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ON PADE APPROXIMANTS ASSOCIATED WITH HAMBURGER SERIES (t)

Dedicated to Professor Aldo Ghizzetti on his 75th birthday

W. GAUTSCHI (z)

ABSTRACT - We discuss three (only loosely connected) aspects of Pad6 approximants associated with Hamburger series: (i) Normality criteria, expressed in terms of orthogonal polynomials (ii) Inequalities for expansion coefficients (iii) Compu- tational methods.

1. Introduction.

A formal power series

(1.1) f ( z ) = ~ + / z t z+/zz z2+ ...

is called a Hamburger series if its coefficients are moments

(1.2) /~k =f_ t k d2 (t)

R

of a bounded nondecreasing function 2(t) having infinitely many points of increase; (1.1) is called a Stielt]es series if (1.2) holds with a distribution d2 supported on the nonnegative real axis R+. There is a well-known connection between Pad6 approximants associated with a Stieltjes or Hamburger series and polynomials orthogonal with respect to the distribution d~. (For a recent exposition, emphasiz- ing this point of view, see [ 1]). The Pad6 approximants on the first subdiagonal of the Pad6 table (3), indeed, are

(1) Sponsored in part by the National Science Foundation under grant MCS- 79.27158AI.

(z) Department of Computer Sciences, Purdue University, West Lafayette, U.S.A.. (3) We arrange the Pad6 table so that fractions with constant denominator degree

~appear in the same row.

112 W. GAUTSCHI: On Padd approximants

(1.30) ] [n- - 1, n] (z )= ~ 2v(") n = 1, 2, 3 ... . ,~.1 1 - - T ~ , (n) g "

where ~'~(") are the zeros of the n-th degree orthogonal polynomial rr, (t; dJ.) and 2~ (") the corresponding Christoffel numbers. More generally, given any integer ]>__0, we h_ave for Stieltjes series, and also for Hamburger series if ] is even,

(1.3i) J I n - 1 -hi, n] (z)=tzo-klZl z-F ... -b~i-1 zi-l-l-z i ~ iv'i(") �9 ~ I 1 - T~,,] (hI Z '

where ~'~,i (') are the zeros of zr,, i ( . )=z r , ( . ; d2i) and 2~,i (n) the associated Christoffel numbers, d2i now being the measure

(1.4) d2i (t) = t i d2(t).

The integer n in (1.3i) may be any nonnegative integer, if ] > 0 , assuming the usual convention that empty sums are zero. Since for Stieltjes series the support of d2 is in R+, the m e a s u r e d~ i is positive definite, hence defines a unique set of (monic) orthogonal polynomials zr,,t, n = 0 , 1, 2 .... (cf., e.g., [7, w 2.2]). The same is true for Hamburger series, if ] is even.

Pad6 tables associated with Stieltjes series are known to be always normal. This, of course, is no longer true for Hamburger series, an extreme example being furnished by a symmetric distribution d2 on a symmetric interval, in which case the formal power series (1.1) proceeds in even powers of z and no entry of the Pad6 table is normal. (The Pad6 table, however, is seminormal in a sense defined by Gragg; see [9, p. 16 and Theorem 7.3]). In Section 2 we formulate and prove a condition, in terms of orthogonal polynomials, for all entries ] [n-- 1 +] , n], n>_ 1, ]_>0, of the Pad~ table associated with a Hamburger series to be normal (Corollary to Theorem 2.1).

Expanding the rational part of (1.3i) in powers of z, one gets

(1.5)

where n

(1.6) /.zk,/'~ = X 2~./") [v~./,o]k, k=O, 1, 2 . . . . . ~'-----1

l [ n - 1 +] , n] (z)=m+tz~ z + ... +/z~,-~+i z ~-~+j + ~ gk,i (~) z k+i, k = 2 n

We have used the fact that

(1.7) Pk,i (") =/~k+i if O_<_k<2n, j___O.

associated with Hamburger serles 113

The expansion coefficients /zk,ff ), when (1.1) is a St]cities series, satisfy the following interesting inequalities,

(1.8) tZk.i(~+I)--l.tk./"~>O for all k>__2n, n = 1 , 2 , 3 . . . . . ] = 0 , 1,2 .... ;

see the proof of Theorem 5.2.7 in Baker & Graves-Morris [2]. This means that the coefficients of some fixed power z k+i in the expansion of the Pad6 approximants down a diagonal [n--1 +], n], n = 1, 2 ..... ]>_0, are monotonically increasing until they become, and stay, equal to an exact moment.

