j = 0, 1, • • • , n , the associated polynomials of P. For m a n y applications we can abbreviate A iu, P) b y A iu). Observe t h a t t h e left side of (1.2) is t h e first divided difference of t h e polynomial tP(t) with respect to t and u. Hence we m a y write t — U 3 3 n (1.3) (tP)[t, u] = An-Au, P)t\ t u. 3=0 I n words, t h e generating function of t h e associated polynomials of P is the first divided difference of tP. Multiplying (1.2) b y t — u, comparing powers of t, and recalling t h a t P is monic, yields t h e recursion A (u) = 1, Aj(u) = uAj_i(u) 0 (1.4) If we define A-i(u) + ay, j = 1, 2, • • • , n. + ay, j = 0, 1, • • • , n. = 0, we can write (1.5) Aj(u) = uAj_ {u) x F r o m (1.5) we obtain t h e explicit formula 3 (1.6) Aj(u) This verifies t h a t Aj(u) r = a,j-rU , j = 0, 1, • • • , n. is a polynomial of degree j . I n particular (1.7) A {u) n = P(u). I t is sometimes convenient to replace t h e generating function for Aj(u), j = 0, 1, • • • , n, by a simpler generating function for Aj(u),j = 0,1, ••• , n - 1. Since (tP)[t, u] = tP[t, u] + P(u), n-l P[t,u] = £ 4n-i-;(t0* '. y=o J (1.8) Since A {u) n = P(u), n-l (1.9) t — u t — u y=o ASSOCIATED POLYNOMIALS AND UNIFORM 279 METHODS Hence the quotient and remainder after division by a monic linear polynomial m a y be calculated recursively. T h e nature of the recursion is such t h a t one m a y work with detached coefficients. T h e process is often called synthetic division. ( F o r historical comments see §14.) From (1.9) it follows t h a t .n—l—j Thus (1.10) Aj(pi) where a ,i,j denotes the j i h elementary symmetric function of p i , p , • • • , p with pi deleted. Hence any result which involves Aj(pi) can always be interpreted as a result involving elementary symmetric functions. 2. Brief survey of results. T h e orthonormality relation given b y (3.5) is of basic importance. I t is used in §4 to provide us with two numerical algorithms for inverting the Vandermonde matrix. A variety of row and column sums to be used for check calculations are provided. I n §5 we point out advantages of writing the representation of an interpolatory polynomial in canonical polynomial form. I n §6, the canonical representation of the preceding section is used to derive formulas for the approximation of linear functionals. E q u a t i o n (6.4) exhibits the approximation as a vector-matrix-vector product. T h e columns of the inverse to the Vandermonde matrix m a y be characterized as the coefficients of the approximations to f (0)/kl. I n §7 we show t h a t the variation of the coefficients of a polynomial with the variation of its zeros is determined b y associated polynomials, (7.1) being the main result. In §8 we give the general solution of an inhomogeneous difference or differential equation with constant coefficients as an explicit function of the initial conditions. T h e calculation of divided differences is discussed in §9 and is applied to §10 to "nonnoisy" computation and to the calculation of Stirling numbers. A number of interesting formulas are derived in §11. A symbolic statement of the Newton relations is given b y (11.7) and a sweeping generalization of an important formula of Euler is given b y (11.14) or (11.16). Consideration of the transformation between powers of t and associated polynomials leads to recognition of the relation between Wronski's aleph function and associated polynomials. T h e main result is given by (12.4). I n §13 we give a simple solution to the classical problem of expressing an arbitrary power of a zero of a polynomial as a linear combination of the first n — 1 powers of t h a t zero. A symbolic solution is provided by (13.5). We end with a historical survey in §14 and a summary of notation in the Appendix. 3. The basic orthonormality relation. Consider the generating function 2 n (k) (3.1) n-l P[t, M] = E A „ _ ( « ) « ' ' . H n 280 J. F. TRAUB Let t —> u. T h e n (3.2) P\u) = E^n-l-i(^)^'. Let pi and p be distinct zeros of P. T h e n n-l k (3.3) 0 = ]C An_ (p<W. w Since pi is a simple zero of P, we conclude from (3.2) and (3.3), (3.4) g^-i-iUW = 1,2, ... , n , = S i k > where is the Kronecker delta function which vanishes except when the indices are equal. Let cc and (5 denote the matrices whose elements are a i j = ^ P ^ ' y ft-* = P* , i = 0, 1, • • • , n - 1, < = 1, 2, • • • , n; j = 0, 1, • • • , n - 1, fc = 1, 2, • • • , w. T h u s 5 is just t h e Vandermonde matrix. Let / denote the identity matrix. T h e n (3.4) m a y be written a(3 = I. $a = I This implies t h a t or E (3.5) *=1 = p 8 i k . (Pi) T h e orthonormality relation given by (3.5) is a basic result which permits us to solve a variety of linear problems. I t is of importance for two reasons. First it permits the theoretical investigation of "inverse problems". Secondly it gives t h e inverse to the Vandermonde matrix in a form well suited for numerical calculation. Two algorithms for the inversion are discussed in detail in the next section. Note t h a t t h e ranges of i, j , k are not t h e same. This is because we have chosen to follow the usual convention of denoting t h e n zeros of a polynomial by p i , p , • • • , p . If we had labeled t h e zeros po, Pi, • • • , Pn-i, the difference in ranges could have been avoided. Two important consequences of (3.5) are n v 2 n (3.6) (3.7) E p£h = E^4=5„-,„, . if = 0 , 1 , ••• , n - 1, v = 0 , 1, ••• , n - 1. ASSOCIATED 281 POLYNOMIALS A N D UNIFORM METHODS Relation (3.6) is a well-known formula of Euler. B o t h (3.6) and (3.7) are special cases of (11.16). 