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1 and hence the denominator of (4) will not vanish in N. s, will then be (k + [3)-1summable to 1.
Chapter 10
The Brezinski-Havie Protocol
10.1. Introduction and Derivation; Sequences in a Banach Space Let s be a complex convergent sequence approaching its limit s in the following manner: s; ~
S
+ L crj~(n) r= 1
(1)
where {f..(n)} is an asymptotic scale. It is surprising that very often in practical problems the form of the functionsj, is known but the values ofthe coefficients A r are not-or are at best, difficult to compute. As a simple example, suppose S is the value of an integral whose nth approximation by the trapezoidal rule is Sn' Thenf,(n) = w > but the An which depend on the higher derivatives of the integrand, may be impossible to compute with any accuracy since, generally, only tabular values of the integrand are known. Many other examples are given in Chapter 3. What the Brezinski-Havie (BH) protocol does, loosely speaking, is to establish a deltoid algorithm that maps s into a sequence S(k) where S~k) has a representation similar to (1), but where the first k terms are accounted for. Thus the convergence of s can be accelerated with no knowledge whatsoever of the coefficients A r • The importance and usefulness of the method can hardly be exaggerated. The idea of the general representation (1) was, apparently, first articulated by Levin (1973), although special cases, such as the Romberg scheme for integration, are classical and form the basis for many of the algorithms in previous chapters. The discovery of the deltoid computation of S~k) and its representation by a ratio of determinants, the real heart of the method, are 175
176
10. The Brezinski-Havie Protocol
due to Havie (1979) and Brezinski, respectively. The latter was communicated to me privately in 1979. We shall conduct the derivation of the algorithm for sequences in a Banach space. Although the formal aspects of derivation can be carried through for sequences in any topological vector space, interesting applications of the algorithm seem to be possible only when the underlying space is metrizable and the dual has a reasonable supply of functionals. For normed spaces the Hahn-Banach theorem guarantees the latter and, of course, the norm provides a convenient metric. In what follows Pil will be a nontrivial real or complex Banach space with norm II· II, and Pil* will be its dual; s will be a sequence with members in fJB, i.e., S E P4s and «l> a convergent (not to zero) sequence of functionals in /JIJ*, i.e., «l> E !?4~;
~
00.
To say «l> is a constant sequence means that
s; <
°
fl(n)···J,.(n) (2)
where
Cn
=
[~
•
1
...
<
It is assumed here and in what follows that ICn I =F
°
(3)
for the values of n under consideration. Note that Ek is not exact for constant sequences nor even for all members of Linlf.}, Although it is possible to define a similar transformation that is exact for constant sequences, it does not have the very nice formal properties of the present one. At any rate, we may simply remark that E k is exact for constants and members of Lin If.} whenever «l> is constant. The following easily demonstrated theorem describes the effect of E k on sequences of the form (1) when the right-hand side converges.
10.1. Introduction and Derivation; Sequences in a Banach Space
177
In what follows let
fjk)(n) =
=
S~k)
=
=
(4)
- s).
Theorem 1. Let (1) hold where c, is in the field of gg and the series on the right converges absolutely, i.e., IArlll.f..(n)11 < 00. Then
L:
S~k)
L: CX!
= Ek(s) +
r~k+
and S~k) =
Proof
Trivial.
<1>., Ek(s»
1
crEk[.f..(n)]
L:
(5)
CX!
+
r~k+
l
(6)
cJ~k)(n).
•
Note Ek(s) = S when is constant. To examine the convergence properties of E; requires recursion formulas for S~k) and R~k).
Theorem 2 S
~
0, (7)
R(k+ I) = fn l(n)R~kL - fikL(n + I)R~k) • fn I(n) - fikL(n + 1) 11.<1>., s)fik~ I(n) - fik~ I(n + 1)'
+ fY~ I(n) it: 1)( ) = I
n
fikt I(n)flk)(n + 1) - fikt I(n + 1}f~k)(n) fikt t(n) - fikL(n + 1 ) '
i ~ 1,0 S k
s
flo'(n) =
n, k ~ 0, R~O) =
(8)
(9)
i - 2, i ~ 1.
Since
+ k»
+ k»
(10)
the first recursion formula is easily demonstrated by applying Sylvester's identity (Appendix, Section A.3). The prooffor R~k) follows by the substitution R~k) = S~k) -
178
10. The Brezinski-Havie Protocol
A computational tableau for computing ftk)(n) will be discussed later along with the scalar case fJI = C(5. Now consider a path P. Provided r~k) = S~k) - S for n large on P does not approach too close, relatively speaking, to the kernel of
<
$'
d(g, $')
= {g E 881<
(11)
= inf Ilg - hll,
(12)
s= Theorem 3.
For n
hE .JI'
<
(13)
+ k sufficiently large on d(r~kl, $')/Hk)1I 2
Then S~k) converges in norm to two following cases:
S
s>
P, let O.
(14)
on P iff S~k) converges to S on P in the
(i)
For n
-> 00.
+ k large, 11
2
I<
[see Jameson (1974, p. 188)]. Now observe that, since <
(15)
+ <
1<
(16)
so it is clear we can find m1 and m 2 , 0 < m1 < m 2 , such that (17)
mlllr~k)1I :o;;.I<
Further, <
+ <
and combining this with
and the assertion of the second case follows on taking n
-> 00.
•
10.1. Introduction and Derivation; Sequences in a Banach Space
179
What is desired, of course, is information about the convergence of S~k) that does not require a knowledge of S~k). The following lemma is the link to a theorem that does this. Lemma (Brezinski). (lnk
Let S
E
f!Jc , and for each k, let
= fikL(n + 1)/fikL(n)
(19)
be bounded away from I, 0 ~ k ~ K. Then S~) converges to S on all vertical paths, 0 ~ k ~ K + 1. Proof
Since (lnk/(I - (lnk) is bounded, Eq. (20) furnishes an easy inductive proof on k that R~k) ~ o. It is then trivial that S~k) ~ S. • Paths where k is unbounded present a formidable problem. Aside from certain easy scalar cases, i.e., the summation by extrapolation formulas of Chapter 3, nothing has yet been accomplished on this problem. We have now proved the following result. Theorem 4. Let the hypothesis (14) of Theorem 3 hold and for each fixed k, 0 ~ k ~ K, let (lnk be bounded away from 1. Then Ek is regular on all vertical paths, 0 ~ k ~ K + 1. Example. Let f!4 = ~z; cPI = cPz = cP;
(22) A sufficient condition for the regularity for E 1 in this case is that (23)
For instance, if Lib n = o(bn), then it is sufficient that an = o(bn). Ek is then regular for this class of sequences.
180
10. The Brezinski-Hiivie Protocol
10.2. The Case
cP Constant
For the case cI> = const the algorithm can be derived in the same way the Schmidt transformation was derived. One assumes that Sn behaves as Sn
=S+
k
L
(1)
crfr(n).
r~l
Taking ¢ of Eq. (1), replacing n by m, n :-: : ; m :-: : ; n + k, and considering those equations and the above as k + 2 equations in the k + 1 unknowns (¢, S), Cb C z , " " C k produces the requirement S
0
f~(n)
(¢,sn)
I
(¢, ftc(n»
s, -
(¢,
Sn+k)
(¢, fl(n
+ k»
(¢, ftc(n
= 0,
(2)
+ k»
but this is clearly equivalent to Eq. 10.1(2) when S = s~kl, cI> = const. Conversely, when s; has the form (I), then E k will be exact, S~k) == S, provided the algorithm is defined. For the study of this algorithm, there are two modes of regularity or accelerativeness to consider. One pertains to weak convergence, i.e., convergence in the seminorm I(¢, .) I.The other is the usual strong convergence, convergence in the norm. The regularity result below, though based on pretty specific properties off,., is often applicable. Theorem 1.
Let s E fJB c and
lim(¢, f,.(n
+
1»/(¢, f,.(n»
= b,
# 1,
1 :-: : ; r :-: : ; k,
(3)
where b, # bj , i # j. Define
(4) (which we assume exists for n sufficiently large) and denote by Am the proposition "lIfm(n)II/(¢, fm(n»
= 0(1)."
(5)
Then along any vertical path.
In
(i) if '7n is bounded, then E; is regular for s in the seminorm I(¢,.) I; (ii) if '7n ----> b, for some j, then Ek accelerates s in seminorm; (iii) if '7n is bounded and Am holds, 1 :-: : ; m :-: : ; k, then E, is regular for s norm;
10.2. The Case I/J Constant
if'1n ---> b, for some j, Am holds, 1 :s; m :s; k, m
(iv)
=1=
181
j, and (6)
then E k accelerates s in norm.
Proof ship r(k) n
~ ~
All these statements are immediate consequences of the relation-
<¢, rn)
v;. + 1(1, bl' b 2 , · · · , bk) 0 fl(n)/<¢, fl(n»
rn/<¢, rn) 1
x
1 1
'1n '1n ... '1n+k -
I
j~(n)/<¢,
1 bl
1 bk
b kI
b~
fk(n» (7)
where v;. is a Vandermonde determinant (see Notation). Details are left to the reader with a hint: For (iv) subtract the first column of the determinant from the jth column. • Example 1.
Let (8)
f,.(n) = x~h,
and let Xn+ I/Xn ---> b, b =1= 1, b =1= O. Now, (Ink = Xn+k+ .t»: This provides a generalization to Banach spaces of the summation by extrapolation scheme of Section 3.3, but here there is no simple deltoid algorithm for the computation of S~k), only for S~k) = <¢, S~k». Example 2.
Let f,.(n)
=
x~h,
X E~,
h
E
fJH,
h ¢'.
x:
(9)
and let the Xi be distinct numbers, none of which is 1. (Ink = Xk+ I and this gives a generalization of the deltoid of Section 3.2. Although there is no deltoid in the general case, S~k) can be written out as a linear combination of the};(n) with closedform coefficients that are Vandermonde determinants. Both of these examples generalize the Romberg and Richardson procedures.
182
10. The Brezinski-Havie Protocol
10.3. The Topological Schmidt Transformation
In the algorithm of Section 10.1 take};(n) = ization of the Shanks-Schmidt transformation when
This gives a generalThe most useful case is
.1Sn + i - 1.
ek(Sn)'
(1)
Jv,. as in Eq. 6.1(8). Although the above is a nonlinear algorithm, its denominator is a scalar. Thus it requires nothing in the way of invertibility from the elements of []d. Obviously it retains the properties of homogeneity and translativity of the scalar algorithm. The first question one asks is, How good is this abstract version of the Schmidt transformation? Insight into this very difficult question can be obtained by answering a simpler question: For what sequences is the algorithm exact? This question is easily resolved. Since, for the scalar algorithm, there is an intimate relation between exactness and regularity, one expects this relation to hold in other Banach spaces. We shall show that, roughly speaking, the topological Ek can be exact only for sequences that are linear combinations of fixed elements of !!l, where all the dependence on n is restricted to the scalars in the linear combination. Unless !!l is finite dimensional, this is clearly a small class of sequences. Although more theoretical and numerical investigations are required, this probably means that E, is regular for a disappointingly small subspace of !!lc. In what follows Greek letters denote scalars. (Note that the arguments used carry through for any topological vector space.) Theorem 1. Let k be fixed. Then S~k) is defined and s~) (¢, r n E ''X''k (see Section 6.3) and
>
k-l
_ S + '" 1...
Sn -
° °
m=O
r m Tn(m) ,
== s,
n ~
0, iff
(2)
where T~m), S m S k - 1, is a basis of solutions of the (scalar) equation = satisfying
;J!lk
TU) = m
J mJ'.
Os m,j S k - 1.
(3)
Proof
(4)
10.3. The Topological Schmidt Transformation
183
Taking ¢ of both sides gives
o = ~~k) =
(5)
Jtv,.(~n' d~n)/Jtv,.(I, d~n)'
By Theorem 6.3, the definition and exactness of ~~k) imply ~n E x:k : Let {r~O), r~l), ... , r~k-l)} be a basis of gIIk where gIIk(~n) = 0, satisfying (2). Then :<
k-l
_
,,:<
L....
Sn -
But this means <¢, r«
r,
where V k- I
Vn E
=
(6)
.
k-I
L rmr~m) + Vn,
(7)
m=O
ff (the kernel of ¢). Putting n
= O. Substituting (7) into (2) gives
o=
(m)
Sm'T n
=0 or
L rm r~m»
-
m=O
= 0,1,2, ... , k - 1 shows Vo, VI' ... ,
II
k r Jtv,.( r~m») m= 0 m Jtv,.( 1)
+
Jtv,.( vn) Jtv,.(1) .
(8)
Now we have g\(d~n)
= Yo
d~n
+
Yl d~n+ 1
+ ." + Yk d~n+k =
0
(9)
with, say, Yk = 1. By Lemma 6.3(3) each term in the sum in (8) is proportional to gIIk(r~m») and hence equal to zero. Thus Jtv,.(v n) = 0, or Vn satisfies (10)
= 0, Vn = 0 for all n. This part is trivial, since (2) shows gIIk(rn) tional to Jtv,.(rn) (=0). • Since
Vo, ... , Vk-l
<=:
=0, and this is propor-
The following, more of an observation than a theorem, is a useful negative criterion. Theorem 2. gIIk = 0 and let
< >
Let ¢, rn satisfy a homogeneous linear difference equation
o~ m ~ k -
1,
where IJn is a nonmaximal solution of f?l!k = O. Then S~k) is not defined. Example 1 (k
= 2).