We show in Section 3 (see Corollary to Theorem 3.2) that (1.8) remains true for Pad6 approximants associated with a large class of Hamburger series, provided ] is restricted to an even integer. Our tool, which produces the assertion rather quickly, is a recent result of Hunter [ 10] on orthogonal polynomials, for which we also give a slightly simplified proof.

Finally, in Section 4, we discuss several methods, all based on ~he representation (1.32/), of computing Pad6 approximants f [n--1 +2], n]. Numerical stability being of particular concern to us, we avoid methods that depart from the moments. We assume, instead, ,that we are given the measure d2(t), or equi- valently, the recurs]on coefficients of the associated orthogonal polynomials, and use these as input to our procedures for generating the moments /zk and Gauss- Christoffel data 7:v,2i (n), 2~,2i (~) required in (1.32/).

2. Normality criteria.

An entry of the Padd table is said to be normal if the, same entry does not occur in any o~her location of the Padd table. The Padd table is normal if each of its entries is normal. Every Padd table associated with a Stieltjes series is known to be normal [13, p. 390]. We now formulate necessary and sufficient conditions for normality in the case of a Padd table associated with a Hamburger series.

The measure d2i(t) of (1.4), when j>O is even, gives rise to (mort]c) orthogonal polynomials zrn.i and polynomials of the second kind,

(2.1) an, i (z) = f zr,~,j (Z)z_ t-- zr''i ( t) d 2i ( t),

It

where lr,,, t and a,,,i are of exact degree n and n - 1 , respectively. We also need the function

(2.2) p., i(z)= f ~ d 2 1 ( t ) , R

114 W. GAUTSEHI: On Padd approximants

where z is assumed outside the support of d2i. We write an and pn for an,o and

On,0. Clearly,

(2.3) rc.,i (z) f d2i (t____)) _ an,j (z)+P.,i (z). �9 - z - - t

R

Also, as is well-known (see, e. g., [4, w 1.4]),

~.,; (z) ~ 2~,/") (2.4) zcn, i (z) -- ~=-1 z--v~,i (n) '

where v,,i <") are ~he zeros of z:.,i and 2~,/") the corresponding Christoffel numbers. Note that the limit pn, i ( O ) = l imo.u(z) , if j>O, formally exists, since by (2.5)

Z ~ 0

and (2.4),

-p . . , (0)= (o) (o)

(2.5) R

_ ~ , ~ , 1 = (o) u -x

if r~n,i ( 0 ) 4 0 , and similarly, with an expression involving r~',,/(0), if zcn,i ( 0 )=0 .

THEOREM 2.1. The entry / [ n - - l + j , n ] , n_>l, j (even)>_O, in the Pad~ table associated with the Hamburger series (1.1), (1.2) is normal i] and only if

(2.6) zrn (0) an (0) ~ 0 in the case j = O,

and

(2.7) ~zn,i (O)pn, i (0)@O in the case j>O.

Proof. Assume first j > 0 . Then, by (1.3i) and (2.4),

(2.8i) f [ n - - l + j , n ] (z )= zr,a* ( z ) [ ~ + t z l z + . . . + l Z i _ l z i - 1 ] + z i ~n,i* (z) zc,,i* (z)

where zrn, i* ( z ) = z n zrn, i ( l /z) and o'n,i* (z) - -z "-1 a, , j(1/z). From (2.4) it can be seen that the zeros of o'n,i are real and alternate with the zeros of rr i (if n > 1 ) . It follows that zrn,i and an, i have no common zeros, hence neither do ~r,,i* and r Consequently, the fraction in (2.8/) is irreducible, if n > l . The same is trivially true (since o'1.i* (z)-=/zi>0) if n = l . A well-known theorem [12, Satz 5.3] then tells us that the entry (2.8/) is normal if and only if its numerator and

associated with Hamburger series 115

denominator polynomials are of exact degree n - 1 +] and n, respectively, and /~z~+s--#2.,j C") ~ 0 [cf. (1.5)].

The last condition is always satisfied, since Markov's formula for the remainder term R, (.) in Gaussian integration (with measure d2i) yields

(2.9) /~z~+s-~z,,J<">= R, (t~) = . f ~, , i (t) d2j ( t)>0. R

Now the coefficient of the power z" in zc,,i* is rc,,i (0), while the coefficient of the power z "-~+i in the numerator polynomial of (2.8i) is zr,,j (0) tzi_~+~r,~, i (0), or, by (2.3),

f ~.,; (o) , ~ _ , - ~ . ~ (o) / ~ - P " , ~ (o)= -p.,; (o).

l

R

Therefore, the degrees of the numerator and denominator polynomials in (2.8/) are exactly n - - l + ] and n, respectively, if and only if z:,,i(O)p,,s(O)+O. This proves (2.7).