4. The inverse of the Vandermonde matrix. F r o m (3.5) we see t h a t the inverse of t h e Vandermonde matrix is given b y the matrix whose elements are (41) a . . = ^n-l-j(pt) E q u a t i o n (4.1) n o t only gives a n explicit formula for t h e inverse, b u t in addition this formula is well suited for h a n d or machine calculation. W e shall describe two algorithms for forming a, given t h e sample points p . As we shall show below, check calculations m a y easily be performed on t h e rows a n d columns of (4.1). If t h e pi satisfy symmetry conditions, t h e n t h e rows of (4.1) satisfy certain s y m m e t r y conditions. A characterization of t h e columns of a in terms of approximation of derivatives is given in §6. T h e case of most importance for applications is t h a t of a* and pi real. F o r t h e purpose of making operation counts, we restrict ourselves t o this case for t h e remainder of this section. W e now describe Algorithm I for calculating a. T h e first step in this algorithm is to calculate t h e coefficients of P from t h e p . Let <j ,j denote t h e ^ t h elementary symmetric function of t h e numbers p i , p , • • • , p . T h e n aj = ( — l ) V , j • . T h e a j m a y be calculated recursively b y t { m 2 n m m (4.2) aj = a -£i,j m m + p (r -i,j-i , ^ o-i.i = m m m = 2, 3, • • • , n, (T ,j = j = 1, 2, • • • , m\ with <r ,o m = 1 , j 0; Pi ; m 0, j > m. A numerical example of t h e recursion is given below. T h e calculation of all t h e di requires \n{n — 1) multiplications a n d t h e same number of additions. For each p , A _i_,-(pt) m a y be calculated b y t h e recursion t n Aj(pi) = PiA^ipi) Ao(pi) = 1. + aj, j = 1, 2, • • • , n - 1; (4.3) Hence Aj(pi), j = 0, 1, • • • , n — 1, m a y be calculated with just n — 2 multiplications and n — 1 additions. Observe t h a t 4 ' 4 A/(pi) = piAj-_i( ) + Aj-i(pi), A,\ i) = 1, P j = 2, 3, • • •, n ; Pi A \ i) n P = P\ ). Pi Hence P ' ( p . ) m a y be calculated with just n — 2 multiplications and n — 1 additions. Let us assume t h a t a division time is comparable to a multiplication time. T h e n a m a y be formed from t h e pi with ( n / 2 ) ( 7 n — 9) multiplications a n d ( 5 / 2 ) n ( n — 1) additions for n > 1. (See also t h e discussion below on t h e decomposition of a into t h e product of two matrices.) We now describe Algorithm I I for forming a. This algorithm is based on t h e 282 J. F. TRAUB fact t h a t (4.3) = pf±t — pi = n (t - P j ) = j=l j Ei - (p^ . n H j=0 Hence t h e A _i_y(pi), j = 0, 1, • • • , n — 1, are given b y t h e coefficients of TTi(t) and these are formed b y t h e recursion (4.2) based on t h e numbers pi, P 2 , • • • , pi_i, p i , • • • , p . T h e calculation of the P'{p%) coincides with t h e calculation of t h e P\pi) in Algorithm I . T h e calculation of a from t h e p* b y Algorithm I I requires n(n + n — 2 ) / 2 multiplications and n(n — l ) / 2 additions. Hence for n ^ 4, Algorithm I I requires fewer operations while for n > 4, Algorithm I requires fewer operations. Observe t h a t in Algorithm I, a is formed by a process of "analysis" whereas in Algorithm I I , a is formed b y a process of "synthesis". T h a t is, in Algorithm I we form t h e coefficients of P a n d then calculate t h e Aj(pi) b y division. I n Algorithm I I we form t h e coefficients of iri(t) and no division is involved. I t is clear t h a t Algorithm I I is numerically more stable t h a n Algorithm I . We can cut down t h e number of multiplications required b y Algorithm I I b y reusing some of t h e common factors obtained in t h e calculation of one iri(t) in t h e calculation of another iri(t). We now w a n t t o show t h a t certain advantages are obtained b y decomposing a into t h e product of two matrices. Let n be t h e diagonal matrix with elements n i + (4.6) n tin = P\pi). 