<¢, rn>
<¢, r n >= (cln
E
ff z if, for example,
+ c z)-1.n , C I i=
0, -1. i= 0, 1.
(11)
184
10. The Brezinski-Havie Protocol
Then
(12) Thus £2 will be exact for sequences of the form
s, =
S
+ ro(1
- n)An
+ rlnAn-l,
A =F 0, 1,
(13)
provided Aro - r l rf- f. Example 2
(k
= 1. Aitken's (j2_process in a Banach space) sn =
Sn -
«1>, !J.sn)/<1>, !J.2sn»)!J.sn·
Theorem 3. Let S EIJc . Then s; in (14) is defined and s; S + Anro for some A E 'Ii, A =F 0, 1.
s; =
(14)
== S iff r0 rf- x: and
If S E .IJc , our previous work, Theorem 10.1(4), states that this algorithm converges provided (15) and
0< a < 1 <
/3.
(16)
The acceleration properties of the algorithm are easily established. Theorem 4.
Let
S E
.r1Jc and
<1>, an+ 2)1 <1>, an+ I) =
Then
p
°< Ipl <
+ 0(1),
1.
(17)
s converges to S more rapidly than s in the seminorm 1<1>, .) I.
If, further,
(18) then
s converges more rapidly than s in norm.
Proof
By Theorem 1.4(1),
<1>, rn + 1)/<1>, rn ) = so Theorem 10.2(1) may be applied.
p
+ 0(1),
(19)
•
Brezinski (1975) has studied this algorithm. It may be used to generalize the Pade table in the following way. Let a E :JBs and Z E (c. Then we may write the formal power series f(z) =
00
L: ajz j
j=O
(20)
lOA. The Scalar Case
185
with partial sums
s;
=
n
L a.z',
(21)
j=O
S~k) will then define a formal rational approximation tofwhose numerator is a polynomial of degree n + k in z with coefficients in [J9 and whose denominator is a scalar polynomial of degree n. For details, see Brezinski (1975). Germain- Bonne (1978) has also studied this algorithm. The topological Schmidt transformation provides a construction for iteration functions for the solution of operator equations. Let f: [J9 -+ flJ and define (22) j~(x) = x.
Take, in (1), ~Sn +k -+
Sn+k
-+
k ? 0, k > 0.
fk+ 1 (Sn) - fk(Sn), !t(Sn),
(23)
Thus the k = 1 case of Eq. (1) produces sn+ 1 = s; -
(24)
for the solution of x = f(x). In contrast with often-used methods such as the generalized Newton iteration scheme, these formulas do not require the evaluation of the (Frechet) derivatives off There are a multitude of other ways the BH protocol can be used to construct iteration functions. One could take «p const,
Jj(n + k) = [h.{sn) - !t-l(Sn)]
-+
fk(Sn),
k ? 0,
> 0,
(25)
and replace S~k) by sn+ 1 on the left-hand side of Eq. 10.1(2).
10.4. The Scalar Case When [J9 is its scalar field and cPn == I (the identity), S~k) = S~k) and there is a deltoid computational scheme for the computation of S~k). Then !t(n) (1)
186
10. The Brezinski-Havie Protocol
and the algorithm becomes n ?:: 0;
(2)
.
F\k + 1)( ) = fikL I (n)f~k)(n + 1) - fn I (n + 1)f~kJ(n) n fnl(n) - ftL(n + 1) , I
i ?:: I,
0.:::; k .:::; i - 2;
i ?:: 1,
(3)
and f~k)(n) = EkU/n», i.e., the ratio (1) with s, replaced by /;(n). The computational tableau of the algorithm is as follows. The S~k) array is filled out in diagonal lines, the kth line being {siO~ I' sp~ 2; si2~ 3, ... , s~- I J}:
S1
S2
To compute this kth diagonal line, k - I subsidiary arrays are needed. Each array has the following form: ith Array
flO)(o) = flO) flO)(1)
=
1;(1)
11 i -1)(0) .ni-I)(l)
f\i-I)(k - i-I)
J;(k - 2) h(k - 1)
fli-I)(k - i)
187
10.4. The Scalar Case
For instance, to compute the diagonal three arrays are needed: Array 1 fl(O)
{S~Oi, s~1), S)2),
Array 3
Array 2 f3(0)
j~(O)
fil)(O)
I, (1)
f~I)(O)
fil)
f2(1)
u»
fl(2) fl(3)
slJ3)} the following
fi1)(1)
f~2)(0)
f~I)(1)
fY)(1)
f3(2) fi 1)(2)
f~I)(2)
f3(3)
f2(3)
Clearly, the amount of computer storage necessary at the completion of the computation of the kth diagonal is k(k + 1)(2k + 0/6 : : :; k 3/3. The computations are probably best done in the following order: (a) (b) (c) arrays;
Initialize slJO) = so,/)O)(O) = fl(O), f)O)(1) = fl(1). Assume the (k - l)th S~k) diagonal has been filled out, k :2: 2. For k > 2 compute new ascending diagonals of each of the k - 2 i.e., DO, for 1 ~ i ~ k - 2, generate flO)(k - 1) = J;(k - 1); compute
(d)
f~j)(k
- 1 - j)
from (3),
1
~ j ~ i-I.
For k > 2 fill out array k - 1, i.e., generate fkO~ 1 and DO, for
l~i~k-l:
generate fko~ I (i) = j~ - I (i); compute fV~ 1 (i - j) from (3),
j
= 1, 2, ... ,i
(i - 1 if i
= k - 1).
(e) For k :2: 2 fill out the kth diagonal of S~k); i.e., generate compute st~ 1-;' 1 ~ i ~ k - 1, from (2). (f ) Go back to (a).
Sk _ 1
and
Note that moving down one diagonal in the S~k) table necessitates adding one more subsidiary array. The Brezinski-Havie protocol for scalar sequences is undoubtedly the most elegant and flexible computational procedure yet discovered for the transformation of sequences. The flexibility of the algorithm lies in the possible choices of L. Generally speaking, the choices are made with a foreknowledge of the kinds of sequences one wishes to accelerate.
188
10. The Brezinski-Havie Protocol
Before discussing particular cases of the scalar algorithm, let us gather together the convergence and acceleration results. Theorem 1. Let, for each fixed k, (Ink [Eq. 10.1(19)] be bounded away from I. Then E, is regular on all vertical paths. Theorem 2.
Let Sn =
S
+
L aJr(n), 00
(4)
r= I
where for each n the series on the right converges absolutely. Then S~k)
=
S
+
L 00
r=k+1
(5)
aJ~k)(n).
Proof Follows from Theorem 10.1(1). This shows E k is exact for constant sequences and Lin{f l , f 2 , ... , fd when f j is independent ofs. •
For the next three theorems, the common hypothesis is limf,.(n
+ 1)/f,.(n) =
Theorem 3.
Let s E rt'c.
b, =I- 1,
1~ r
~
b, =I- bj ,
k,
i =I- j.
(6)
(i) If h; = 0(1), then E; is regular along vertical paths. (ii) If h; ~ bj for some j, then E k is accelerative along vertical paths. Our last two results require the following. Lemma fJk)(n) ~ jj(n) (bj - b l ) ... (bj - bk) , (1 - bl)· ··(1 - bk ) Proof
Left to the reader.
Theorem 4.
j>k~1.
•
Let k
~
o.
(8)
k
~
o.
(9)
Then
Proof
(7)
Left to the reader.
•
10.5. The Levin Transformations
Theorem 5.
189
For some A let Iail < Ai and let (6) and
f..+ l(n) =
o(f..(n»
(10)
hold uniformly in r. Let the representation (4) hold. Then k ? O.
(11)
Proof Requirement (10) guarantees the absolute and uniform convergence of (4) for n > no- It is easily seen that r~k) ~ a k + I f~k+ I), and the previous theorem may be invoked. •
Example 1
A E [ -1,1).
(12)
Then S~I)/Sn+ I
= 1/(..1. - I)n = 0(1).
(13)
The conditions of Theorems 1, 3, and 4 are satisfied but not those of Theorems 2 and 5 since there is no a l =I 0 such that An/n = alAn + .... Example 2. If};.(n) = ~Sn+k-l' the result is the Schmidt transformation but, interestingly, the algorithm for computing the transformation is not the s-algorithm. Work remains to be done in assessing the relative computational advantages of the two algorithms. 10.5. The Levin Transformations In 1973, Levin gave a general transformation of series that is enormously useful in numerical analysis and that has been the subject of a wide literature. Essentially, the transformation is a special case of the previous transformation, although Levin did not develop an algorithm for computing S~k) efficiently. There are a number of useful cases of his algorithm, and I will examine each of these in turn. After making the assumption 10.4(4) Levin effected the specialization Jj(n) = x~-l(n,
Xm =I x n ,
Expanding by minors, one finds S(k) n
=
t
Sn+m
m=O (n+m
n~k,m)1
r=O r*m
_1_
m=O (n+m
k
, m ) = "(x n (k n L, n+r
i
m =I n.
-
Xn+m )-1 ,
(1)
n~k,m), (2)
190
10. The Brezinski-Havie Protocol
where it is assumed, of course, that all quantities are defined. Furthermore, (3) There are a number of ways to choose the x, and (n that make sense. One is as follows. If the sequence s converges rapidly enough and, say, the terms IQjl are ultimately monotone in such a way that we may write
(4) then a good choice would seem to be (n = Qn+ r- (h n can be multiplied, of course, by any constant without affecting the algorithm.) For this choice of hn , we wish to analyze the acceleration properties of the algorithm using the theorems of Chapter 5. Let, as usual, (5) Note that S~k) depends on Sn' s, + 1> ••• , s; +k + 1 and is translative and homogeneous. Thus we need to consider the functions (6) An application of the Smith-Ford (1979) theorem gives
Theorem. some Un let
In (2) let (n =
Qn+l' Q n
'I. O. Let
SE
.Plp , 0 <
Ipi < 1, and for (7)
where
I
k
p-mn(k.m)
# O.
m=O
(8)
Then S~k) accelerates the convergence of s along any vertical path.
Proof
=.
±
(1
+ Pn+l + Pn+lPn+2 + ... + Pn+l···Pn+m_l)n~k.m)
m=1 X
L k
(
)-1
Pn+1Pn+2'''Pn+m
1
m=O Pn+ IPn+2'"
n~k,m) Pn+m
(9)
10.5. The Levin Transformations
191
Thus
(10)
and
(11)
By elementary properties of interpolation sums,
L n~k.m) == O. Thus
gn(pe) =.1/(1 - p).
(12)
g(pe) = 1/(1 - p),
(13)
By uniform convergence, and this concludes the proof.
•
In this proof it was assumed x is independent ofs. However, x may depend on s if the dependency is such that the homogeneity and translativity of the transformation is maintained, e.g., x, = Llsn •
10.5.1. The t-Transform In his analysis, Levin took (n = an, but, as Smith and Ford point out, (n = a n + 1 makes better sense and simplifies the convergence analysis. One natural choice of the x, would result by assuming that s converges as
(1) where
Vn
is a Poincare asymptotic series, in other words, by taking
x, = lien
+
1).
(2)
In what follows, it is assumed that an "1=. o. This choice and dividing numerator and denominator by common factors amounts to the choices and
Jr~k.m) = (n + m + ll- 1(_1)m(~)
192
10. The Brezinski-Havie Protocol
in Eq. 10.5(2). Then _ (k) _ L~=o (sn+m/an+m+ l)(n + m + l)k-l( _1)m(~) tk(Sn ) = Sn '\'k k i m k) L,m=O (1/a n+ m+ l)(n + m + 1) (-1) (m
(3)
This is called the Levin t-transform.
k ?: 1, is accelerative for C(f, along any vertical path. In Theorem 10.5 take Un = nk- 1 and then n(k.m) = (_1)m(~).
Theorem 1. Proof We have
tb
Ik m1 n(k, m) = (1l-)-k #- O. m=O P P
•
(4)
tk turns out to be regular for any path for another large and important class of sequences.
S
Theorem 2. Ifs E along any path. Proof
C(fc
and a is alternating, S~k) is defined and converges to
We can write
=
() =
~_~n
First, assume n
-+ 00
(n
+m+
l)k-l
1an + m + 1 I
(k).; ~
L.. r =0
m
(n
+ r + IlIan+ r + 1 I
1
(k) r
(5)
.
on P. In this case, use k
Ir~k) I s sup IrmiL Jlmk m~n
m=O
=
sup IrmI -+ o.
(6)
m~n
If n is bounded on P, use Theorem 5.2(1). Conditions (i) and (ii) of that theorem are satisfied. We can majorize Jlkm by throwing away all the terms in its denominator except the last, so k! lan+k+ll(n+m+l)k-l Jlkm < .,------------, - Ian + m + lin + k + 1 m! (k - m)!
~ Ian +k + 1 I (n + m + Il- 1 k" - k+ 1 Ian + m + 1 1m! en+ 1 ' so Jlkm
=
k
-+ 00,
n bounded,
0(1) in k along P and this establishes convergence along P.
t does not work well on monotone sequences.
(7)
•
10.5. The Levin Transformations
Theorem 3.
Let Re () < -1, an '" n/l(co
Co "1=
0, and
+ c fn + C2/n2 + ...).