The proof in the case ] = 0 follows similarly from

(2.8o) f [ n - 1, n] = o',*(z) z:,* (z)'

the degree condition (for the numerator polynomial) now reading ~, (0)=~0 in place of p,,i (0) +0 . []

REMARKS.

1. Theorem 2.1 holds also for n = 0, ] > 0, if the condition in (2.7) is replaced by/zi_1 ~0 .

2. By virtue of (2.4), (2.5), ~he conditions (2.6) and (2.7) can also be written in the form

(2.63 rr. (o),=,z ~ # 0, i = 0,

(2.7') r:,,i (0) l/zi_l-- ~ k~'J<") ) ,=l ~ l =~0' j (even)>0.

3. The condition zr,,i (0)=~0, ] (even)_>0, implies the existence of zr#,i+l; indeed,

116 W . G A U T S C H I : On Padd approxlmants

1 [ zr~+t,i(O) ] tin,i+1 (t) = --i-- ~n+Li (t)-- ZCn.j (0) 7Cn, i (t) .

The representation (1.3i+0 therefore holds for f [n+j, n], ] (even) >__0, provided ~r,,i+x has n! distinct zeros.

4. The usual criterion for normality of l [ n - l + ] , n], ] ( even)> 0, in terms of determinants, is [12, Satz 5.4]

(2.1o) An-l,] An d An-l,/+1 And-l=[=O,

where

(2.11)

[gk gk+l

A,,k - det]~.:+, t g.:+:

[ / z k + n /zk+n+t

�9 �9 " g k + n

�9 �9 �9 / ~ k + n + l

o . �9 � 9 �9

�9 �9 �9 ~Zk+~

, k > - - I

(with the understanding that /z_l=0). The conditions implied by (2.10) can easily be recovered from (2.6), (2.7). It suffices, first of all, to show z1~-1.i+1 z1,,i_1 ~=0, since from the theory of the Hamburger moment problem, A~_~,i>0, z~,i>0 if j > 0 is even (cf., e. g., [13, p. 325]). The representation of orthogonal polynomials in determinant form,

(2.12) zrn.i ( z ) -

-[.tj " " " gj+n-t 1 ] -- 1- -~ldet l [zi+1 "'" gJ+" z

dn_l, i . . . . . . . . . . . .

Lgi+~ . . . ~j+2~-t z ~

(see, e. g., [7, Eq. (2.2.7)]), yields

( - I ) " rc..i (O)=zl._~.j+,/za._,.i, ]_0 .

and, for ] > 0,

r -p , , j (0)= lrr,,i (t) tJ-t d2(t)

. /

R

rgi "'" gj+n-t t t-I ]

_ 1 . f de t [ g j + t ~ [ "'" tzi+n tt d2 An_l, i . . . . . . . . . . . .

R L~j+ ,~ . . . / z i + e n - t t ~+n-1

(t) - ( - 1)" A,,.t-i/A.-~,j.


The condition rc.,t(0)=~0 thus is equivalent to A~-1,i+1~-0, while p.,j(O)~=O (for j > 0 ) is equivalent to A..i-~@O. Finally, if j = 0 ,

a, (0) - - f zr, (t)t-- re,, (0) d2 (t) =

R

1! nll I 1]I = .. 1 f de~ /~1 * ' " /-/n t - - d e t /z~ . . . /z, 0 d2(t) A n - l , O . . . �9 . . . . . . . . �9 . . . . . . . . . . . .

R I_IZ~ . . . IZ~- i t" [ l Z . " . ttz~-1 0

. - - g . - 1

f IfZl . - - /z,z

o] 1 d2(t) . . .

cn-1

= ( - 1)" A._I/A,,_,,o,

so that a,, (0)~=0 is equivalent to A . _ I ~ 0 . As a consequence of Remark 4, note that ~. (0):~0, all n>__ 1, is equivalent

to A.,_I~0, all n___l, while n.,i (0)~:0, all n_>l , all ] (even)>0, is equivalent to An, k~=O, all n>_O, all k (odd)> 1. Since A . . i > 0 whenever ] is even, it follows that the conditions

(2.13) ~.(0)=[=0, all n>_l,

and

(2.14) zr,,/(O)=~O, all n ~ l , all j (even)>O,

together are sufficient, and also necessary, for (2.10) to hold for all n_> 1 and all j_>0. This establishes:

COROLLARY TO THEOREM 2.1. Every entry / [n-- 1 +L n], n>__ 1, in the Padd tcble associated with the Hamburger series (1.1), (1.2), regardless of whether ]>_0 is even or odd, is normal i/ and only if (2.13) and (2.14) hold.