1 T h e n n" h a s elements 1/P'(p;) on its diagonal. Let y be t h e matrix with elements (4.7) Jij = A _ ( p i ) . n W Then (4.8) a = ITY This decomposition of a h a s a number of advantages. As we shall see, a is often used t o form a vector-matrix-vector product, faM. (See (6.4).) R a t h e r t h a n using n multiplications t o form a from n a n d y , we can form y M and then form n ( y M ) in just n multiplications. A second advantage of this approach is t h a t if t h e pi are integers, all elements of y a n d n are integers, whereas t h e elements of a would not in general be integers. Because of t h e reduction in t h e number of multiplications and because of t h e integral property, t h e accuracy of t h e calculation is improved b y t h e use of t h e decomposition. If t h e matrices n and y are stored in a machine or in a table, one vector suffices t o hold t h e information in n. Observe on t h e other hand t h a t a can be produced once and for all, and to use t h e decomposition a = n f requires n extra divisions for each M to which oc is applied. As a numerical example, we choose t h e pi as —2, — 1 , 0 , 1 , 2 and calculate a b y Algorithm I. T h e calculation of t h e ay, Ay(pi), and P'(pi) is performed with _ 1 - 1 - 1 ASSOCIATED POLYNOMIALS AND UNIFORM 283 METHODS detached coefficients. T h e rows of y are obtained by writing the elements enclosed in parentheses in reverse order. T h e P ' ( p . ) are enclosed in circles. Note that A ( ) is calculated as a check; it should be zero. See Table 1. Hence n Pi n = (24, - 6 , 4 , - 6 , 2 4 ) , (4.7) Y = 0 0 4 0 0 0 0 24 0 0 (4.8) 2 4 0 -4 -2 -1 -4 -5 -4 -1 2 -16 0 16 -2 2 -1 0 1 2 -1 16 30 16 -1 TABLE -2 -1 1 1 (1 1 1 (1 1 1 (1 1 1 (1 1 1 (1 1 1 1 1 1 1 0 -2 -2 -2 -4 0 -1 -1 -1 -2 0 0 0 0 0 0 1 1 1 2 0 2 2 2 4 -2 -3 -3 -2 0 -5 4 -1 8 7 -5 1 -4 2 -2 -5 0 -5 0 -5 -5 1 -4 2 -2 -5 4 -1 8 7 2 2 -1 -5 0 2 2 -14 -12 0 4 4 2 6 0 0 0 0 0 0 -4 -4 -2 -6 0 -2 -2 14 12 1" 1 1 1 1_ -2 4 0 •4 2 1 -4 6 -4 1 1 0 2 0 4 -4 0) 24 @ 4 -4 0) -6 Q> 4 0 4) 0 © 4 -4 0) -6 0 -4 0) 24 ® 0 4 0 00 0 -2 1 -1 0 0o 1 0 0 0 0 0 1 0 0 0 2 284 J. F. TRAUB Observe t h a t t h e rows of y and a enjoy certain s y m m e t r y properties and t h a t certain of the elements are zero. W e show t h a t this is characteristic for symmetric arrangements of the zeros of P . First we draw some conclusions for t h e case t h a t P ( 0 ) = 0. I n particular, let p = 0. T h e n it m a y be shown t h a t k JiO = dikdn-l ; (4.9) (4.10) i = 1, 2, 71, i = 1, 2, , n. Furthermore Jkj ~ dn-l-j j (4.11) Q>n—1—j (4.12) 0, 1, • • • , n - 1, 0, 1, • • • ,n 1. - dn-l Observe t h a t a -i 0 since double zeros are not permitted. Assume now t h a t n is odd and t h a t the pi are symmetrically placed about t h e origin. This is an i m p o r t a n t case in t h e applications. Arrange t h e pi in increasing order. T h e n it m a y be shown t h a t n ru = ( - l ) V f i - , ; , j = 0, 1, • • • , n - 1, i = 1, 2, (4.13) = Jkj — Q>n-l-j , J <Xij = ( — l) Oi +l-i,j n (4.14) Oikj = , j 1,2, 0,1 Gn—l—j dn-l 1) ,i(n- 1) fc = i ( w + 1). Observe t h a t P h a s only odd powers of t. T h e application of (4.9), (4.10), (4.13), and (4.14) shows t h a t t h e occurrence of zeros in t h e first column and middle row of (4.7) and (4.8) is not accidental. Now let n be even and let the zeros of P be symmetrically arranged about t h e origin. T h e n (4.15) < ; = (-iy yn i-ij, +1 7 (4.16) + 3 an = (-l) a i-i , n+ tj j = 0, 1, • • • , n - 1, i = 1, 2, • • • , i n , j = 0, 1, • • • , n - 1, i = 1, 2, • • • , \n. A number of check formulas m a y be derived. First, note t h a t A (pi) = 0. Hence in performing t h e recursion to form t h e elements Aj(pt), j = 0, 1, • • • , n — 1, it m a y be worthwhile to perform one more recursion so as to obtain A (p») and compare with zero. I n addition we obtain row and column check sums for y and a. F r o m (11.18), n n (4.17) 4„_ (p.) = 7a = U + l ) a » - w , w i=i i = 0, 1, ,n - 1, i=i which provides us with a column sum check for y. F r o m (3.7), x (p.) ^ , . . , l , (4.18) = i=i r {pi) a = 5 i=i which provides us with a column sum check for a 0 ? i = 0 . . . , n - l , ASSOCIATED POLYNOMIALS AND UNIFORM METHODS 285 T o obtain row sum checks we proceed as follows. Let p , 5 * 1, i = 1, 2, • • • , n . From n-l — t = E 3=0 Pi An-t-jipi)?, it follows t h a t Z (4.19) = E An-i-jM 7iJ = ^ , — i = 1, 2, where C = E a,. k=0 On the other hand, suppose one of the pi is unity. Let p = 1. T h e n n-l (4.20) Z 7iJ = P (l)«<* , i = 1, 2, • • • , n . i=o T h e corresponding row sum check for a is k , n-l E «.i (4.21) C 1 p't \ 1 „ n if none of t h e p is unity and t n-l E (4.22) oLij = 8 i k 3=0 if = 1. From (3.6), Pk n -j n E p77-s = E n « (4.23) =0. If n is even and the zeros are symmetric with respect to the origin, t h e n E (4.24) = E = 0. i=l 5. Interpolation. Let p i , p , • • • , p be n arbitrary distinct numbers and let fi ? h> • • • , /n be n arbitrary numbers which are not necessarily distinct. Let P ( t ) = H?=i (2 — p.