(8)
Then r~k) ~ -
Proof
+ 1)( -
k! n/l+ ICO/(O
193
k
()k'
~
o.
(9)
Exercise. •
10.5.2. The u-Transform
The t-transform was designed to be used on rapidly convergent alternating series.The u-transform is designed for monotonic series and has the following heuristic basis. Consider
1
n
s;
=
k~O (k + 1)'"
IX > 1.
(1)
Then, according to the work in Chapter 1 [Theorem 1.7(3)], Sn -
S ~
-(n
+ 1)1-«/(1
- IX),
(2)
or s, - S ~ C(n + l)an + i- Since the above sequence is such a typical one, it makes sense to take (n = (n
+ 1)an+ 1
(3)
and, as before, x; = (n + 1)-1. (This is not the precise choice Levin made= nan' x, = n-1-but seems preferable since the transform is now defined for all n.) The result is called the Levin u-transform: (n
Uk ( Sn )
_
(k) _
- Sn -
L~=o(sn+..Jan+m+'>(_1)m(n + m + 1)k-2(~) "k m k 2 k • L,.m=O (1/a n+ m+ 1)( -1) (n + m + 1) (m)
(4)
Theorem 10.5 gives immediately the next result. Theorem 1. The vertical path.
Uk
transform, k
~
1, is accelerative for
Cfij,
along any
It is ironic that, despite its derivation, it has not been established that Uk is regular for monotone series. (In fact, I suspect this is not true.) Nevertheless, a result in the previous section continues to hold.
Theorem 2. If S E Cfijc and a is alternating, to S along any path.
S~k)
is defined and converges
194
10. The Brezinskj-Havie Protocol
For certain kinds of monotone series, however, the columns of the utransform give excellent results. These are series whose general term has a Poincare type of asymptotic expansion. Theorem 3.
Let Re 0 < -I, an ~ n8(c o
Co f=
0, and
+ cdn + cz/n z + ...).
(5)
Then for some m ?: k, (6)
Left to the reader.
Proof
•
If s behaves as
0< IAI < I,
(7)
n
(8)
then it is easily shown that ---> 00,
for both t k and ui : Thus the Levin transformations enhance exponential convergence algebraically. Levin defined another transform, called the v-transform, by taking e 1(sn), the Aitken 15 Z-iterate, as an estimate for s; i.e., in Eq. 10.4(1), fj(n) = [a n+ l/(Pn - I)JX~-l, x, = (n + 1)- 1. Thus the u-transform is defined by (k) _
Sn
-
L~~o (sn+m(Pn+m - l)ja n+m+l)(n + L:~~o «Pn+m - I)/a n +m+ l)(n + m
m+ It+
1
(
_1)m(~)
I)k 1( _1)m(~.)
,
(9)
Obviously this idea can be elaborated ad absurdum, since any sequence transformation can be used for (n' Smith and Ford, however, think that v has special advantages, and consider it one of the best practical transformations. 10.5.3. Exactness Theoremsfor t and u t and u turn out to be exact for a surprisingly large and varied class of sequences. To explore this matter, we first demonstrate an exactness result for a general case of the scalar algorithm 10.4(1).
Theorem 1. Let k ?: I and jj(n) = an + 19/n) where the gj are linearly independent and independent of s. Let rn' an f= 0 and let the denominator
tn.s,
The Levin Transformations
195
of Eq. 10.4(1) vanish for no value of n. Then S~k) == Sn' n ~ 0, iff s, =
n- I
S
+ (so - s) fl (1 + r= 1
'r-
I
n
),
~
0,
(1)
where T is a nontrivial member of Lin[gl' ... , gk]. Furthermore if gl(n) == 1 and gj(n) is an asymptotic scale, the transform is exact for S E ~c only if (2)
for some j, 1 :s; j :s; k, and some E E
re
N'
Proof =: S~k)
-
S
r;
=
fl(n)
fk(n)
+ k)
rn+ k fl(n
.h(n
fl(n) x
fl(n
+ k)
j~(n)
+ k)
f~(n
-I
+ k)
(3)
Let Vn
=
n-I
fl (1 + -; 1 ).
(4)
r=O
Differencing (1) shows (5)
and if an is defined and nonzero, then r; = an+1T n and S~k) - S = 0, n ~ 0. =: We must have
Tn
# 0,
So
# s. Thus rn/an+ 1 = r, or
(6)
and by linear independence of the gj'
C1
# 0. This can be written (7)
for some r, E Lin[gl' ... , gkJ, or rn+1/rn = (1
and taking products gives (1).
+ Tn-I),
(8)
196
10. The Brezjnskt-Havie Protocol
To prove the second part of the theorem, write 1
+
I/!n = 1 + [CI
=
(l
+ c 2g 2(n) + ... + ckgk(n)]-1 + I/cl)(l + 8n ) , CI =1= () E C{/ N,
-1.
(9)
Note that C1 =1= 0, since otherwise s is not convergent. Substituting (9) in (8), taking products, and using Theorem 1.4(2) gives (2) (with j = 1). The case C1 = - 1 is handled similarly. • (Exactness for Euler Series).
Theorem 2
Sn
Then for t k , S~k)
=
S
+
=s for x
n
I
k'x",
=1=
1, k 2
k=O
Let (10)
IX
+
1. For
Uk>
S~k)
=
S
for k 2
IX
+
2.
We shall prove the first statement. According to (7), t is exact iff
Proof
.I kax n
k=O
k
[c + +
= (n + 1yxn + 1
Co
(n
1
1)
+ ... +
(n
Ck-l 1)k
+
] I
(11)
for some constants CO, C l' . . . , Ck _ l' By the work in Section 1.7 this is certainly possible provided x =1= 1 and k - 1 2 IX. (Notice that in this case the asymptotic expansion for rn terminates.) • Theorem 3. If t k is exact for some sequence s, then Urn is exact for s when m> k. If Uk is exact, Urn is exact, m 2 k. If t k is exact, t rn is exact, m 2 k. Proof
Trivial.
•
Theorem 4. Let s; = S + (so - s)Pn' U 1 is exact for P« = (a + 1)n/nl; is exact for P» = (a + 1)n/(b + l), or c"; U3 is exact for the previous sequences and Pn = c-n(c + l ),; or (b/a)"(a + On/(b + On' t l is exact for c"; t 2 is exact for Pn = c-n(c + 1)n, en, or (b/a)n(a + 1)n/
Uz
(b
+ n,
Proof Left to the reader. (Assume that the parameters are such that all quantities are defined and denominators of tor U are never zero.) • 10.5.4. Numerics
Diagonal modes of convergence seem to always be preferable with the Levin transforms. Table I shows the effect of t and u on typical sequences. t sums alternating series well but not monotone series; u sums both. Further, both sum the divergent 1 - 1! + 2! - 3! + .... In almost every case, the
197
10.5. The Levin Transformations Table I S~':
Levin Transforms
s; = (GAM). k
2 4 6 8 10
s. = (LN 2).
u
u
0.645 0.577621 0.608 0.577268 0.594 0.577216 0.588 0.577215661 0.585 0.577215664926 y = 0.577215664901533
0.692 0.694 0.693144 0.693161 0.693147186 0.693147203 0.693147180584 0.693147180437 0.6931471805598 0.693147180559951 In 2 = 0.693147180559945
s; = (FAC). (divergent)
s; = (IT
u
0.571 0.595 0.596399 0.596346 0.596347283
u
0.615 0.598 0.596368 0.596341 0.596347823
fo e-'
1.521 1.586 1.611 1.624 1.630
1.639 1.644676 1.644931 1.644934081 1.644934067
I). u
1.369456 1.368860 1.368812 1.368808438 1.368808134
1.369734 1.368882 1.368813 1.368808559 1.368808143
~
1[2/6 = 1.644934066846
=
The root of x 3 + 2x 2 + lOx - 20 is 1.368808107
=
0
performance of u is spectacular, and it is clear why Smith and Ford call it one of the three best all-round practical summation methods (along with the O-algorithm and Levin v-transform). However, no method can do everything; apparently the t and u methods perform less satisfactorily on iteration sequences, convergent or divergent. True they both sum (IT l ), well diagonally, but Sn itself is rapidly convergent also. For (IT 2)n both methods fail. For t and u, respectively,
Sb1 5 ) =
1.2596,
Sb1 5 ) =
1.2606
and the data seem to indicate s~) converges in either case, although it is not clear to what. (sn has two limit points, 0.549 and 1.293.) The GBW algorithm is the only one I know of that will sum this kind of sequence. The example of 7.1(4), (LUB)n, can be used to show neither t nor u is regular for any path P = (n, k), k > O. For t the sequence s~) is (1, 1.3 (exact), 3 (exact), 0.955, 0.876, 1.45, 1.316, ...) and for u it is (1, 1.214, 1.583,0.870, 0.904, 1.917, ...).
198
10. The Brezmski-Havie Protocol
10.6. Special Computational Procedures: The Trench Algorithm Two of the drawbacks to the BH protocol are computing time and storage space. However, if a certain relationship prevails among the Jj(n), a very efficient algorithm due to Trench (1965) for the inversion of finite Hankel matrices can be used to compute S~k). It is surprising that whenJj(n) = L\sn+ j - I ' yielding the iterates of the Schmidt transformation, the result is even more efficient than the s-algorithm for the computation of s~kl, requiring only one-third as many operations. Recall that S~k) results from solving the system
=
Sm
S
+
k
L crlr(m),
n s; m :::;; n
1'= I
+ k.
(1)
Differencing gives k
L c, L\f,.(m),
L\sm =
r=
1
n:::;;m:::;;n+k-l.
(2)
+r-
(3)
Now assumef,. has the property
f,.(n) = g(n
1).
The system may be written k- I
L\sm+n
=
L
C
1'=0
r+,Dr+m, 0:::;; m :::;; k - 1,
Define
i.i
0:::;;
Dj
== L\g(n + j).
i: k.
(4) (5)
The algorithm for the inversion of the H, proceeds as follows. Let
H; I
=
0
[bl'l],
s
i,j :::;; k,
(6)
and assume H;;' I is known, 0:::;; m :::;; k for some fixed n. (The algorithm generates a diagonal of S~k).) Initialize as follows: Y_I = 0,
= 0, uo, - I = 1, ui , U-I,i = U i + I , i - I = 0,
U i• - 2
= 0,
I
U i + I,i
i # 0;
(7)
= 1.
Then compute Ak Ur,k
=
=
k
L Dj+kUj.k-1
j= 0
(Ak-_1'Yk_l -
k
h =
Ak-1Ydur,k-l
L
j= 0
Dj+k+ IUj,k-I'
+ Ur-1,k-l
0:::;; r s; k.
-
Ak-\Ak Ur.k-2'
(8)
10.6. Special Computational Procedures: The Trench Algorithm
199
Next compute k+ 1
=
Ak+ 1
L D j + k + lUjk>
(9)
j=O
and finally O~i~j~k+l O~j
S~) may be computed from (1):
(k) _ Sn -
Sn -
f [ l(n),
f2(n), ... , };.(n)]H - 1
l
k- 1
j
L'1sn L'1Sn + 1
:
Llsn + k -
.
(10)
(11)
1
Brezinski (1976) has shown that for the special case of the s-algorithm, the computation in (11) can be bypassed and the algorithm becomes more compact. An important fact is that cj = jth component of {H;_\(Lls n , ••• , L'1Sn+k_l)T},
and thus if it is known that kind
Sn
(12)
has a complete asymptotic expansion of the
s, '"
S
+
L c, f,.(n),
r= 1
then Eq. (12) gives a lozenge algorithm for the computation of each cj , j ~ 1; i.e., just label the left-hand side of (12) cJ~~.
Chapter 11
The Brezinski-Havie Protocol and Numerical Quadrature
11.1. Introduction; The G-Transform
The sequence transformations discussed in the previous chapters can be put to obvious use to compute approximate values of s = S~ g(x) dx. If any sequence s of approximants to s has been obtained (for instance, by the application of some standard numerical quadrature rule to progressively finer subdivisions of [a, b]), then any acceleration method can be applied to s. We shall not belabor this obvious approach. Brezinski (1978) discusses this approach exhaustively and gives many numerical examples. One problem in such an ad hoc approach is that there is usually no clear way of finding those functions for which the method yields exact answers and thus of characterizing the class of functions for which the method should be expected to provide good answers. However, there is another more intuitive way of proceeding. The underlying philosophy ofthis method, which is applied to infinite integrals (b = 00), was first set forth in a paper by Gray and Atchison (1967) and developed in subsequent papers by Gray, Atchison, and Clark. Their algorithm came to be known as the G-transformation, and is a special case-vertical convergence in the second column of the S~k) array-of the algorithm to be developed in this section. The latter method is one to which the BH protocol can easily be applied. The derivation will be informal; convergence theorems will come later. Suppose we have a way of computing approximate values of G(t)
=
fg(X) dx, 200
t
z
a,
(1)
11.1. Introduction; The G-Transform
20 I
for a sequence of values of t. Let p be a fixed number> 0 and write
c, =
G(t
gj = g(t
1=
+ jp), + jp),
(2)
LX) g(x ) dx
We have
= Goo.
L oo
G(t) = 1 -
Go = G(t),
j 2 0,
g(x
+ r) dx,
t 2
(3)
Q.