3. lnequafities for Padr expansion coefficients.

We assume in this section that

(3.1) d2( t ) = w(t) dr.

118 W. GAUTSCHI: Or/ Padd approximants

where w (t) is a nonnegative weight function on a symmetric interval I: --oo<__-a<__t<__a<~, a>O, continuous on the open interval ( - - a , a ) , and such that all moments #k in (1.2) exist and /zo>0. We denote the associated (monic) orthogonal polynomial of degree n by =, ( .) = rcn (. ; d2), its zeros by v~ ("), and the Christoffel numbers by 2~ ("). Note that z" =, ( l /z ) is a polynomial of degree < n, equal to 1 at z = 0. Let

1 (3.2) z" rcn (1/z) = c~ + c1(") z + c2 (n) z 2 + . . . . co (n) = 1,

be the Maclaurin expansion of its reciprocal.

Tm~oga~M 3.1. (Hunter [10]) (a) I f w ( t ) / w ( - - t ) is strictly increasing on L then

(3.3) Ck~")>O, k = 0 , 1 , 2 .. . . ; n - - 1 , 2 , 3 ....

(b) I f w ( t ) = w ( - - t ) on L and n>_2, then

(3.4) ek (") > 0 if k is even, ck (") = 0 if k is odd.

(c) I f w ( t ) / w ( - t ) is strictly decreasing on L then

(3.5) ( - 1 ) k r k = 0 , 1 , 2 . . . . ; n = 1 , 2 , 3 . . . . .

PROOV (cf. Hunter [10]). Let

w ( t , a ) = c r w ( t ) + ( 1 - - a ) w ( - - t ) , 0<_a<_l,

and let z-~ (a)>v~ (U)>...>Vn (a) be the zeros of rrn ( . ; w (t, r Consider

1 n 1

z ~ =n ; w (t, a) dt ,~=l 1 - r~ (u) z

(3.6)

where

r":':J I I 1 =q (z) 11 ,-1 [ 1 -- r~ Ca) z] [ 1 --'r,+1_~ (a) z] '

q ( z ) = t 1' 1

n el:en,

n odd.


(a) We prove that q (z) (if n is odd) and each product in curled brackets in (3.6) (if n>_2) has a Maclaurin expansion with all coefficients positive if 1/2 < a < 1. Since w (t, 1) = w (t), hence r J '~ = -c~ (1), v = 1, 2, . . . , n, the assertion (3.3) then follows immediately from (3.6) with 6 = 1.

Write, for short, z~=v, (6), v~*=r.,+a_~ (09. Since w (t, 1/2) is an even function, the zeros -r~ (o-) for r 1/2 are symmetric with respect to the origin. Fur- thermore, each zero z~ (a), under the assumption of (a), increases monotonically on 1/2<_6<_1 (see, e.g., [10]). If all z~ (o') are positive, the assertion is obvious in view of

1 i _~ . z = 1 + z , z+z-,2 za+ .... r~>0.

It suffices, vherefore, to consider pairs of zeros r~, -r,* (if n_>2) such that

- -a < z ' J _ < 0 < T , <a , [r~*[<l:~.

Write ~-~*= -- 7~ v~, 0_< 7 , < 1 . Then

(3.7)

1 1 (1--~'~ z) (1 - -~* z) (1 --'r z) (1 +g~ ~'~ z)

1 1--[v~ (1--y~) z+7~ r~z Z2] = 1 + [ ' ' ' ] + [ ' ' ' ]2+ ''" '

where the content of the brackets on the right is the same as in the denominator immediately to the left. Since v~>0 and 0_<7,<1, the coefficients of z and z a in these brackets are positive and nonnegative, respectively, hence (3.7), when fully expanded in powers of z, can produce only positive coefficients. The same is true for q(z) if n is odd, since by the monotonicity of the zeros, v(~+~)n ( 6 ) > 0 for V2<a<_ 1.

(b) In this case of symmetry, w (t, a ) = w (t) is even and 7 , = 1 in (3.7); the expansion of (3.6) contains only even powers of z, each, as before, with a positive coefficient, and q (z)---1.

(c) Applying part (a) to the weight function w ( - t ) and its associated (monic) orthogonal polynomials ( - - 1 ) n z r , ( - z ) yields positive coefficients in the expansion of [ ( - z ) " z r , . ( - 1 / z ) ] -1, .hence alternating coefficients in the expansion of [z ~ zr~ ( l / z ) ] -1. []

We are now in a position to prove the following theorem for the expansion coefficients/zk (") =/&o (") in (1.5).