-) - T h e n there exists a unique polynomial of degree n — 1, I -i(t, f, P), called the interpolatory polynomial, such t h a t P'( P i ) 1=1 2 n n (5.1) h ^ i p i J i P ) = f i y i = 1,2, ,n. T h e symbol / on the left side of (5.1) symbolizes t h e n given numbers / 1 , / , . . . , / , and does not imply the existence of a function / . Any of the coefficients of I n - i m a y be zero and indeed I - i could be identically zero. There are two well-known representations of t h e interpolatory polynomial, those which bear the names of Lagrange and Newton. One other representation has certain advantages over the two conventional representations. This is a 2 n n 286 J. F. TRAUB representation in the canonical form for a polynomial, as a linear combination of powers of t. Such a representation is advantageous if one w a n t s to perform operations on the interpolatory polynomial, which is often the case in practice. Writing the interpolatory polynomial in canonical form is hardly a surprising step. B u t the advantages of doing so seem t o have been largely overlooked. Let n-l 3=0 Thus 3 fi = dn-i-jpi , i = 1, 2, • • • , n. 3=0 By t h e orthonormality relation (3.5), (5.2) d . _ 1 ^ - g i=i / . % ^ r {pi) Let a be the matrix with elements _ A -i-j(pi) n j = 0, 1, • • • , n - P'iPi) 1, i = 1, 2, Let f be the vector with elements /» ,i= 1,2, • • • , n, and let t be the vector with elements t ? = t\ j = 0, 1, • • • , n — 1. T h e n 3 n n—1 /„_!(*,/,P) = D ZWy, i=o i=L or (5.3) I ^(tJ,P) = fat. n Observe t h a t a depends only on the sample points pi , while f depends only on the fi . T h e matrix a m a y be easily calculated as discussed in §4. For certain " s t a n d a r d s e t s " of sample points this calculation m a y be done once and for all and tabulated. For any set of / * , the coefficients of I -i(t) m a y then be computed by a vector-matrix multiplication. As we shall see in the next section, the form (5.3) is well suited for t h e approximation of linear functionals. R a t h e r t h a n using the set 1, t, • • • , t ~ as the basic set, we can use Ao(t), Ai(t), ••• , A _ i ( 0 . L e t n n l n (5.4) / N - L ( * , / , P) = £ 3=0 T h e n from the orthonormality relation, C N _ W Aj(t) . ASSOCIATED POLYNOMIALS AND UNIFORM 287 METHODS 6. Approximation of linear functionals. Let L be a linear functional from a function space with elements / to t h e real or complex numbers. Let / ( p . ) = /»• . One general method for approximating Lf is b y applying L to a n interpolatory polynomial for / . M a n y well-known approximate integration and differentiation formulas are of this type. Let M - denote the moments of L, 3 s (6.1) Lt = M j = 0, 1, • • • , n - j7 1. W e introduce t h e notation (6.2) Q(L,f,P) If we approximate Lf b y LI -i(t, n = LI ^(t,f,P). n / , P ) , we write (6.3) Lf~Q(L,f,P). From In-lit, f,P) = fat, we have immediately t h a t (6.4) Q(L,f,P) = f«M, where M denotes the vector with elements M , Mi, • • • , M -i. Observe t h a t M depends only on L, a depends only on t h e sample points, and f depends only on / . These formulas are conventionally given as a linear combination of /* . T h a t is, t h e matrix-vector product aM is formed and Q is t h e n expressed as a scalar product. T h e generalization of (6.4) to the case of confluent sample points will be discussed in another paper. T h e columns of the a matrix m a y be characterized by the approximation of one particular set of functionals. Let 0 (6.5) Lf = k n , k = 0, 1, • • • , n - 1. Then Lt k J = Sjk, j , k = 0, 1, • • • , n — 1. 3 For k fixed Lkt = M is a column vector with one nonzero element and aM^ is just the kth column of a. Hence k n faM = ^2f(pi)a ik , i=l (6.6) ——— ~ 2^f(pi)oLik • k! i=i T h u s we can characterize the columns of a as the coefficients in the approximation of f (Q)/k\, where t h e approximation is obtained by applying L to the interpolatory polynomial. (k) k 288 J. F. TRAUB As a simple numerical illustration of t h e results of this chapter we t a k e t h e sample zeros as —2, — 1, 0, 1, 2. We calculated in §4 t h a t " 2 0 0 24 16 24 0 - 3 0 0 16 _ 16 - 2 0 -2 4 0 -4 2 - 1 16 - 1 1 -4 6 -4 1 Let Lf = / ( 0 ) . Then, M = 0 r 1 -8 0 8 -1 aM = — 12 0 0 0 (6.7) /'(O) ~ T W ( - 2 ) 8/(-l) + - 8/(1) - /(2)]. This is a special case of t h e preceding discussion on t h e columns of a. Let Lf = f f(t) dt. Then 15" " 7 0 32 aM = — 20 45 0 12 32 7 48 (6.8) [ 7 / ( - 2 ) dt + 32/(-l) + 12/(0) + 32/(1) + 7/(2)]. 45 Let dt. Lf J-3 Then M (6.9) dt ^ = = 5 5 0 15 0 81 [ll/(-2) - aM = 10 11 -14 26 -14 11 14/(-l) +26/(0) - 14/(1) + 11/(2)]. ASSOCIATED POLYNOMIALS AND UNIFORM METHODS 289 Equation (6.8) is an example of a closed integration formula while (6.9) is an example of an open formula. 