Now assume that a quadrature formula with equally spaced nodes is available for the evaluation of the above integral; in fact, for the purposes of the derivation, we assume the integral can be represented exactly for all t by such a formula, so that 1 - Go =
k-I
L c.q..
(4)
ro:=.O
Replacing t by t + mp, 0 ~ m ~ k, yields a system of k + 1 equations in the k unknowns Co, CI, ... , ck- I' For consistency, the augmented determinant of the system must vanish and that determinant can be solved for I. For general integrands, of course, the result will no longer be exact, but we can use it to obtain an approximation W) to 1 that looks like
,
I(k) --
Go go Gk
gk~ 1
?/k
?i2k ~ 1
Thus
nO) =
G(t),
,
-
1(1) -
/
1
1
?Jo
Yk-I
a,
Y2k- I
G(t)g(t + p) - G(t + p)g(t) , .... g(t + p) - g(t)
(5)
(6)
There are several ways the above formula can be used. One could assume, for instance, that G is tabulated at equally spaced points, to + jh, to 2 a, j 2 O. The BH protocol can then be applied by taking f,.(n) = g(t o
+ (n + r
- l)p),
s; = G(to
+ np),
(7)
and so S~k) yields an extrapolation to 1 in terms of the known values G(to +np), n 2 O. Of course, to may be chosen larger than a, the advantage being that 9 may have a singularity near Q and that the necessity of tabulating 9 near a can be avoided. Alternatively, one may take to = a and define, arbitrarily 1)(0) = 0 when 9 is singular at a.
202
11. The Brezlnskt-Havie Protocol and Numerical Quadrature Table I k
sn-.
0 1 2 3 4 5
-0.45939 -0.49006 -0.49634 -0.49829 -0.49909 -0.49948
k
sn-.
6 7 8 9 10
-0.49968 -0.49979 -0.49986 -0.49990 -0.50003
Example 1
+ ev'X)2, - !,
g(x) = -efiI2J~(l G(t)
= (1 + ev'l)-1
I =
50'" g(x) dx
=
-!,
x> 0, (8)
to = a = 0,
p = 1.
Here G is known explicitly, but, surprisingly, not more than ten or so tabular values are required to determine I to almost five places despite the fact that 9 is singular at zero. Thus we may assume, for the example11(0) = 0, that 11 values of G are known and tabulate the 11th ascending diagonal of S~k) (see Table I). Example 2 g(x) = -e-X(x G(t)
+
x> 0,
1)lx 2 ,
= e-tlt - lie,
I =
J'"
(9)
g(x) dx = e-
I
= 0.367879441,
to = a = p = 1.
The sixth ascending diagonal is tabulated in Table II. In this example double precision (16 significant figures) was used, and Sb1 5 ) is accurate to 16 significant figures. This indicates the method has great numerical stability, at least when applied to monotonic integrands. Table II k 0 I 2
5-'
k
S~~k
-0.367466316 -0.367863093 -0.367878981
3 4 5
- 0.367879339 -0.367879477 - 0.367879363
Sl')
203
11.1. Introduction; The G-Transform Table III
Example 3.
k
S\k~ _ k
k
S\k~ -k
0 2 4 6
1.04471 0.99818 1.00015 0,99968
8 10 12
0.99996 0.99967 0,99996
This has an oscillatory integrand, corresponding to
I = Then
f
OO
sin x - x cos x
o
x
G(t)
=
2
dx = 1.
(10)
1 - (sin t}/t.
(11)
Some elements on the 13th ascending diagonal are tabulated in Table III. The error in #5) is 1 X 10- 6 . Obviously, the algorithm was not designed for integrands that decay algebraically or logarithmically. For J~ x-l(ln x)" 2 dx, as another example, Sb1 5 ) = 1.262, while the true value of the integral is l/ln 2 = 1.443. We now look at the exactness problem for this algorithm. Theorem 1. For some complex constant do, d 1 , ••• , db let A. E f:(}c be a sequence of roots with negative real parts of the exponential polynomial H(A) = do
k-l
+ AI
r=O
(12)
dr+ le"rp •
Then if (13)
where Pm(t) is a polynomial of degree less than the multiplicity of Am' infinite sums being allowed subject to convergence conditions, the transformation (5) is exact for each t; i.e., Ilk) == I, t > a, provided the denominator of (5) does not vanish. Proof
Define !£(f) = dof(t)
k-l
+ I
r=O
dr+d'(t
+ rp).
(14)
If g satisfies the equation !£(g) = 0, then, by integration between t and
do[G(t) - 1]
k-l
+ I
r=O
dr+lg(t
+ rp) =
0,
t > a,
00,
(15)
204
II. The Brezinski-Havie Protocol and Numerical Quadrature
°
so the numerator of the determinantal expression of I~k) - I will vanish. Let ,10 be a root of H(A) of multiplicity m. We need only show 2(ti e AOI) = for O:5:j:5:m-l. We can write eAIH(A)
d
k-l
= co eAI + "c _ 1... r+ 1 dt r=O
eA(I+rp )
(16)
O:5:j:5:m-l,
(17)
'
so
which was to be shown.
•
Corollary 1 (k = 1). Let f E L(O, iff g(t) = Me-at, M -# 0, Re a > 0. Corollary 2.
00).
Then 1~IJ is defined and exact
For some complex constants db ... , db d,
+ dk -# 0, let A be a sequence of roots with negative real parts of
°
k- 1
"d 1... r+ 1 e Lrp .
r=O
Then
IlkJ, k
~
+ ... (18)
1, is defined and exact for
=
g(t)
L: Pm(t)e
Am l
(19)
,
where Pm is as in Theorem 1.
Proof Completion of the proof, which requires Heymann's theorem to guarantee the nonvanishing of the denominator of Ilk!, is left to the reader; see Section 6.3. • The following result on accelerativeness is easy to demonstrate.
Theorem 2. Let D(t) denote the denominator of I:k ) and Mr(t) the rth cofactor of the first column of D. Let Mr/D be bounded. Let I exist, g be bounded, and 1 :5: r :5: k.
(20)
Then lim {(Wl - l)/[G(t) - I]} = 0. t~oo
Proof
Left to the reader.
•
(21)
1l.2 The Computation or Fourier Coefficients
205
Let us take as an example the important case k = 1, III _
~
-
G(t
+ p)
- G(t)g(t + p)jg(t) . 1 - g(t + P)jg(t)
(22)
If g(t + p)jg(t) = A + 0(1), 0 < A < 1, the hypotheses of Theorem 2 are satisfied ~ in fact, in this case the conditions are necessary and sufficient for the accelerativeness of Ipl; see Gray and Atchison (1967). The algorithm is most suitable for integrands that behave exponentially. Obviously iff = o(t-a), the conditions of theorem are not satisfied; in fact, for k = 1, one has (23)
An algorithm suitable for cases in which f behaves algebraically can be obtained by making an exponential substitution in (2)-(6). This amounts to taking in the BH protocol f,(n)
= topn+r-lg(topn+r-I), to
~ a ~
1, p > 1.
(24)
However, these equations offer no clear computational advantage over (7), since tabular values of G for very large t are required. An exactness theorem analogous to Theorem 1 is easily established for the new algorithm. Details are left to the reader. Theorem 2 remains unchanged. For the important case k = 1, these results show the algorithm is exact for functions f(t) = Mt- a, M #- 0, Re IX > 1, and accelerative if f(t) = O(t- a), Re IX > 1. The papers by Gray, Atchison, and Clark detail many other properties of the k = 1 algorithm. 11.2. The Computation of Fourier Coefficients Suppose it is required to compute the Fourier coefficients I(m)
=
L
f(x) cos(2nmx) dx,
(1)
and that a sequence s of values of the trapezoidal sums (2)
is known. Further, assume that Romberg integration (Section 3.1) has been applied to Sn to produce a value of 1(0) accurate to as many figures as are required of 1(m).
206
II. The Brezinski-Havie Protocol and Numerical Quadrature
The BH protocol, combined with a method due to Lyness (1970, 1971) can be used to attack this problem. To be accurate, we should speak of a "class" of methods, since Lyness's theory has a great deal of flexibility, which allows one to take advantage of additional data, i.e., a knowledge of the derivatives off Here only the simplest form of his algorithm will be used. (It seems a pity that Lyness's work, uncomplicated and beautifully ingenious, has received almost no attention from the authors of books on numerical analysis.) Supposefhas the Fourier series development f(y)
=
1(0)
+
2JI f
f(x) cos[2nk(x - y)] dx.
Let y assume the values jim and sum from j = be expressed
2
°to m -
(3)
1. The result may
00
I
k=l
l(km) = rm ,
(4)
[For details, see Luke (1969, Vol. II, p. 215).] Now, the Mobius inversion formula (Hardy and Wright, 1959, p. 237) states that, subject to certain convergence conditions, the sum m
~
(5)
1,
may be inverted to yield 00
r; = I
k= 1
ilk G k·m,
m
~
1,
(6)
where ilk is the Mobius function, ilk
=
f~
1
(-1)'
k = 1 if k has a square factor if k is the product of r prime numbers.
(The first ten values of ilk are + 1, -1, -1,0, -1, applied this formula to the sum (4) to obtain l(m) =
1
(7)
+ 1, -1,0,0, + 1). Lyness
00
2 k;/krk.m.
(8)
This is the series from which we wish to compute l(m). We show how the BH protocol can be applied to the partial sums of this series. Let 1 n+ 1 lim) = -2 I ilkrk'm, k=l
n
~
0,
(9)
11.3. The tanh Rule
207
and define R; =: I n(m) - l(m) =
1
2
L 00
k;n+2
(10)
Ilkrk'm'
From the fact that (11) fj(n) =
L 00
k;n+ 2
Ilk
k2 j
'
(12)
However, (13)
so I
fj(n) = (2j) -
n
+
1
k~1
Ilk k2j
'
n ::::: 0, j
>
1,
(14)
and to complete the BH protocol one takes 1 n+ 1 s, = I n(m) = -2 L Ilkrk'm, k;l
rn =
T" =
T" -
1(0),
~n in f(~), n k;O
1(0)
=
f
(15) f(x) dx.
[The numbers (2j) are extensively tabulated; see e.g., Abramowitz and Stegun (1964).J One would expect, based on the representation 10.4(1), that rin ) = Oin" 2k- 2), n -> 00. (This has not been proved, of course.) The original series, Eq. (8), converges only as n- 2 . Iffhas derivatives, i.e., if the values of c., c 2 , ••• , c 2 r + I' are known, these may be used in an obvious way to make the process even more efficient, with T" minus the first several terms in the series (11) taken for T". 11.3. The tanh Rule The basis of the tanh rule is the approximation of a doubly infinite integral by means of a trapezoidal approximating sum. Thus the quadrature process is similar to the methods based on cardinal interpolation. However, there is an important difference, one that changes completely the nature ofthe
208
11. The Brezinski-Hiivie Protocol and Numerical Quadrature
error term: The infinite sum is truncated at ± N(h). The problem is, how should N be chosen to obtain optimal results? Following Schwartz (1969), we make a change of variable in the finite integral J~ 1 g(x) dx. Let ljJ be a reasonably smooth function that is monotone and maps ( -1, 1) into ( - 00, (0).
~ hrt_f'(rh)g(ljJ(rh)).
flg(X)dX = f:oog(ljJ(t))ljJ'(t)dt
(1)
How should ljJ and h == hen) be chosen? Schwartz suggested ljJ(t) = tanh(!t) (hence the name "tanh rule") and h = nj2FJ. For integrands 9 in Hardy class H 2 , Haber (1977) has computed the asymptotic form of the error norm and has shown that for the above choice of ljJ, the choice of h is optimal. [The functions in the Hardy class H 2 are iO 2 functions analytic in N for which I f(re ) 1 dO is bounded as r ---+ 1.] Let 9 E H 2 and define
gJr
( )= h Sn
9
It can be shown that S that
i
s(g) =
r= -n
-
f
1 g(x)
(2)
dx,
g(tanh(nh/2)) 2 cosh 2(nh/2) ,
h=
nj2FJ.
(3)
s; is a bounded linear functional on H 2 • Haber found (4)
Note that this seems to be considerably inferior to the bound obtained for the trapezoidal rule in Section 3.4. However, there the sum is not truncated and the class of functions is smaller. Haber's computations seem to indicate that a good choice for the BH protocol is Jj(n) = e-(Jr/J].lJri/(n
+ l)U- 1 )/ 2 .
(5)
The function g(x) = (l - x 2 y is in H 2 provided Re a> I = rca + l)fi/rca + ~) and
-i.
Then
n 2: 1 (6)
n 2: 1,
and
So
= O.
11.3. The tanh Rule Table IV BH Protocol Applied to 1
7
Ci
k
2
4 6 8
8
12
=
Sk
-t 1_ 13 =
(tanh rule)
2.611931003 2.586166070 2.586239244 2.586715520 2.586937436 2.587032111
2.587109559 s~)
2.266890051 2.563060233 2.586139159 2.587082817 2.587108878 2.587109544
2
=
IX
=
Sk
fO o
-
209
x 2 )' dx
i, 1_ L4
(tanh rule)
2.440806880 2.399070105 2.396475368 2.396260717 2.396257569 2.396267876
=
2.396280467 )'(kl
'0
2.048670072 2.371528094 2.395295728 2.396255106 2.396279761 2.396280440
Table IV displays Sk versus s~), i.e., vertical versus diagonal, convergence for the choice (5) and the cases (X = -t and (X = -t. Clearly, the BH protocol is a powerful tool to use in conjunction with the tanh rule.