120

(3.8)

(3.9)

(3.1~)

and

(3.12)

(3.1o)

(while, trivially, O'--#k (1) =l~k (2) ---... =#k if k is odd).

(c) I f w ( t ) /w (--t) is stricdy decreasing on I, then

0 ~ k (1) ~ / ,~k (2) ~ ,,. ~ [ - L k ( [ k /2 ]+ l ) - ' - [,Lk ([k/2]+2) "--" . , , - - [ . s

W. GAurscm: On Pad~ approximants

Trmom~M 3.2. Let

/zk(~)= Z 2J~)[~-~(')] k, k = 1 , 2 , 3 , . . . ; n = 1 , 2 , 3 . . . . .

(a) I f w ( t ) /w (--t) is strictly increasing on I, then

0 ~ ,Uk (1) ~ ,Uk (2) ~ ... ~ ~k ([M2]+I) - -[ . l .k ([k/21+2) - - . , , - - ' ~ k �9

(b) I f w (0 - - w (-- t) on I, then

O---/~k(1)<~,~(Z)<~...<~/-Lk ([k/2]+l)-- /Zk (tkn]+2) ---...'-/.tk i] k is even

~/r . , . -~-/-~k ([k/2]+2) - " ].L/c ([k/2]+l) ~ ~ k ([kl2]) ~ . . . <~/..s (2) < ~ k (1) ~ 0

PROOF. By (1.5), (1.6), with ]----0, we have

(3.13)

With h =f, zc~ (t) d2(t) Z

if k is even

i] k is odd.

oo I [ n - - l , n] (z)=/zo+/~1 z+ . . .+ / z~_ l z2~-z+ 2: / Z k ( " ) z k.

k~2n

denoting the normalization factor for the orthogonal

polynomial re,, it is known (see, e. g., [11], where --z is used in place of our z), and easily verified, that

l [n, n + l ] (z)--I [ n - - l , n] (z )= hrt

zrr~ ( l /z ) zr,+l ( l / z )

By (3.13), this can be rewritten in the form

.Z t . . (n4-1) - - , , (n) ~:.z__ l ' ~ (3.14) z=0,,-2,,+~ e'2n+V- - - z" zr~ ( l /z ) z ~+1 zrn+l ( l /z )

Now in the case (a) we apply Theorem 3.1 (a) to the expansion of both [z" rr, ( l / z ) ] -1 and [z "+1 rr,+l ( l / z ) ] -1 on the fight of (3.14) and conclude that the product expansion has all coefficients positive, hence/.t("+u--/z (") 2n+~ ~n+z > 0 for

all l>_0. This proves all <<inner>> inequalities in (3.9). The outer inequality on the left,


].~k(1) --/~i(I) [" ~i(1)] k ~> 0,

follows from the posifivity of the Christoffel number 2, (') and from 71(1)>0, which in turn is a consequence of flae monotonicity property for the root ~'~ (~r) (used in the proof of Theorem 3.1). The equalities on the fight of (3.9) follow from (1.7). Parts (b) and (c) of Theorem 3.2 follow similarly from Theorem 3.I (b) and (c). []

COROLLARY TO THEOREM 3.2. The inequalities (3.9) - (3.12) o / T h e o r e m 3.2, under the appropriate assumption on w, hold also/or the quantities ltk,/n) defined in (1.6), provided ] is even.

PROOF. If W satisfies one of the assumptions (a), (b), (c) of Theorem 3.2, then wi (t)=t i w (t), J even, satisfies the same assumption. []

EXAMPLE 3.1. lacobi distribution w ( t ) = ( 1 - - t ) ~ ( l + t ) # on [ - 1 , 1]. Here,

w (t) (1 + t ~ - ~ w C - t ) - \TL7-} '

which is strictly increasing if a<fl, equal to 1 if a- - r , and strictly decreasing if a>fl. Accordingly, we have (3.9) if a<fl, (3.10) if a=fl, and (3.11), (3.12) if a>fl.

EXAMPLE 3.2. The special cases a=+_1/2, fl=+--V2 of Example 3.1 yield interesting trigonometric inequalities, the simplest of which (for a=f l=-- l /2) are

(3.15) - - L c ~ < - c o s ' n ,=, \ 2n ]1 n + l ,--1 t

k = 2 , 3 , 4 .... ; n = 2 , 3 . . . . . k.

EXAMPLE 3.3. A translate of the logistic distribution:

1 ~-(t-~)D Ig(t)= ~ [1..[_e_(t_t0/~] 2, - - o o < t < ~ , 9 > 0 .