7. The variation of the coefficients of a polynomial with the variation of its zeros. Let n P(t) = On-jt', a = 0 1, j=0 n P(t) = I I (t - Pi) = PC*, P I , ••• , P n ) . Denote b y Aa& t h e change in dk due t o a change Ap* in p*. W e have P(t, Pi , * ' ' , Pi + Api , • • • , p ) = (J — n P i ) • ' • (J — — P(2, pi , • • • , Pi , Pi — Api) • • • (t — • • • , p) n — Api p) n Pit) t — pi Hence 1 ap = - Z r Wr(p<)r- - , where AP = P(t, pi , • • • , pi, + Ap* , • • • , pn) - P(t, pi , • • • , pi , • • • , p ). n On t h e other h a n d AP = ZAa n r + 1 1 r t - - . r=0 Hence (7.1) = —ApiA (p ), Aa +i r r i = 1, 2, • • • , n, t r = 0, 1, • • • , n — 1. T h u s A ( p ) m a y be characterized as t h e ehange in a +i due t o a change of — 1 in p i . N o t e t h a t (7.1) holds in t h e large, t h a t is, for arbitrary changes Ap* . Taking t h e limit as Ap —> 0, r t r t (7.2) = - A r ( p i ) . dpi T h e differential form only is given b y Brioschi [2] without proof. Since t h e matrix with elements dpi/da i is t h e inverse of t h e matrix with elements da +i/dpi, we can conclude from (3.5) t h a t r+ r ^ OPi d a r n—1—r _ + 1 pi P'(p,) ' or (7.3) dpi da pi r n—r P'( ) Pi E q u a t i o n (7.3) has been obtained b y Brioschi [2] a n d applied b y Olver [21, p . 412]. 290 J. F. TRAUB 8. Solution of inhomogeneous difference and differential equations with constant coefficients. Consider the linear difference equation (or recursion formula) n (8.1) £ an-J(\ + j) = g(\), X = 0, 1, • • • , with the initial conditions V* /(m) = , Using t h e translation operator we m a y also write , , P{E)f{\) = g{\), EJiO) = y, , (8.2) m = 0, 1, • • • , n- 1. n = 0, 1, • • • , n 1. I t m a y be shown t h a t the solution as a n explicit function of the initial conditions is given b y (8.3) /(X) = E »„-!-,-E j=o g(\-i-j)t^h:. i=i (pi) + E i=i r (pi) j=o r This formula was given b y T r a u b [27], the proof based on a division algebra. This formula m a y also be derived by any of the standard transform techniques. A direct derivation based on polynomial interpolation is given b y T r a u b [28]. T h e fact t h a t tne solution satisfies the initial conditions follows immediately from (3.5). T h a t the solution satisfies the difference equation m a y also be verified. Certain solutions of the homogeneous equation are particularly simple. Let g(\) = 0. Let y - i - j = 8jk and let the corresponding solution be labeled/^(X). T h a t is, the starting sequence has a one in the (n — 1 — k)th position and zero elsewhere. T h e n n Jk\A) — —7577—\— J t-i r (pi) or using (1.6), (8.4) f (\) = A (EM\), k where k Q(X) = U. P'iPi)' We t u r n to differential equations. Let P(D)f(x) D f(0) (8-5) = g(x), , = W' ( j where D is t h e differentiation operator. T h e n (8.6) f(x) = E 3=0 y ^ E e"* i=i + r (pi) f gix - v) Jo E 4r~\ i=i 1 de (pj ' ASSOCIATED POLYNOMIALS AND UNIFORM 291 METHODS As in the difference equation case treated above, one m a y verify t h a t (8.6) satisfies the equation and initial conditions of (8.5). T h e fitting of initial conditions usually involves determinants. We have avoided the determinants b y inverting the Vandermonde matrix in (3.5). 9. Divided differences. I t is well known t h a t repeated synthetic division of a polynomial P b y a fixed number u results in the calculation of P (uo)/jl j = 0, 1, • • • , n. This fact lies at the heart of the Ruffini-Horner method for solving algebraic equations [3], [9]. This method depends on calculating from the coefficients of a given polynomial the coefficients of a second polynomial all of whose zeros are less t h a n those of the given polynomial by an amount Uo. Let P(t + u ) = Z(t). T h e n ij) 0 J Q (t) p = ± Z W ^ o ) t\ J! 3=0 Let pi be a zero of P . T h e n 0 = P(pi) = Z( - Pi u ). Q u Hence the polynomial with reduced zeros has coefficients P \uo)/jl, which coefficients m a y be computed b y iterated synthetic division by u . We generalize this result by investigating synthetic division by a sequence Uo, Ui, • • • , u -\ . T h e results m a y be used to perform "nonnoisy" calculations on polynomials as .shown in §10. Divided differences are usually recursively defined as follows. Let u , U\, • • • , u -i be a sequence of numbers. Let 0 n 0 n Pit] = r. Pit, Uo , Ui D , • • • , P(t), ! P[t, Uo , Ui , ' ' ' , Ui-i] — PWo , Ui , ' ' * Ui] Ui\ = , t — Ui i = 0, 1, • • • , n — 1, with P[t, U- ] t = P[(\. If P is a polynomial of degree n, t h e n the ^th divided difference, P[t, u , Ui, • • • , Ui-i], is a polynomial of degree n — i and is symmetric in all its arguments. We introduce the simplified notation 0 Po(t) = P(0, Pi{t) = P[t, uo, Ui, • • • , Pi(t) = i = 1, 2, • • • , n. Then (9.1) , t — Ui-i or (9.2) P (t) t = Pi-l[t, Ut-i]. % = 1, 2, • • • , n, 292 J. F. TRAUB Let (9.3) m = n — i. F r o m (9.1) a n d (1.8), m-l (9.4) p (t) = p [t, i+1 t u<\ = E w p,)r- . A j ( u i , 3=0 T h a t is, t h e coefficients of t h e (i + l ) s t difference of P(t) with respect t o t, u , u i , • • • , w» are given b y the associated polynomials of the ith difference of P . I t follows t h a t if, for convenience, we define t h e symbols 0 A - i ( U i , Pi) = 0, P_i) A j ( u - i , = a,-, then A y ( W i , Pi) = UiAj-tiUi , Pi) + , P;_i), (9.5) i = 0, 1, • • • , n , j = 0, 1, • • • , m, where m = n — i. F r o m this i t follows t h a t (9.6) A j ( U i , = E A (^_!, Pi) f c i = 0, 1, • • • , m. P ; _ i W " * , fc=0 Furthermore (9.7) A m ( v , i , Pi) = P[w , 0 w i , • • • , w,-]. This generalizes (1.7), A ( w , P ) = P(tto). n 0 E q u a t i o n (9.7) gives t h e generalization of the comments a t the beginning of this section concerning t h e calculation of derivatives b y detached coefficients since if Uo = U\ = • • • = Ui, the ith divided difference is just P ( u ) / i l . 