Chapter 12
Probabilistic Methods
12.1. Introduction
Historically, the construction of summability methods has been based on the philosophy and techniques of classical analysis. Actually, the problem of accelerating the convergence of a sequence is more at home in a probabilistic setting. A formulation in terms of prediction theory or recursion filtering, for instance, immediately suggests the minimization of the expectation {E(lrnl)} of the transformed error sequence if the original sequence is interpreted as a sequence of random variables.t By assuming certain distribution functions for the {sn} and performing this minimization, one is led naturally to a class of methods for transforming sequences. Of course, the methods will depend on the parameters of the chosen distributions. If these parameters are unknown, any well-known estimation technique can be applied. Each estimation technique provides a different summation method. Although the construction of summation methods has not traditionally been based on probabilistic techniques, the methods themselves have been put to extensive probabilistic use. For example, Chow and Teicher (1971) represent the strong law as a trivial special case of the following Toeplitz summability. Let {X n}:'= 1 be independent identically distributed random variables with finite first moment. Suppose (1) «.> 0, n > 0, t Good sources for the theory of probability and stochastic processes needed in this chapter are Papoulis (1965) and Miller (1974). 210
12.2. Derivation of the Methods
211
and 11 ;::::
0,
(2)
diverges. Define the transformed sequence {1;,} by
1;, =
n
s;; 1 L a.x;
11 ;::::
j=O
O.
(3)
If 1;, - C, -+ 0 almost surely for some centering constants {Cn}, then {X n} is called an-summable with probability 1. Note that the strong law is obtained by using C, = EX,
11;:::: 0,
(4)
the common mean of the underlying distribution, and
11;:::: O.
(5)
The summation methods to be derived here are nonlinear and nonregular. They are simple to use. They are useful for summing classical series and also for summing "statistical" series whose terms are realizations of random sequences. Numerical examples of both kinds of applications are included here. The advantages the methods hold for statistical applications are clear: For series defined by complicated experiments in which obtaining data is difficult and expensive, the use of the proper summation method based on an appropriate probabilistic assumption can result in practical advantages. Finally, we shall show that for one large and important class of sequences, the methods are regular, namely, the sequence space of partial sums of alternating series whose terms in absolute value are monotone decreasing. No other nonregular method has been shown to be regular for this sequence space. 12.2. Derivation of the Methods To motivate our derivation, suppose that the series Lk='O a k is a realization of the following" experiment": Let {xdk'= 1 be a sequence of independent random variables with and where
Ipl < 1 and q <
E(xD 00.
=
q,
k ;:::: 1,
(1)
k ;:::: 1,
(2)
212
12. Probabilistic Methods
Defining ak
=
n k
aO
j= I
(3)
k ? 1,
Xj'
one finds that (4) Since (5) it follows that n ? 0,
(6)
(7)
E(s) = ao/(1 - p), and n
= 0, 1,2, ...
(8)
Now,
-aop
= -1-
-p
Pn(P),
n ? 0.
°
(9)
for n ? 1.
All methods will have the property that E(r n ) =
Definition. The summation method U is called Esadmissible if the characteristic polynomial has the form
(10)
°
where k b k 2 , ... are positive integers, 1 degree j in A., and dil) #- for any j.
~
k;
~
n, dp.. ) is a polynomial of
Clearly, IE(r n)I is minimized if and only if U is E-admissible. Perhaps the simplest example of an E-admissible method is n ? 1,
(11)
which leads to the following very simple choice.
Method I flnk fln.n-l
= 0,
°
= -p/O - p),
~ k ~ n - 2; flnn =
1/(1 - p).
(12)
12.2. Derivation of the Methods
213
For this matrix U, s; for each n is the expected value of s given So, Sl,"" Sn' Does there exist a U that minimizes both IE(r n) I and E(r~)? The answer is yes: It can be found as follows. Let k-1 (13) Wnk = L Ilnj' i s k :s; n, j=O
Then (14)
Now n
E(r;) = E(r~) - 2k~1 wnkE(rnak)
but for k :s; n, E(rnak) = E(-
I
j=n+1
aja k) = -a5
L
= - ao2(q)k -
n
f
j=n+1
)2 ,
(15)
pj-kqk
v:-1 , pl = - ao2(q)k - P 1- P
(16)
OCJ.
P j=n+ 1
E(r;) = E(r;)
(
+ E k~l wnkak
)2 .
(n + 2a1°_pn+1 Ln Wnk(q)k ~ + E L wnkak p k= 1
P
k= 1
(17)
For the last term, ECt1 WnkakY = E(t1 kt WnkWnlakal) 1
n
" 2 k - ao2L. wnkq k=l
_
Let
F= E(r~) - A(k=i
1
n
k- 1
k=l
1=1
+ 2ao2 L. " Wnk "L. WnlP k-l q.1 Wnkpk- 1 +
~). 1- P
(18)
(19)
Then (20)
214
o
12. Probabilistic Methods
Setting
of/ownj =
0 for 1 S j S n gives
=
W
=~~
~~ 1- P
-2-' -
2aoq)
or nk
r:':'
)..pj-l
w nj
-
j-l LW nk -
q
k= 1
pn+1-k (p2 ) --1 , I-p q
2sksn.
(21)
(22)
Since I E(r n ) 1 is a minimum, Pn(p)
= 0 = Wnl + (w n2 - w n1)p + ...
+ (w nn -
and this implies
~
Wn. n_l)pn-l
L. WnkP
k-l
+ (l
- wnn)pn,
_pn
(24)
= -1--'
k= 1
P
-
(23)
We can now determine Wn1 = IlnO and then, from (22), all the Ilnk: Wn1 = IlnO
- p" - ~ k- 1 = -1-L. WnkP . - P
(25)
k=2
This method minimizes both IE(r n) I and E(r;): 2 IlnO = + (n - 1)(pq -
Method II.
1)l
1-!~ [1
Ilnl
r'
= (1 _ p)
q-
(p2
Ilnk
= ( pq2 _ 1)pn-\
11
=
nn
) 1 [1
_1 (1 _p3). I-p
+ (n
- l)p]
+
pn 1 _ p;
(26) 2 S k S n - 1;
q
We shall be concerned with yet a third E-admissible method, namely, that arising from the choice Pn(A) = (A - p)"/(l - p)n,
n
~
0,
(27)
i.e., the choice that forces Pn(A) to vanish as strongly as possible at x = p. Method III.
This yields the weights Ilnk
=
(~)( - vr: /(1
- p)".
These are related to the Euler means of Section 2.3.2. (There p < 0.)
(28)
12.2. Derivation of the Methods
Let p E Cf/. Then Methods I-III define regular methods iff s 0, respectively.
Theorem.
p
215
i= I, Ipl < 1, and p Proof
Application of Theorem 2.2(1) to the weights of Methods I and
II is trivial. For Method III, note that for fixed k Illnk I '" M knkip 1 _ p In , so that lim n _
00
Ilnk =
°iff Ip/(l -
n i p In
Mk
independent of n,
p) I < 1, that is, Re p < n
k~olllnki = 11 - plnk~O
l
(29)
Further,
(n)k Ipl _k = IIIIp I-+pi1 In.
(30)
Now, the triangle with vertices {O, 1, p} in the complex plane has legs of length 1, Ipl, and 11 - pl. Thus [Ipl + IJ/ll - pi> 1 unless p is real and negative, in which case the ratio is 1 and ~ 0 Illnk I = 1. •
Lk
To obtain the final summation formulas, note that in almost all applications the variables {xdf~ 1 are identically distributed with the first moment p unknown. One could then estimate p by the method of moments
n
~
1,
(31)
or by the maximum likelihood estimate. The first estimate provides the more useful results. Thus, for instance, for Methods I and III one has (32) and (33) respectively, and a similar formula is obtained for Method II with p in (26) replaced by P« and q by qn = n- 1 Lk~ 1 aUa~-I' The form of the sequence of expected errors (8) is a fortunate consequence of the derivation since many of the sequences encountered in practice are at least approximately a constant plus an exponentially decreasing term. The above methods (like many nonlinear methods) cannot be applied to certain sequences. For instance, if s, = s, + 1 for some value of n, then an+ 2/ an + 1 is undefined and so is Pn+ l' The problem in definition is not resolved by considering only subsequences of {sn} containing no adjacent duplicate members since the possibility that P« = 1 is still not obviated.
216
12. Probabilistic Methods
In fact, it is possible to manufacture examples of convergent infinite series where P« = 1for an infinite number of n, for instance, by folding together the two absolutely convergent series
1,11 < 1,
(34)
to obtain ,1+ d
+
,12
+d
2
+ ....
(35)
The sequence an/an-I' n = 1,2,3, ... , is then e, so that
P2n
Aje, e,
2n ak 1 ( = -21 L -= -2 n e + -A) =
n k=
1
ak- I n c
Aje, ... ,
(e + -A) , e
(36)
n
= 1,2,3, ... , (37)
and for the choice (38)
e=I+~, P2n = 1, n = 1, 2, 3, .... 12.3. Properties of the Methods
For Method I there is a simple necessary and sufficient condition for convergence. We have
Theorem 1.
Let s E
f(ic.
S=s
Proof
Trivial.
Then for Method I iff
lim an/( 1 - Pn) =
o.
(2)
•
Method I is not regular because of its lack of definition for certain convergent sequences, but ifboth {s.} and {sn} converge then limn s, = limn sn so long as {Pn} is bounded away from 1. Corollary. For any convergent alternating series L:'=o an' Method I preserves convergence. Further if Ian I is monotone decreasing, then s; lies between Sn- I and Sn for all n.
Proof Since -1 < an/an- 1 < 0, -1 < P« < 0, and < 1, Eq. (1) immediately yields the corollary. •
1<
(l - Pn)-l
12.3. Properties of the Methods
217
Further, Methods I and II preserve convergence when {sn} is the sequence of partial sums of any series, Lk'=o ai, for which Raabe's test (Knopp, 1947, p. 285) is applicable. Let an > 0 for all n, and let s
Theorem 2.
n(an+da n -
n s.
with
E C(jc
(3)
-Yo
Then Method I is regular for s if - y < - 1 and Method II is regular for s if -y < -2. Proof
We may rewrite (3)
+
n(an+ dan - 1
lin)
:os::. -
f3 = y - 1,
f3 < 0,
(4)
so that (5) or
(n - l)a n
nan+ 12. f3a n .> O.
-
(6)
Thus ultimately {en - l)a n } is monotone decreasing. Therefore lim nan + 1 = lim nan
n--+ co
n- 00
exists. Now from (7)
we can conclude
ak!ak -
1
:OS::.l - YI(k - 1),
1 n ak 1[NL-+ ak Pn=-I-:os::In nk=l ak-l
n
k=l
ak-l
k=N+l
(8)
Y)J
(1 - - , k- 1
(9)
or
Pn
:os::
[n - yin n
+ M(n)]ln,
(10)
where {M(n)} is a bounded sequence. Here we have used the fact that n
1
I k_ k=N+l
+ 0(1).
(11)
1 - P« 2 [y In n - M(n)]ln,
(12)
1 = In n
Thus
218
12. Probabilistic Methods
and for n large enough, the right-hand side is positive. Thus
an
--<. 1 - Pn -
nan , yin n - M(n)
(13)
or lim aJ(1 - Pn) = O.
n->
(14)
00
This gives the result for Method I. Now
IPnln~.
[1 + M(n) ~ yin nT
=. exp [ n In ( 1 + =. ex p[n(M(n)
M(n) - yin n
~
yin n
n)J '
+ e~n»)
l
(15)
by a Taylor's series argument, where {sen)} is a null sequence. Thus
nlPnln {3n 2 - i' --<. , 1 - Pn - yin n - M(n) and since 2 - }' < 0, lim
IPnl n/(1
- Pn) =
and the result is established for Method II.
°
(16)
(17)
•
Both Methods I and II preserve convergence for series for which the ratio test shows convergence provided one further stipulation is added for Method II.
Theorem 3.
Let
SE
fll c with lim an/an-l < 1.
(18)
Then Method I is regular for s. If in addition (19)
the same holds for Method II.
219
12.3. Properties of the Methods
Proof Since Pn is the Cesaro means of an/an-I' lim an/an-I ~ lim Thus, for every
I:
o, ~ rrm: », ~ rrm: an/an-I'
(20)
> 0, lim an/an-I -
E
<. Pn <. Iilli an/un-I n-r ca
+ E.
(21)
By virtue of (18), Pn <. r < 1,
or or
(22)
(1 - Pn) >. 1 - r > 0,
(23)
(1 - Pn)-I <.(1 - r)-I.
(24)
Theorem 1 may be invoked to show regularity for Method I. Using (19) and (22) now gives (25)
-1 < rl <. Pn <. rz < 1,
so
(26)
Thus (27) Many sequences encountered in practice can be expressed (at least approximately) as a constant plus a linear combination of exponential terms. More precisely, one can define a sequence space Ye as follows. Let!/' be the space of complex sequences (28) Define Ye =
{SISn = S+ J/~u~), Ar E N, 1 > IAII > IAII, 2 ~ j
~ k,
u(r)
E
Y} (29)
Note Ye is a generalization of CC Ek(N) [see Eq. 2.2(12)]. Theorem 4.