An elementary computation will show that we are in the case (a), (b), (c) of Theorem 3.2 depending on whether /z > 0, /z = 0, o r / z < 0, respectively.

4. Computational methods.

Instead of considering as input data the moments g~ in (1.2), which, as is

122 W. GAUTSCHI: On Padd approxirnants

well-known, give usually rise to ill-conditioned problems, we assume here that we are given the measure d),(t), and that it be required to generate from it the desired Pad4 approximant. We concentrate on the approximant f [n - - 1 -F2j, n], n _ l , j _ 0 , referring to Remark 3 of Section 2 for the case of f [ n + 2 j , hi. The measure d2, being positive definite, generates a set of (monic) orthogonal polynomials r~k ( ') = rck (. ; d2) satisfying

(4.D

=~+~ (z) = ( z - a~) ~ (z) - / ~ , zck_, (z),

~r-1 (z) =0 , re0 (z)= 1,

k--0, 1, 2 . . . . .

where ak, [3k are real numbers and flk>O. Although [30 is arbitrary, we find it

convenient to define [3o= [d2(t). J

/

R

Given the measure d2, a number of (usually stable) methods are known for generating the coefficients ak, ilk, k=O, 1, 2 .... ; see Gautschi [6]. We assume, therefore, that we are given the first n+] of these coefficients: ak, flk, k=O, 1,2 . . . . . n+j- -1 . (For ~classical>> measures they are known explicitly). From these, it will be possible to compute not only the moments/zo,/~ . . . . . /,tzi_l, but also the Gauss-Christoffel data 2,,2ff ), -r,.2j ("), all in a stable manner. Together, they determine the desired Pad4 approximant f [ n - - l + 2 L n ] according to (1.32i). Our motivation to proceed in this manner derives mainly from two considerations: First, we wish to maintain a high degree of numerical stability in generating the approximant in question, and also obtain it in a form condu- cive to stable evaluation. Secondly, it is desirable to employ mathematical soft- ware which, by now, ought to be part of the standard computing repertoire.

Basically, the problem is to compute the recursion coefficients ak.2i, [3~.2i associated with the measure d22i (t)'--t2id2(t), given the recursion coefficients ~Z=ak,o, flk=flk,o. There are several methods of accomplishing this; we discuss three of them based, respectively, on Christoffel's theorem, Chebyshev's algorithm, and the OR algorithm. It suffices to consider j = 1, since the general case can be treated by repeated application of the special one.

4.1. An algorithm derived from ChristoffeI's theorem. Our first algorithm obtains from multiplying the measure d2(t) twice by t, each time employing Galant's algorithmic version [3] of Christoffel's theorem to generate the new recursion coefficients. The result is (cf. also Eq. (4.1) in Gautschi [5] , where z=0) :


(4.2)

A

e_t--O, qo=~

ek--r~+t/qk A ^

q~= qk+ ek-- ek-1 ^

flk = qk ek_l I qk+l = (lk+l -- r A ^

ek=q~+, ek/qk ^ ^ A A

a~=qk +ek

k=O, 1, 2 . . . . . n - - 1,

^ ^

^ A

where ak, rk denote the recursion coefficients for the measure dt2 ( t )=t td t ( t ) .

In (4.2), flo is set equal to zero. If we wish to adhere to our convention rio---

22 (t)~ we must redefine fl0 as

R

A A

(4.20) fl0 =flo (fit + ao=).

This is obtained by representing /2 in terms of the orthogonal polynomials zrk,

(4.3) t2=co rCo (t)+cl rh (t) +c2 ~2 (t),

comparing coefficients of equal powers on the fight and left, and making use of ~rt ( t ) = t - ~ , re2 ( t )=( t -a t ) ( t -ao ) - rx . One finds

(4.4) 0o--,81-I-c~o 2, c ,=~oq-al , c2--1,

f from which, by orthogonality, rio= 2d2(t) = co d2(t)=(fll+~2)flo, as claim-

R R

ed in (4.20). The algorithm (4.2), (4.2o) produces the first n of the desired recursion

coefficients in terms of the first n + 1 given coefficients. In the general case of d22i, one needs ~he first n+j recursion coefficients ak, ilk, k=O, 1 . . . . . n+]--1, in order to produce ak,2i, rk,2i, k = O, 1 . . . . . n-- 1, by/-fold repetition of (4.2), (4.2o).