10. Applications of divided difference calculations. T h e importance of (9.5) and (9.7) lies in t h e fact t h a t these results permit t h e calculation of divided differences of polynomials in a "nonnoisy m o d e . " By this we mean the following. If the points u , U \ , • • • , u i are close together, then the calculation of P [ u , U\, ( t ) 0 0 • • • , Wi+i] / i n n (10.1) i + 0 by nr P[uo, , fl u i , „, • • • , Ui+i\ i P[U0 ,Ui , ' • • , Uj\ - P[Ui , U j , , Uj+i] = Uo — Ui+i leads t o the loss of accuracy due t o t h e subtraction of quantities which are close together. This is the case when divided differences are being used t o approximate the derivative a t a point. This is easily corrected b y performing t h e calculation using (9.5) a n d (9.7). W e must add one word of caution. T h e use of (9.5) a n d (9.7) permits u s t o avoid the subtraction of (10.1). Of course, it is possible t h a t (9.5) also involves a ASSOCIATED POLYNOMIALS AND UNIFORM 293 METHODS subtraction of close quantities. A simple example is given when Uq ~ — a\. However, this would be an unusual rather t h a n a typical circumstance. We apply this to t h e solution of polynomial equations b y iterative methods. One common method is to use t h e iteration formula _ P(Xj) with P(Xi) and P'(Xi) calculated b y detached coefficients. T h i s variation of t h e Newton-Raphson method is often called t h e Birge-Vieta method. F o r monic polynomials, t h e Birge-Vieta method requires 2n — 1 multiplications per iteration. Another possibility is to use the formula (10.2) x = Xi - i+1 P ^ P[Xi , X i ) and to use a t t h e next iteration the two points a t which P has opposite signs. This method is commonly called regula falsi [26]. Let P(a) = 0. As x. —> a, P[x , a\_i] becomes harder to calculate accurately by t h e usual formula { (10.3) P b < , x ] - P M Z P i X " ) X{ X%—i B y calculating P(xi) b y synthetic division and P[x , b y a second synthetic division, t h e difficulty is avoided. T h e number of multiplications is 2n — 1 as in t h e Birge-Vieta method. Regula falsi has the virtue of bracketing the zero at each step. As a second application we consider the calculation of Stirling numbers of t h e second kind, s(n,i), defined b y n s n x = Z (> > M t n where 3=0 I t follows t h a t s(n,i) n = ( x ) [ 0 , 1, ••• i = 0, 1, t h a t is, t h e nth row of t h e Stirling number matrix m a y be obtained using repeated synthetic division b y 0, 1, • • • , n. This is a more efficient way of obtaining a number of rows of this matrix t h a n b y using either the explicit formula for Stirling numbers of t h e second kind or their recurrence relation [11, p . 169]. 11. Some interesting formulas. A number of interesting formulas are direct consequences of the interpolation formulas of §5. T h e y stem from t h e fact t h a t if there exists a polynomial / , of degree less t h a n or equal to n — 1, such t h a t f(pi) = fi > t h e In-iit, / , P) = f(t). A number of the formulas to be derived are n 294 j. F . TRAUB well known while others are new. Define (11-1) s(j) : = ± p/, i=l 3 (n.2) o ( ; ) = Z ' n n—l+j and recall t h a t (H.4) A {t) = 3 Taking/, = P ( ) Pi Z a ^ r f . in (5.5) and using (5.4) yields (11.5) P\t) = Z s i J ) A n + j i t ) . 3=0 Comparing coefficients of powers of t in (11.5) yields r (11.6) = X ) a -js(j), (n - r)a r r r = 0, 1, • • • , n — 1, which is equivalent to t h e Newton relations. E q u a t i o n (11.6) m a y be written in t h e symbolic form A (E)s(0) (11.7) = (n - r r)ar. T a k i n g / i = pf, fc = 0, 1, • • • , n — 1, in (5.5) and using (5.4) yields (H.8) f 2 f l O ' 3=0 = *Mn-w(0. + I n particular, we have (n.9) r 1 = Z u i j l A n + j i t ) 3=0 as the counterpart of (11.5). Comparing coefficients of powers of tin (11.9) yields 3 (11.10) £ r=0 a _«(r) = y 8 j t 0 , j = 0, 1, • • • , n - This m a y be written in symbolic form as (11.11) Aj(E)a(0) Let R (t) m = 5y, . 0 be a n arbitrary polynomial of degree m S n — 1, m 1. ASSOCIATED POLYNOMIALS AND UNIFORM 295 METHODS This m a y be written n-l (11.12) R (t) j Hgm-jt = m 3=0 if we define (11.13) Q-i = 0, I t follows from (5.2) w i t h / ( 0 = R (t) A Z 1 - ) - ^ f w ( p 1 - m. j , m = 0, 1, • • • , n - 1. that m (11.14) i = 1, 2, • • • , n - ° = g -j, m Let P ( 0 be monic. T h e n m A -i-j(pi)R {pi) E *-i ( P * ) n (11.15) _ m . p j = 0, 1, ••• , n - 1, m = 0, 1, ••• J. m = 0, 1, • • • , n - 1. m = 0, 1, • • • , n - 1. I n particular, if i2 (2) is monic, m (11.16) Z § ^ = i=i r (pi) This includes (3.6)*and (3.7) as special cases. Taking R (t) = A {t) in (11.14) yields m m with t h e convention of (11.13). Hence Z (11.17) P ^p// ^ = i=l f (Pi) Taking R (t) m = P\t) a 2m-n+i, yields n 12 .A„_i_y(pi) = 0 " + l)a„_i_y, j = 0, 1, • • • , n — 1. 