Proof
Let
Method II is regular for Ye; Method I accelerates Ye. uUl E
Y. By an application of Theorem 1.4(2), (30)
220
12. Probabilistic Methods
Now, an + 1
k
L A~+lu~)[1 + o(1)J
=
r=1
-
k
k
r=1
r=1
L A~u~) = L A~u;;l[(Ar -
Thus .
hm
n-e co
rUn An (r) 1 .fn<TI -_ .hm
I
lUn
n-+oo
Uo I(I) Iexp [( n In I-, I + C (r)
A
Uo
(r) n
r
1\..1
(I)
Cn
)J
_ -
1)
0,
+ 0(1)].
(31)
2 :s; r :s; k, (32)
by virtue of the fact that In IArlAl I < O. Thus
lim an + dA~u~1)
= Al - 1.
(33)
Also, (34) Dividing these two limits gives lim an+ dan
= AI'
(35)
Since p is the Cesaro mean sequence of the sequence {anla n- d, it also converges to AI' and so (1) gives
sn =
s
+
±
An - l u (rn -)
r= 2 r
I
[(1 + ~) + 1 _ Al
O(1)J
+
AnI - 1U (nI- ) I 0(1) .
(36)
Thus (37) n~
00
From (32), (38) Dividing these limits shows that
= 0,
(39)
~ = ~ [1 + 0(1)J,
(40)
lim [(Sn - s)/(sn - s)J and this is the desired result. For the result for Method II, note that
I - P«
1 - Al
and this, used in (25), implies the method is regular for s.
•
12.3. Properties of the Methods
Note that all these methods map the partial sums of So
=
s; = 11(1 - x),
1,
I:=
n 2: 1.
0
221
x" into (41)
This property is shared by other nonlinear transformation, for example, the Shanks ef transformation (Shanks, 1955). In fact, a transformation related to Method I was mentioned in passing in Shanks (1955, pp. 25-26) under the name "geometric extrapolation." This transformation is defined by
sn =
Sn -
lim.(anlan-l)sn-l
n 2: 1.
-"---"-'-"-'-~"--"~
I - limn(anla n- 1 )
(42)
Method III is, for certain classes of sequences, the most effective method of all. We now make this more precise. Lemma.
Let
(- 00,0].
WE ~
be bounded and belong to a bounded subset S of
(43) it follows that
lim s,
Proof
Let sup
ZES
= s.
I-zI l-z
= d
(44)
< 1.
(45)
Note also that
\1-zl- 1 :::;; 1,
Z
E
S.
(46)
Since (47) one can write
Choose N such that WnE S, n > N, and ISn - s] rk is bounded, Irk \ < C
= Irnl < s, n> N. Since
222
12. Probabilistic Methods
or (50) Now (51) for fixed k, so lim
n-e co
(n)d. k
(52)
= 0,
taking lim sup term by term gives
rrm If.1 < s,
(53)
or, since c: was arbitrary, or
Theorem 5.
lim f.
= O. •
(54)
Method III is regular for all alternating series
for which sup bk/bk- 1 = M <
00.
(55)
k~l
Proof
In the lemma, let ak = (- 1)kbk, aJak-l = -bk/b k- 1 <0,
(56)
223
12.4. Numerics
so that P« < O. Also, 1
IPnl = -
I n
b,
nk=lb k- 1
s
(57)
M.
•
Thus P« belongs to a bounded set and the proof of the theorem is complete.
12.4. Numerics
As the reader will see, the three methods derived in the previous section have very unusual properties. We first discuss their application to probabilistic sequences. It must be kept in mind that, although the methods are used on individual random sequences, each method was designed for a space of random sequences. The space (and consequently the method) is identified by its parameters P (and perhaps q). The concept of applying one of the methods to an individual sequence is ambiguous. It really makes more sense to talk about the average value of the transformed sequence over a great number of trials s~, in other words,
'" N 1
-
N
.
-, L. sn'
(1)
i=1
This should, in some sense, be close to the expected value of Sn' Now suppose that {x.}~ 1 is a sequence of independent beta-distributed random variables. Recall the random variable x is beta distributed when its probability distribution function is the incomplete beta integral F(rx, [3, x*)
=
Pr(x
s
x")
= B(rx,1 [3)
IX' t 0
a
-
1
(2)
1(1 - t)fJ- dt
with corresponding probability density function 1 a f(rx, [3, x) = B(rx, [3) x - l (1 - x)fJ- 1 •
(3)
Note that E(x) =
and E(x 2 ) =
II o
I
I
o
t!(rx, [3, t) dt =
+ 1, [3) a ([3) = --[3 B o; o: +
B(rx
t 2f(rx, [3, t) dt = B(rx + 2, [3) = Bt«, [3) (rx
rx(rx + 1) + [3)(rx + [3 +
(4)
1)
.
(5)
224
12. Probabilistic Methods
Since each Xk is beta distributed with parameters (a, {3), (4) and (5) hold for each X k. When the Xk are independent
n21.
(6)
Taking {3 = 1 for the numerics, define sn(a) = 1 + tn(a) = 1 From (6),
so
Xl
+
+
Xl
+ ... + XIXZ ••. X n' + ... + (-ltx l,x z,···,x n·
XIX Z
XIX Z -
1 - (a/a + It E(sn) = I - (a/a + 1) ,
E(s)
while E(tn) =
=
a
+
E(t) = (a
+
1)/(2a
(8) (9)
1,
1 + ( -It(a/a + It+ I 1 + (a/a + 1) ,
so
(7)
+ 1).
(10) (11)
Computer generated values of Xk will be selected as follows. Pick Y I' Y2, ... , Yn from a uniform density on [0, IJ; then for k = 1,2, ... , n the value Xk is defined by (2), i.e., (12) It is possible that the sample values x, are atypical of beta-distributed random variables since the distribution function controls only the likelihood (probability) that particular values are obtained and not the possibility that particular values occur. For instance, when a ~ 1 the distribution function for X k is skewed toward X = 1 (see Fig. 1), so that one "expects"
Fig. I
12.4. Numerics
225
Table I Effect of Methods on Two Typical Probabilistic Sequences"
s, n
Sa
3 6 9 12 15 18
3.168 4.616 5.495 6.071 6.461 6.754
6.872 6.415 7.294 6.971 7.324 7.437
"s(a) and t(a), a
= 1'7 = 0.4737.
I
in
II
II
n
In
6.889 6.523 7.347 7.022 7.355 7.457
29.95 8333
3 6 9 12 15 18
0.019 0.881 0.160 0.695 0.228 0.596
= 8; s = 7.2766, t = 0.3360.
0.466 0.500 0.504 0.438 0.455 0.420
Expected value s
= 9;
II
III
0.475 0.487 0.515 0.473 0.468 0.406
0.479 0.471 0.483 0.492 0.493 0.484
expected value
most simulated values for X k to lie near 1 (and hence for the sequence Sn to converge more slowly than for smaller values of (X). However, it is still possible to generate values for Xk anywhere in [0, 1]. Even when (X is small, say, < (X < 1, so that any individual sequences sand t must converge rapidly, the inevitability of producing Xk arbitrarily close to 1 prevents one from concluding that lim an/an-t < 1 and thus using Theorem 12.3(3) to demonstrate regularity of the methods for such sequences; this occurs in spite of the fact that, for all practical purposes, the sequences converge exponentially. In fact, since t n is composed of alternating monotone decreasing terms the difference It - t n I may be bounded and t computed confidently to any desired accuracy. Table I elaborates on this strange phenomenon. For the given individual sequence t n the methods seem to be summing t to its expected, rather than its actual, value. Of course, this is precisely what one would expect, and indeed demand, of a method to be applied to a sequence of experiments arising from a fixed probability distribution. But this bizarre
°
Table II Effect of Methods on Various Analytic Sequences
(PI 2)n n
5 10 15 20 25
1.5213 1.5735 1.5945 1.6058 1.6130
(LN 2)N
II
III
1.5116 1.5671 1.5780 1.5951 1.6052
1.5323 2.9197 -1747.9
0.68586 0.69619 0.69143 0.69427 0.69235
(FAC)n
II
III
III
0.66994 0.70758 0.68257 0.70158 0.68608
0.69327 0.693147 0.693147 0.693147 0.693147
0.608398 0.61003 0.60863 0.60798
226
12. Probabilistic Methods
kind ofnonregularity is clearly a hazard if the method is to be applied to any individual series, in particular, to an analytic sequence. The effect these methods have on analytic sequences is capricious; some examples are given in Table II. Methods I and II produced only so-so results on every analytic sequence on which they were tested. Method III is a disaster on monotone sequences but performs very well on certain sequences alternating about their limits. It even seems to sum to the usually ascribed value the sequence (FAC)" = 1 - I! + 2! + - ... + (- 1)"n!.
Chapter 13
Multiple Sequences
13.1. Rectangular Transformations
Given a double sequence {snd, define the transformed double array
{snd componentwise by
n,k 2 0.
(1)
The transformation is completely characterized by the four-dimensional array of weights U = [p7f], s i :s;; n, o « j :s;; k. (2) Obviously, convergence in the {snd array can be path dependent. Let Snk be located at the point (k, -n) of JO x - J . We shall be concerned here with only two modes of convergence, horizontal, limk _ oo Snk, and vertical limn _ oo Snk' (This designation differs slightly (rom the convention previously used for array transformations s -> S(k l, but here it is more useful to think of {Snk} as a rectangular, rather than a triangular, array.) We shall assume that
°
lim Snk =
k-
fln,
n 2 0,
00
k 20. Definition. regular if
(3)
The transformation defined by (1) is called horizontally lim
k-oo
Snk
=
fln'
227
n 20,
(4)
228
13. Multiple Sequences
and vertically regular if k
~
(5)
0.
The material in the remainder of this section is due to Higgins (1976). Theorem 1. The transformation defined by (1) is horizontally regular iff (i) given n
0,
~
n
k
L L IlliY I ~ R n
;=0 j=O
for some positive number R; independent of k; (ii) given n ~ 0, lim
°
k
L 1l~1 =
bnr ;
k-oo j=O
(iii) given n
0, j
~
~
0,
n,
~ i ~
lim lliY = 0.
k-oo
Proof <=: By virtue of (3), one can write sij = fii
Thus
Snk - fin
=
n
+ £ij' where
lim j _
oo
£ij = 0.
k
L L Ili'f(fi; + £i) -
;=0 j=O
fin
(6)
For n fixed, condition (ii) guarantees that the first two terms on the righthand side of (6) can be made arbitrarily small for large k. Now separate the remaining term as n
k
n
n
J
L L Ilijk£ij = L L lliY£ij + L
;=0 j=O
i=O j=O
° ° °
k
L lliYeij'
i=O j=J+ I
(7)
°
Since n is fixed and lim j_ 00 eij = for ~ i ~ n, given s > it is possible to pick J such that IeijI < e/2R n for ~ i ~ n andj > J. Thus for k sufficiently large, the second term on the right-hand side of (7) has modulus less than e/2 when condition (i) applies. Now with nand J fixed, define M
=
max leijl,
O::s;isn
o ,.:,j,.:,J
(8)
13.1. Rectangular Transformations
229
and pick K so large that k > K implies
°
Ifl71 I < c/2M nJ
°
(9)
for ~ i ~ nand ~ j ~ J. Condition (iii) has obviously been used to guarantee (9). Thus for k sufficiently large, both terms on the right-hand side of (7) can be made arbitrarily small so that the three conditions of the theorem imply limk~GO (Snk - f3n) = for each n. =>: First, let r be fixed and apply the transformation to the array
°
I,
Snk
obtaining
a, =
-
Snk
{
°
n = r, k 2 otherwise
= { 0,
(10)
r> n
k
fl,)'
"nk
L...
j~O
(11)
n 2 r.
By the horizontal regularity,
l5 n, = lim Snk = lim
k
I
(12)
n 2 r.
fl~J,
k-r sx» j==O
k-oo
Now let r vary to demonstrate the necessity of condition (ii). Second, for the necessity of condition (iii), fix i andj and define the double arrays Dij = (dnk ) , where dnk = l5 inl5 j k • Applying the transformation, we obtain n 2 i, k 2j otherwise.
Thus for i
~
n, · 1im
k-co
flijnk
- = liim = liim Snk k-rJ:)
k-oo
Snk
=
(13)
°
(14)
by horizontal regularity. Varying i andj, we obtain the necessity of condition (iii). Third, the necessity of condition (i) will be demonstrated by contradiction. Suppose there is an integer n for which k
n
lim k~GO
I I
i~O j~O
Ifl71 I = + 00.
Then there must be at least one integer i such that k
lim
I
1
k~GOj~O
fl71 1 = + 00.
°
~ i ~
(15)
nand (16)
230
13. Multiple Sequences
If i = n, define the Toeplitz transformation with matrix (c k ) by k 2 0,
O:::;;j:::;; k.
(17)
This Toeplitz matrix represents a nonregular method since L~ 1 Ickj I is unbounded. Thus, there is a sequence {x) with lim, Xj = x and L'=o CkjXj does not converge to x as k goes to infinity. Consider now the double array that has zero entries except for row n, wherein lies the sequence {xj}, and apply the transformation of the theorem to obtain as
k
~
co.