A serious deficiency of this algorithm is the fact that it breaks down whenever a o = 0 (division by zero in eo=rl/qo), which happens, for example, if d l is a symmetric measure. Also, considerable loss of accuracy is observed in cases where d2 is enearly symmetrical, i.e., the coefficients ak are relatively small. The two algorithms that follow are more robust in this respect.

124 W . G A U T S C H I : O n Padd approximants

4.2. Modified Chebyshev algorithm. Another implementation of Christoffel's theorem, using modified moments, has been described in Gautschi [4, p. 123], The ~modified moments>> of d22(t)=t2d2(t) with respect to the polynomials zck (.)=zrk (. ; d2), in view of (4.3), are

t" f vk= l ~'k (t) d).2 ( t)= I ~k (t) t 2 d2 (t)

d / R R

=ic~ .f~z~ 2 (t)d~(t)

i0 R if k > 2 .

if k<_2,

Since f=z d,~(t)=flofll ... ilk, and taking note of (4.4), one finds

R

(4.5) vo=/~o ~+ao2), vl=/~op, (ao+~,), v2=/~oB~&,

va=0 for k > 2 .

Given these modified moments of d22, the modified Chebyshev algorithm produces the desired recursion coefficients for d22 in terms of certain quantities ~k,t generated recursively from the vk (cf., e. g., Gautschi [6, w 2.4]). The algorithm, in fact, simplifies considerably, since vk=0, k>2 , which implies ak, k+3=0, all k>_0. Using the notation blk=~k,k, V k "--" •k,k+l, W k = ak, k+2o the algorithm can be written in the form

Initialization:

(4.6o)

/20=~0 ( ~ I - ~ t X 0 2)

w - l = 0 ) if n > l ^

^ *2o a o = t Z o + ~

tto A ^

Po=uo

Continuation: for k=l , 2 . . . . . n - - l :


(4.6)

uk = wk-1 - (ak-~--ak) ~k-1--/?~-1 W,-2+/?, Uk-l

2 '/3k " - - - ( a k - 1 - - a k + l ) W k - 1 -[- flk+l Vk-1

142k--~k+2 kl)k-1 (if k < n - 1)

(~ k "-" (~ k -]- - - g k Uk-1

Uk ilk=

Uk-1

Given the recursion coefficients ak, ilk, 0 < k < n, for d2, this determines uniquely

and unequivocally (since Uk=ak, k>0) the recursion coefficients ak, ilk, 0 __< k ___ n-- 1, for d22. As before, the coefficients ak.2j, flk,zi can be obtained by repeating the process/-times.

4.3 OR algorithm. Golub &Kautsky [8, Corollary 1 to Lemma 4] recently observed that the recursion coefficients for d22 (t)=t2d2(t) can be obtained from those for d2(t) by applying one step of the OR algorithm (with zero shift) to a symmetric tridiagonal matrix. More precisely, if the first n recursion

coefficients ak, ilk, k = 0 , 1 . . . . . n--1, of d2~ are desired, one applies the OR step to the (symmetric, tridiagonal) lacobi matrix of order n + 2 belonging to d2 [with diagonal elements ak, k = 0 , 1 . . . . . n + 1, and first side-diagonal elements

[/f12, k = 1, 2 . . . . . n + 1] and discards the last two rows and columns in the result. The matrix of order n so obtained is the lacobi matrix for d22. Using the square root free implementation of the OR algorithm, described in Wilkinson [14, p. 567], one is led to the following algorithm (in the notations of [ 14], except that

aj is replaced by ai-1 and b i by ~l/-~-t_l):

uo=o, co=l , /?o=/~o ~,+ao~,l

(4.7)

~k -" tZk-I ~ Uk-I

~g~2/Ck-12 if Ck-14:0 Pk2= (Ck-z2 flk_l if r 2 ~k-l"-Sk-12 (pk 2 " ~ k ) i f k > l

s~ ~ = ~k/(p~ +/~k)

Uk=Sk ~ (~'~+ak)

ak-1 = y~ + U~

k = 1, 2 . . . . . n.

126 W. GAUTSCHI: On Padd approximants

As before, /-fold repetition of this algorithm, with n suitably increased, yields the recursion coefficients for d~,2i ( t ) = t zi d2(t).

Numerically, Algorithm (4.7) appears to produce slightly more accurate results than the algorithm in (4.6), but otherwise they are comparable. Algo, rithm (4.2), as already observed, looses accuracy in nearly symmetric situations.