1 T a k i n g ft (<) = ^ ^ o ^ yields ro A ( n . 1 9 ) £ »-yy i=l . r (Pij Pi — 1 = i, P l . ^ i , i = 0 , i , . . . , » - i . 12. The inverse transformation and Wronski's aleph function. T h e exphcit formula for Aj(t, P ) , 3 (12.1) Aj(t, P) = 12 aj-tt, k=0 a = 1, 0 j = 0, 1, • • • , n, 296 J. F. TRAUB shows t h a t t h e matrix of t h e transformation between t h e set t\ j = 0 , 1 , • • • , n, and t h e set A -(t, P ) , j = 0 , 1 , • • • , n , is given by 3 "1 ai a ai _a a -i 1 2 (12.2) a = 1 " n dn-2 n 1_ Observe t h a t t h e transformation is specified b y t h e n numbers ai, a , • • • , a . Since | a | = 1 , t h e transformation is nonsingular. Hence t h e associated polynomials form a basis in t h e space of nth degree polynomials. This is t h e case whether or n o t P h a s multiple zeros. Furthermore t h e matrix of t h e inverse transformation is of t h e same form as ( 1 2 . 2 ) . I t is triangular, h a s t h e same element along a n y diagonal, a n d is determined b y just n numbers. I n t h e case t h a t t h e zeros of P are all simple, t h e formula for t h e elements of t h e inverse matrix m a y be obtained as follows. Let 2 n n-l+v WU) = X 0, 1, • ^77 9 =1 P'iPi) This is t h e formula for Wronski's aleph function of order v for t h e numbers Pi y Pi j " j Pn « T h e aleph function is sometimes referred t o as t h e homogeneous product symmetric function of order v (see [ 1 8 ] , [ 6 ] ) . I t is known t h a t o)(v) satisfies t h e difference equation (12.3) X a -jo>(\ n + j ) = 0, 0, 1, We show t h a t if P h a s only simple zeros, then (12.4) * > = £<*(j If j ^ n — 1 , this result follows from j = n. W e have f = t - r 1 = X«(n - k)A (t), 0, k (11.8). 1, Hence we need prove it only for k)tA (t) k = X"(™ - 1 - k)[A i(t) k+ - ajfc+J = X«>(™ - k=0 k)A (t), k k=0 where t h e last line was obtained using ( 1 2 . 3 ) . This completes t h e proof of Hence for t h e case of simple zeros, t h e inverse of ( 1 2 . 2 ) is given b y 1 «(D 1 co(2) co(L) 1 (12.5) o(n) n. aj(n — 1) co(n — 2 ) (12.4). ASSOCIATED POLYNOMIALS AND UNIFORM METHODS 297 We have used the fact t h a t co(0) = 1 which is a consequence of Euler's formula (3.6). F r o m (11.10) and (12.3), A; X (12.6) <*k-fl»U) So , = * = 0, 1, • • • , n. )fc 3=0 This equation m a y be written in operational form as = A (E)u(0) (12.7) k * = 0, 1, • • • , n. 8o, , k E q u a t i o n s (11.10) and (12.3) m a y be used to recursively calculate the , in terms of the ay, = 0, 1, • • • , n. T h e first few oj(v) are given b y cx>(v), v = 0, 1, • • • j co(O) = 1, co(l) = — a i , co(2) = a* — ai, o)(3) = — a\ + 2aia2 — a%. F r o m (11.10) and (12.3) it follows t h a t < 1 2 - 8 > m m ( 5 " ' - , , " E q u a t i o n (12.8) is the basis for methods used to calculate t h e dominant zero of polynomials. W e defer to another paper a general discussion of such methods. F r o m (12.8) we h a v e Observe that P(t) may be considered n l a , a „ _ i , • • • , a . On the other h a n d [t P{l/t)Y w(v), v = 0, 1, • • • . n 0 the generating function for is the generating function for 13. A result concerning powers of zeros. I t is clear t h a t if p is any zero of P, then an arbitrary power of p» m a y be expressed as a linear combination of 1, p t , * * * > P ; ~ \ with coefficients which are polynomials in t h e coefficients of P. T h e problem of calculating t h e coefficients of this linear combination is a classical one. t n We m a y obtain t h e solution simply as follows. Let 1- Zftypr ', P.'= (13.1) x = 0, 3=0 Our orthonormality relation immediately yields (13.2) p w j = V Z Pi Aj(pi) or 3 (13.3) 0 r i = Z aj-r r=0 Z -^n-\ i-1 t (.pi) = T , aj~ Q ( v r r=0 + r). i , . . . . 298 J. F. TRATJB Since Q(v) = co(v — n + 1) is a polynomial in t h e aj, this is the desired result. Observe t h a t (13.3) m a y be written in t h e symbolic form (13.4) ft, = Aj(EMv). F r o m (13.1), w Pi = tpr^AjiEMv), 3=0 and hence (13.5) P : = P[E, Mv). Pi I t is clear t h a t a n arbitrary power of m a y also be expressed as a linear combination of j4 (pt), Ai(pi), • • • , A _i(p»). Let P i 0 w n-1 Pi = E 3=0 a ^ « - i - i ( p » ) - Then a vj (13.6) = Q(v + j), -! n Pi* = T,A +j(piMv+j). n 3=0 I n symbolic form, we have P! = T, A +j( i)E Q(v) j n P = P[E iMv), jP 3=0 which is the result obtained in (13.5). 