(18)
Therefore the method is not horizontally regular if i = n. If i < n, consider the Toeplitz transformation whose matrix (Ckj) is given by nk k 20, O:::;;j:::;; k. (19) Ckj = J1ij' By Hardy (1956, p. 43), k
L Ickjl = + co k-oo j=O fiiii
implies the existence of a bounded sequence {y i} with the property that {L'=o CkjYj}k"= 1 does not converge as k ~ co. Now apply the transformation of the theorem to the double array (Snk), which has all zero entries except in row i, wherein lies the sequence {yJ. Clearly, (20) but limk_oo Snk = 0; thus if i < n, the method is not horizontally regular. The necessity of condition (i) is now established by contradiction. • The proof actually substantiates some stronger results regarding the transformation (1). For example, note that with n fixed, the three conditions of the theorem are necessary and sufficient for lim k_ oo Snk = Pn. Also, conditions (i) and (iii) are sufficient conditions that the transformation (1) map a double array with all row limits zero to a double array with all row limits zero. The three corresponding necessary and sufficient conditions for vertical regularity can be obtained by analogy. The existence of higher-dimensional analogs of these methods and of this theorem also is clear. An extension of these ideas that has practical application but will not be pursued in this book is the following.
231
13.1. Rectangular Transformations
Suppose f: JO -+ JO and g: JO -+ JO with either limn fen) = limn g(n) = 00 (or both). Call the double array (f, g)-convergent to s if lim
=
S!(n). g(n)
00
or (21)
S.
The natural question is, What are necessary and sufficient conditions on the weights Ili} in (1) to ensure that if (Snk) is (j, g)-convergent to S then (snd is (j, g)-convergent to s? If the double array (Snk) has lim Snk k~oo
= /3,
n 2:: 0,
(22)
then condition (ii) of Theorem 1 can be somewhat relaxed and still maintain horizontal regularity of the transformation (1).
Theorem 2. Suppose that the double array (Snk) enjoys property (22) and that conditions (i) and (iii) of Theorem 1 are satisfied. If n
lim
k-v co
then
n
lim
k-r o:
Proof
L ;;0
k
'Illi}
=
1,
n 2:: 0,
(23)
lliJSij
= /3,
n > 0.
(24)
j;O
k
I I ;;0
j;O
Obvious in view of the proof of Theorem 1.
•
This theorem aids in the design of transformations of the double arrays. If we assume that the better approximations to the limit /3 appear for the larger indices nand k, the weights Ili} should put more mass on the larger indices i and j than on the smaller indices. Therefore, let fbe a function from JO x JO to P/I+ satisfying f(Il,j) > f(v,j)
for
11 > v
and all j,
(25)
fU,Il) > f(i, v)
for
11 > v
and all
(26)
i,
and 'I);O f(O,j) diverges. We choose the weights Ili} by Ili}
= f(i,j) l,to
.t
o
f(ll, v),
(27)
which leads to 'Ii;o 'I); 0 Ili} = 1 for all nand k, so that (23) is satisfied. Condition (iii) of Theorem 1 is satisfied in view of the divergence of the sum 'I); 0 f(1,j)· Condition (i) is satisfied because of the positivity off Thus, the weights (27) define a transformation for which limk~ 00 Snk = P for all n implies lirn.., 00 snk = /3 for all n. These weights will be computationally useful only when the functionfreflects the input double array.
232
13. Multiple Sequences
13.2. Crystal Lattice Sums
An important class of multidimensional sums arises in the theory of crystal lattices, specifically in the computation of the lattice energy per atom of a given crystalline material. Let f: JP -+~, JP = J x ... x J, M P = (ml"'" mp), Ap=(al"'" ap), Op=(O, 0, ... ,0), IIMpl1 =(mi + ... + m;)1/2. [Where there is no chance of misunderstanding, we omit the subscript p.) The sums of interest are generally of the form S=
M~A
f(M)
11M - A11 2 s '
S E
CfJ.
(1)
f is usually quite simple, typical examples being f
= 1,
(2)
although in so-called phase modulated sums (Glasser, 1974) more complicated functions occur. Sometimes the mj range over only even or odd numbers, but it is not useful to develop a special notation to deal with such cases. It is not at all obvious when (1) converges. The following theorem is often applicable. Theorem. Let M - A#-O and let f be bounded. Then (1) converges and represents an analytic function of s for Re s > p/2, convergence being obtained regardless of the order in p-space in which the terms are added up.
Proof The most elegant demonstration uses the theory of theta functions. This proof is given in Section 13.2.2. •
For a discussion of the physical context in which such sums arise, see the classic treatise by Born and Huang (1954). We shall take an approach with these sums that is fundamentally different from the procedures used previously in this book to accelerate the convergence of series or sequences. The techniques given here will not be general, but will very much depend on the specific character off This is, of course, very much in contrast to the previous work-for instance, the fact that the remainder sequence possessed an asymptotic series of Poincare type-where only the general form of the sequence or series was of interest. The present kind of endeavor might be called the analytic approach to sequence transformations. The arguments used will depend on known properties of mathematical functions, such as theta functions, and on the application of a powerful formula from classical analysis, the Poisson summation formula.
13.2. Crystal Lattice Sums
233
13.2.1. Exact Methods Definition.
Let f be locally L(O, 00) and let the integral .A(f; s) = {"" x·- 1f(x) dx
(1)
converge for Re s = to, Re s = t 1, to < t t- .A is called the Mellin transform
off
Clearly the integral converges for IX
< Re s < {3
(2)
where IX = inf to and {3 = sup t l' (2) is called the strip of absolute convergence of (1). The Mellin inversion theorem states that iffis of bounded variation in a neighborhood of x E (0, 00), then, for any IX < C < {3, f(x+)
+
f(x-) _ _ 1 l' fC+iR «cj, ) . 1m JI't, S X 2m R~"" c-iR
'---'-----'--=--'-----'- -
2
-r
s
d
S,
(3)
Usually.A may be continued analytically into a larger region q; of the complex s plane, for instance, f(s)
-. = a
f"" x
.-1
0
e
-ax
d
X,
Re a> 0.
(4)
°
Here IX = 0, {3 = 00, q; = Cfi - {a, -1, -2, ... }. The theta functions for x > are defined as follows.
(5)
For some of the many beautiful properties of these now almost forgotten functions, consult Whittaker and Watson (1962), Hancock (1909, Vol. I), or Bellman's more recent book (1961), which is compulsively readable. A good collection of formulas is in Abramowitz and Stegun (1964). The following notation is standard:
B/O, q) = BlOlr),
(6)
234
13. Multiple Sequences
Thus 8i(0Iix/n) corresponds to taking q = e- x in 8i(0, q). Formulas such as the following can be found in Hancock (1909, Chapter XVIII):
(7)
Similar formulas exist for OJ, etc.; see Hancock or Jacobi (1829), who gives a list of 47 such relationships. It can be shown that 8/0Iix/n) has an algebraic singularity at x = 0; hence Mellin transforms of Oz, 8 3 - 1, 84 - 1, etc., have a half-plane of convergence. The Mellin transforms of theta functions generally involve meromorphic functions such as Riemann's zeta function, defined for Re s > 1 by (s)
=
1
L s' n 00
(8)
n: 1
We shall need the formulas (1 - 2 1 - ' )( s)
=
00
(_1)"-1
n: 1
n
L
"
Re s > 1; (9)
Re s > 1. Another useful function is Re s > 1,
(10)
which satisfies the relationship L(1 - s)
= (2/n)'r(s) sin(ns/2)L(s).
(11)
Obviously, L(s) can be expressed in terms of the generalized zeta function 00
(s, a)
1
= n:O L (n + a)S'
-a ¢ JO,
Re s > 1.
(12)
13.2. Crystal Lattice Sums
235
The Mellin transforms of powers of the theta functions can be found from such formulas as (7). For instance,
A[8~(01~)J = 4AL~o(-lte-(n+1/2)(2k+l)XJ (-It
00
= 4r(s)
n.~o (n + t)S(2k + I)' 2s
00
= 4r(s)L(s)n~o (2n + I)S = 4(2
S -
1)r(sK(s)L(s). (13)
Mellin transforms of products of theta functions can be found by using the
Landen transformations,
8iO, q)83(0, q)
= ¥1~(0, ql/2),
8iO, q)8iO, q) = te-"i/48~(0, i q I / 2),
(14)
83(0, q)8iO, q) = 8i(0, q2),
and the formula
A{f(ax); s} = a-sA{f(x); s}.
(15)
Table I gives some of the Mellin transforms that can be found this way. Table I Mellin Transforms Involving OJ
=
Ii/Ol ixln)
f(x)
O2
2(2 2 '
04
2(2 1 - 2 ' - 1)[(s)(2s) 4(2' - 1)r(s)(s)L(s)
03 O~ O~
O~
(0 3 (0 4
- I
l)r(s)(2s)
1)2 1)2
_
_
0 304 -
-
I 1)(04
0 20 304 0'2
OJ - I
01-
4r(s)(s)L(s) 1)[(s)(s)L(s) 4r(s)[L(s)(s) - «2s)] 4(1 - 2 1 - 2')[(S)[(2s) - L(s)(s)] 2>+ [(2' - 1)[(s)(s)L(s) 22-'(2 1 - ' - l)f(s)(s)L(s) -2 2 - ' [(s )[r ' ( 2s ) + (I - 2 1-')(s)L(s)] -2'+ [r(s)L(2s - I ) 16(1 - 2'-')(1 - 2-')[(s)(s)(s - I)
4(2 1 - ,
- I
020 3
(0 3
-
2[(s)(2s)
I
O~O~ O~O~ - I 8~0~
-
I)
8(1 - 2 2 - 2')f(s)(s)(s - I) - 8(1 - 21-»(1 - 2 2 -')f(S)(5)(S - I) 2>+ 2f(s)L(s)L(s - I) _2 3-'(1 - 2 2 - ' )(1 - 2[-')r(s)(s)(s - I) 2 2 +' (1 - 2 1 - 5 )(1 - 2-')f(s)(s)(s - I)
236
13. Multiple Sequences
To see how these formulas can be used to obtain closed-form expressions for lattice sums, consider
~, = L,
1 r(s)
--
-00
= - 1
r(s)
foo x
e
s-1 -(m 2+m 2+m 2+m 2)x
0
I
2
foo xS-1[e~(Olix/n) 0
3
4
dX
1] dx
= 8(1 - 22 - 2 S)((S)((s - 1).
(16)
[Later it is shown that this sum converges for Re s > p/2 = 2. Since ((s - 1) has a pole at s = 2, the result is sharp.] As another example, consider
= L(s)((s)
(17)
- ((2s).
A short table (Table II) lists two-dimensional sums determined by Glasser. Table II cc
S =
L' J(m,n)
J
S
(m 1 + n 2 ) - ' (_l)m+n(m 1 + n 2 ) - ' (_I)n+ '(m 2 + n 2 ) - ' [(2m + 1)2 + (2n + 1)2r'. m, n Z 0
4«s)L(s) -4(1 - 2'-2')(s)L(s) 22-'(1 - 2'-')(s)L(s) 2-'(1 - r')(s)L(s) 2(1 - 2-' + 2'-1')(s)L(s)
(m 2
+ 4n 1 ) - '
13.2. Crystal Lattice Sums
237
Certain other related sums have been obtained, i.e.,
L' (m + mn + n ) - S = 6(s)g(s), 00
2
2
g(s)
=
-00
I
00
n=O
[(3n
+ 1)-S - (3n + 2)-S] (18)
(Fletcher et al., 1962, p. 95), and (19) whose derivation is rather complicated (Glasser, 1973b). Obviously, the following case can be expressed by a single sum:
L (ml
mj?l
+
m2
+ ... +
mp )
-r
s
=
L (k + Pk 00
k=O
1) (k 1 y' +P
(20)
The difficulty in computing odd-dimensional sums by the use of theta functions is that most of the known theta function identities involve an even number of theta functions. Glasser (1937b) uses a number-theoretic approach to obtain additional sums, and the theory of basic hypergeometric series (Glasser, 1975) can be used to deduce the five-dimensional sum
L
ml?:O;m2,"';m5~
(m 1m2 + m1m3 + m3m4 + m4ml + m2mS)-S
1
= (S)(S - 2) - (2(S - 1). (21)
(The region of convergence of this sum cannot be deduced from the theorem of Section 13.2.) 13.2.2. Approximate Methods: The Poisson Summation Formula
Many approximation techniques have been developed to deal with lattice sums, beginning, perhaps, with Born's and Huang's approach, which uses values of the incomplete gamma function. That approach is not very adaptable to general values of s. Other approaches (van der Hoff and Benson, 1953; Benson and Schreiber, 1955; Hautot, 1974) use methods that convert the sum to a multidimensional sum involving the modified Bessel functions K v • This might, at first glance, seem to be compounding the problems. However, the transformed sums converge with extraordinary rapidity, and often the contributions at just a few lattice points serve to give six- or eightplace accuracy. Several approaches are possible, including one (Hautot, 1974)using Schlornilch series. My own preference is to begin with the following striking result, which can be found in any book on Fourier methods [e.g., Butzer and Nessel (1971, p. 202)].
238
13. Multiple Sequences
Let f
Theorem.
E
L( -
F(x) =
00,
(0),
Loooo e-iX~f(t)
dt,
X E
(1)
!Jll.