This is illustrated in Table 4.1, where the maximum relative errors in ~zk, ilk, k=O, 1 . . . . . n - - l , are shown in the case d 2 ( t ) = ( l + t ) ~ d t on [ - -1 , 1], with e = . l , .01 . . . . . . 00001, n=20, using Algorithms (4.7), (4.6) and (4.2) based, respectively, on the OR algorithm, Chebyshev's algorithm and Christoffel's ~heorem. (Numbers in parentheses indicate decimal exponents. The computa- tions were performed on the CDC 6500 computer, which has a machine precision of approx. 5.55• 10 -25 in single precision).

OR Chebyshev Christoffel e err a err fl err ~z err fl err a err fl

.1 7.49(-14) 2.48(-14) 1.05(-12) 9.38(-14) 1.28(-11) 1.24(-12)

.01 8.18(-14) 1.67(-14) 7.00(-13) 7.03(-14) 1.89(.9) 1.88(-10)

.001 6.82(-14) 2.25(-14) 4.47(-13) 6.60(-14) 2.97(-7) 2.89(-8)

.0001 8.79(-14) 1.74(-14) 9.07(-13) 6.92(-14) 1.05(-5) 1.02(-6)

.00001 9.30(-14) 1.57(-14) 7.80(-13) 6.97(-14) 3.39(-4) 4.90(-5)

TABLE 4.1. Numerical performance o/ the algorithms (4.7), (4.6) and (4.2) in the case d 2 ( t ) = ( l + t ) " dt on [ - 1 , 1], n=20.

Once the recursion coefficients ~Zk,2~, ~k,:i have been obtained, it is a simple matter to produce from the corresponding Iacobi matrix the Christoffel numbers 2~,2/") and the nodes z~.2/") required in (1.3zi). For appropriate methods see, e. g., Gautschi [4, w 5.1]. Likewise, the moments /zk, k=O, 1 . . . . . 2i--1, can be computed by applying the j-point Gauss-Christoffel quadrature rule (associated with the measure d2) to the integrals in (1.2).

For Stieltjes series the appropriate algorithm is the Cholesky LR algorithm, Golub & Kamsky [8, Theorem 3], or ~he closely related algorithms in Galant [3] and Gautschi [5].

Acknowledgment.

The author is indebted to Richard A. Askey for reminding him of the results in reference [10] and to William B. Gragg Jr. for useful discussions.


R E F E R E N C E S

[I] G. D. ALLEN, C. K. CHUI, W. R. MADYCH, F.J. NACOWICH, P. W. SMITH, Vad~ approxi- mation of Stielt]es series, J. Approx. Theory 14 (1975), 302-516.

[2] G. A. BAKER, ~R., P. GRAVEs-MoRRIS, Padd approximants, part [ : basic theory 1981. Addison-Wesley, Reading, Mass.

[3] D. GALANT, An implementation of Christoffel's theorem in the theory of orthogonal polynomials, Math, Comp. 25 (1971), 111-113.

[4] W. GAUTSCm, A survey .o] Gauss-Christoffel quadrature formulae, in r B. Christoffel, The Influence of his Work on Mathematics and the Physical Seiences~ 1981. P. L. Butzer, F. Feh6r, eds. Birkh~iuser, Basel, 72-147.

[5] W. GAUTSCm, An algorithmic implementation of the generalized Christoffel theorem, in: ~Numerische Integration>~ 1982. G. H~immerlin, ed., 8%i06. Internat. Set. Numer. Math. 57, Birkh~iuser, Basel.

[6] W. GAUTSCm, On generating orthogonal polynomials, SIAM 1. Sci. Statist. Comput.. 3 (1982), 289-317.

[7] A. GmZZ~TTI, A. OssIclm, PoUnomi ortogonali e problema dei momenti, Pubbl. Istit. Mat. Appl. Fac. Ing. Univ. Stud. Roma 231, Rome, 1981.

[8] G. H. GOLUa, l. KAUTSKY, Calculation of Gauss quadratures with multiple lree and fixed knots, Numer. Math. 41 (1983), 147-163.

[9] W. B. GRACe, The Pad~ table and its relation to certain algorithms of numerical analysis, SIAM Rev. 14 (1972), 1-62.

[10] D. B. HUNTER, Some properties of orthogonal polynomials, Math. Comp. 29 (1975), 559-565.

[11] I. KARLSSON, B. VON SYDOW, The convergence o/Pad~ approximants to series o] Stieh]es, Ark. Mat. 14 (1976), 43-53.

[12] O. PERRON, Die Lehre yon den Kettenbriichen, 1957, II, 3rd ed., B. G. Teubner, Stuttgart.

[13] H. S. WALL, Analytic theory of continued fractions, Chelsea, Bronx, N. Y., 1948.

[14] J. H. WILKINSON, The algebraic eigenvaIue problem, Clarendon Press, Oxford, .1965.

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