14. Historical survey. T h e recursion formula (14.1) Aj(u) = uAj_ {u) x + a,, j = 0, 1, ••• , n, A-i(u) = 0, and P{u) = A (u) show t h a t a polynomial m a y be evaluated b y "nesting.'' T h e recursion formula and n (14.2) ^ = ^ + E 4 , - H W t — u t — U j=0 lead to synthetic division. T h e nested evaluation of polynomials, synthetic division, and a certain method for approximating t h e roots of polynomial equations are each commonly called Horner's method. Indeed what we have labeled associated polynomials are sometimes referred to as Horner polynomials. These appellations seem incorrect, as in all these m a t t e r s Horner was anticipated b y others. As pointed out b y Ostrowski [22], Newton (1711) observed t h a t t h e polynomial y* — Ay + 5y — 12y + 17 could be evaluated b y nesting. Horner (1819) [9] noted t h a t the associated polynomials, introduced b y their recursion, were used b y Lagrange in his Theory of Functions. As to the method (1819) for solving 2 ASSOCIATED POLYNOMIALS AND UNIFORM METHODS 299 polynomial equations, which depends on evaluating the derivatives of a polynomial b y repeated synthetic division (see §9), Cajori [3] points out t h a t the same results were obtained b y Ruffini in 1804. Cajori recommends naming this the Ruffini-Horner method. ( D e Morgan [5, p p . 66-67] observes t h a t both men h a d been anticipated b y Chinese mathematicians.) T o sum u p this section of our history, although Horner's work was at first not appreciated b y the Royal Society [5, p . 188], he is today credited with results of which he was not t h e originator. Lagrange (1775) [14] introduced associated polynomials b y giving t h e m in t h e explicit form 3 k=0 H e uses t h e m to obtain a solution of the homogeneous linear difference equation with constant coefficients in the case t h a t the zeros of the indicial polynomial are distinct. T h e result is presented without proof or verification, and his generalization to the case t h a t the indical polynomial has confluent zeros is incorrect. (Lagrange finally corrected this error in 1792.) T h e associated polynomials should perhaps be called Lagrange polynomials except for the confusion with the polynomials in Lagrange's interpolation formula which bear his name. Scattered results on associated polynomials appear in a number of 19th century works. These include Schroder [24], Laguerre [15], Netto [19, p. 97], and Weber [29, pp. 34, 156-163]. (Obreschkoff [20, p. 50] calls the associated polynomials Laguerre polynomials.) Associated polynomials are implicit in Cauchy's [4] t r e a t m e n t of the solution of linear differential equations. A prescription for the numerical inversion of the Vandermonde matrix is given b y Kowalewski [13, pp. 2-6] who obtains the results from Cauchy's interpolatory relations. Kowalewski's prescription is the basis for Algorithm I I , b u t Kowalewski does not concern himself with the numerical details of the prescription. Since then a number of authors have independently rediscovered variations of this method. See Stojakovic [25], Macon and Spitzbart [17], H a m m i n g [8, pp. 124-128], Legras [16, pp. 98-104], Parker [23], and Klinger [12]. Gautschi [7] has studied the norm of the inverse Vandermonde matrix. T h e importance of the Vandermonde inverse for the approximation of linear functionals has been pointed out b y Legras [16] and H a m m i n g [8]. See also Bragg [1]. H a m m i n g calls the inverse matrices "universal matrices". None of these authors has studied the numerical aspects of the inversion. Our contribution to the theory has been a unified approach which has led to m a n y new results. We have taken the generating function as t h e basic definition and the orthonormality relation as the basic result. Acknowledgment. I t is a pleasure to t h a n k m y colleague A. J. Goldstein for a number of illuminating discussions. Appendix. Notation. 1. P(t) pa) 2. = Z"-o W " , = N ? - I (t - oo = 1. p o . 300 J. F. TRAUB 3. P[t, Uo, Ui, • • • , u^i] is t h e i t h divided difference of P(t) with respect to 4. P (t) = t t l o , tti , • • • , M<-J, J r 5. (*P)[*,u] = E ? = o i » - i ( w , P ) l = 0 if j * k, 6 Po(0 = P(0- Z"-o^»-i(ti)^. dik = * u ^ if j = *. o M /= L , 87.. / s( w X += l E ) P= / ,tf/(X), ( X£) -= g D/(X). 9. a : 10. = ^ ' f f l a i j n:?7iy = f P (pi)8ij \ ; c o ( , ) = Z ^ Pa = p/« T h e n «0 = / . r 7;; : = A _i_y(p;). n Then a = n - 1 ^ - REFERENCES [1] L. H. A matrix approach to numerical integration, Amer. Math. Monthly, 71 BRAGG, (1964), pp. 391-398. } [2J F. BRIOSCHI, [3] F. CAJORI, Intorno ad alcune questioni d algebra superiore, Ann. di Sci. Matem. Fis., (1854), pp. 301-312. Horner's method of approximation anticipated by Ruffini, Bull. Amer. Math. S o c , 17(1911), pp. 409-414. [4] A. L. CAUCHY, Sur la determination des constantes arbitraires renfermees dans la inte- grates des equations differentielles lineares, Oeuvres Ser. 2, vol. 7, Academie des Sciences, Paris. [5] A. D E M O R G A N , A Budget of Paradoxes, vol. I I , London, 1872. [6] Encyklop. d. Math. Wissensch., vol. 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