Then, iff is of bounded variation, 2n
L 00
k=-oo
f(x
+ 2kn) =
lim
n
X E
n-cok=-n
where, at points of discontinuity, f(a)
Proof
L eikxF(k),
= -t[f(a+) +
See Butzer and Nessel (1971).
211,
(2)
f(a-)].
•
There follows a list of formulas that will subsequently be of use. For the computation of the integrals involved, consult Erdelyi et al. (1954, Vol. I). f(t) = e- at2 cos bt, a E g~+, b e .OJ;
f
e- a(x+2kn)2 cos[b(x
+ 2kn)J
= _1_
- 00
f
2J"1W -
00
eikxe-(k2+b2)/4a COSh(bk). (S-l) 2a
f(t) = Itl±IlK(altJ),
f
eikX(k 2 + a 2)+11-1/2 =
-00
a E!Jll+;
2J1r + 1/2)
(S-2)
(2a)±Ilr(±/l 00
x
L
[x
-00
+ 2knl±IlKialx + 2knl).
(By analytic continuation and use ofthe well-known asymptotic properties of K Il , one finds that these sums are convergent and equal when Re( ± /l) > 0.) f(t) = Jt 2 + a 2 - l e - b-/tT +a' , a, b E ,OJ+; n
L 00
L 00
[(x + 2kn)2 + a2r1/2e-bv'{.x+2kn)2+a2 =
-00
-00
eikxK o(aJb 2 + k 2); (S-3)
a, b E .0/1.+ ;
L 00
~ e- b-/(;:;: 2kn)'+-a2 ab_ oo
L eikxJb 2 + k 2- 1 00
=
-00
x K 1(aJb 2 + k 2);
(S-4)
a, b E ]1+;
}br.3 a ±I'b 1/H Il L 00
-00
L 00
=
-00
[(x
+
2kn)2
+ a 2J±Il/2-1/4K±I1_ 1/z(bJ(x+2kn)2+ a2)
eikx(k2 + b 2)+11/2K,,(aJb 2
+ e),
/l E
t.{j.
(S-5)
239
13.2. Crystal Lattice Sums
We are now in a position to complete the proof of the theorem in Section 13.2. Let (3) (Without loss of generality we may assume that A = 0.) Then
S
=
9=
1
r(s)
Jo 9 dx, (00
Xs- 1
L
(4)
f(M)e-IIMII2x.
(5)
Imjl
We get
IJ =
9 ~ CXReS-l[83(0Iix/n)p and, by (S-I) with b
h,
(6)
x =0
=
h = O(xRes-l-PIZ),
X
--+
O+,
(7)
while as x --+ 00, h = O(e- ax ) , a > O. Thus, under the stated conditions, h is integrable and, by dominated convergence, limNr'oo Sexists. • Expansion (S-2) will be the principal tool. Let k a =
multiply both sides by and sum:
f
00
[IIMpll z
+ JZJs
23/Z-S~
=
r(s)
-
Il,
f
00
take the upper sign on [x
+ 2m
--+ O.
fl, fl--+ S -
t,
nls-l/ZeiX(m,+· .. +mp-Il
[IIM;_ll1 z
X KS-l/z<JIIMp-IIIZ Now let 15
mp ,
(mi + ... + m;-l + ()Z)l/Z,
eix(m'+"'+m p
eix(m,+ .. ·+m p)
-+
+ JZJS/Z 1/4 + JZlx + 2mpn l).
.
(8)
The result can be written [x
+ 2m nl s- 1/ z 11M p_lll s - 1/ Z
"---_ _~p_~_ eix(m,
x KS-l/z(IIMp-ll1lx
00
+ ... + mp -
i )
cos kx
+ 2mpn l) + 2k~I~'
As it stands, this holds only for x '# 2jn, j term must be peeled off and the relationship
E
J. For x
--+ 0+,
the mp
(9)
=0 (10)
240
13. Multiple Sequences
used. The result gives the original sum as a sum over one lower dimension plus a rapidly convergent series of Bessel functions.
~ IIMpll-2s = 2(2s) + y0tr~(S~ 1/2) +
2n s r(s)
f:
IIM p_t11 1- 2s
1m Is-I/2
!;: IIMp:llIs 00
1/2 Ks_I/2(IIMp_11112mpnl).
(11)
The Besselfunction expansions on the right of (9) and (11), still expansions over p-space, converge with great rapidity. Also, for the values of x of greatest interest, the cosine series on the right of (9) can be evaluated in terms of zeta and related functions. For other values of s, it can be dealt with by the asymptotic techniques of Section 1.6. In many cases, s is an integer. The series on the right then becomes a series of exponentials. (An example is given later on.) In any event, the Bessel function K; can be considered a known quantity, its computation today being standard software. For s = t in the case of a three-dimensional sum, there is convergence provided x '1= 2jn, j E J. The (m l , m2) sum can be expressed in terms of exponentials by (S-3), i.e.,
L 00
L 00
mt=l
eiX(m 1 +m,) Ko(aj
mi + mD = n L 00
m2=-OO
m2=-cIJ
x
[(x
+ 2m2n)2 + a2r
(eJ(X + 2m,n-j'-+ a L
ix _
1)-
I,
1/2
(12)
and this can be used in an obvious way in (11). The same applies, with (S-4), when p = 3 and s = l As an example of how an error analysis of these sums proceeds, let us examine (11). Assume the Bessel function sum is truncated, with all points inside the hypercube sup Ixjl = N
(13)
o s i s»
included. Let RN =
I
Imj[=N+I
1m Is - I/2 11M P II s- I/2 Ks_I/2(IIMp_11112mpnl). p-I
(14)
For an analysis of R N , we shall need several preliminary results useful with sums of this kind. Lemma 1. Let
IX,
n > 0, f3 > IX/n. Then
L kae- Pk :::; nae - Pn(1 00
k=n
ea/n-p)-I.
(15)
241
13.2. Crystal Lattice Sums
By calculus one finds that
Proof
x' Letting x
~
(rx/(jYe-ae bx ,
a, (j, x >
o.
k, (j --> «[n, and substituting the result in (15) proves the lemma.
-->
•
--!-,
For A ~ 1, Re v >
Lemma 2.
I
K.(A)eA r(Re v + 1) AV ~ jr(v + -!-)I KRe.(l) =
I Proof
(16)
A(
(17)
CV •
This follows immediately from the integral
Kv(z)e= -- = ZV
- r
v
2
+ -1)-1 2
Re v>
-1-,
foo e 0
-zt[t (I + -t)]V-I/2 dt
Re z >
2
'
o. •
(18)
Lemma 3 (19) Proof
(20) so (mi
and the lemma follows.
+ ... +
m;)1/2 ~ (l/p)(m l
+ ... + mp )
(21)
•
A straightforward application of all these results shows that for s ~
1< 2s+p+I/2rr2S-I/2cs_I/z{N + 1)2S-1 exp{-[2rr(p
R
I
N
r(s)
-
x {I - exp[ -2rr(N
+
2s + p+ 1/2 rr2s-I/2 r(s)
K s-
+
1)2}
1)/p]}I-P
x {I - exp[(2s - 1)/(N >::::
- 1)/p](N
-!-
+
1 / 2 ( 1)
1) - 2rr(p - l)(N
+
exp{ -[2rr(p - 1)/p](N
l)/p]}-I
+
1)2}, N
--> 00.
(22) For instance, if N = 2, the truncated sum will contain 26 terms if p = 3. The exponential term above is 4.2 x 10- 1 7 . If only seven terms are taken (N = I), the exponential term is still only 5.3 x 10- 8 .
242
13. Multiple Sequences
The case s = 1 of (9) is particularly important. It gives eix(m, + ... +mpl
v- 2 2=n -oomt+···+mp 00
'\' 00
L.
x
L.
eix(m'+"'+m p'
l )
-00
(ml ... ·.mp-,l"O
e-j;;'T+"'+~--;lx+2mp"l
00
cos kx
Jmi + ... + m;-t
k=t
k2
+ 2L
(23)
a rapidly convergent series of exponentials. Obviously the forgoing procedure is easily modified to account for sums with denominator 11M - Ails, A = (at, ... , a p ) . For many special cases, see Hautot's paper. 13.2.3. Laguerre Quadrature
This is an elementary but very accurate method for hand computations. It can be applied for certain functionsfwhen s - 1 - 1P is a value {3 for which the abscissas and weights for the Laguerre quadrature formula for xfJe- x have been tabulated, e.g., {3 = 0, -t, -1-. -1, etc (Concus et al., 1963.) This is illustrated for f == 1.
(1) h(x)
=
exx P/ 2[03(0Iix/ny - 1].
The integral on the right is easily evaluated by Laguerre quadrature, since the series for 0 3 converges with great rapidity. For example, let P = 2, s = l
(2)
Laguerre quadrature with just three abcissas yields S = 9.0352, while the true value is 9.0336.
Appendix
A.I. Lagrangian Interpolation
Let x, yEC(;'s, and denote by p~k)(Z) the polynomial of degree k that at assumes the values Yn, Yn+l'···' Yn+k' respectively. (It is assumed the x j are distinct.) Then X n, Xn-b .•• , Xn+k
(k)( ) _ " k
~
Z -
~h+m
m=O
Il k
Xn+i
Z -
(
i=O X n + m i*m
)
Xn+i
.
(1)
It is easily shown that p~k) satisfies the recursion relationship (k+1)_ Pn -
(
(k)
) X n-ZPn+l-
Xn -
(
) (k) X n+k+l- ZPn
Xn+k+ 1
,
n,
k>O
-
,
(0)_ Pn - Yn,
n 2:: 0, (2)
by putting z = Xi' n :s; i :s; n + k + 1. Another useful expression for p~k) comes from expanding the determinant
Yn+ 1
1 1 1
Yn+k
1
p~k)
Yn
Z
Z2
Zk
2
x kn
X n+l
xn X~+ 1
X~+l =0.
Xn+k
X~+k
X~+k
Xn
243
(3)
244
Appendix
Let
Uj E~,
Vm( u 1,
and denote the Vandermonde determinant Vm by
Uz,""
um)
=
Ul
ui
ui
Uz
u~
ui
Um
Z Um
n n
m-l
=
m
i=O j=i+ 1
m Um
(Uj -
uJ
(4)
Expanding the determinants (3) by minors of the first column and using (4) shows that the determinantal expression is the same as the sum (1).
A.2. The Formula for the s-Algorlthm The proof of Eqs. 6.7(1)-6. 7(3) depends on two determinantal identities. It will be very useful to use Aitken's shorthand notation for determinants, writing only diagonal elements. For instance,
al a3a4a 7 b 1b 3b4b 7 d 1d 3d4d 7 el e3e4e 7
'
(1)
and so forth. The two identities are the obvious generalizations to n x n determinants of
which relates determinants with different first rows, and
!albzC3d41IalbzC3esl-lalbzC3dsllalbzC3e41 = lalbzc3d4esllalbzc31, (3)
which is an expression of the cross product of determinants whose last rows and columns differ in a certain way [see Aitken (1956, p. 108, No.2; p. 49, No.8)]. First, Eq. 6.8(1) is true when k = 1 for
(4)
A.2. The Formula for the ,,-Algorithm
Next consider the case k = 2m, m
~
245
1. Let
1
(5)
and -1
Q'n
=
L\sn+m
(6)
1
L\sn+ Zm
We must show these are the same. Rearranging the elements of the first gives
Qn
=
L\ZSn+ 1
L\ZSn+m
L\Zsn
L\ZSn+m
L\ ZSn+Zm_1
L\ ZSn+m_1
L\sn + 1
L\sn+m
L\sn
L\ ZSn+1
L\ZSn+m
L\Zsn
L\ZSn+m
L\ ZSn+Zm_1
L\ ZSn+m_1
L\ZSn+ 1
L\ZSn+m
L\ ZSn+m_1
L\zSn+zm_z
L\sn + 1
L\sn+m
L\ZSn+ 1
L\ZSn+m
L\ ZSn+m_1
L\ZSn+Zm_Z
(7)
246
Appendix
and using the determinantal identity (2) above one gets Eq. (8).
Qn =
I
I
~Sn+l
~sn+m
~sn
~2Sn+l
~2Sn+m
~2Sn
~2Sn+m_l
~2Sn+2m_l
~2Sn+m_l
~2Sn+ 1
~2Sn+m
~2Sn+m
~2Sn+2m_l
~Sn+l
~Sn+m
Ss;
~Sn+ 1
~Sn+m
~2Sn+ 1
~2Sn+m
~2Sn
~2Sn+ 1
~2Sn+m
~2Sn+m
~2Sn+2m-l
~2Sn+m_l
~2Sn+m_l
~2Sn+2m_2
~Sn
~Sn+m
~Sn+l
~Sn+m+ 1
~Sn+m-l
~Sn+2m-l
~Sn+m
~Sn+2m
~Sn
~Sn+m
~Sn+ 1
~Sn+m+ 1
~Sn+m
~Sn+2m
~Sn+l
~Sn+m
~Sn+m
~sn+2m-l
(8)
The second quantity [Eq. (6)] may be written
Q'
n
Dn
= (-It
~sn
~sn+m
~Sn+l
~Sn+m+l
~Sn+m-l
~sn+ 2m-l
~sn+m
~Sn+2m
Dn
=
~Sn+ 1
~Sn+m+l
~Sn+l
Ss;
~Sn+m
~Sn+ 2m
~Sn+m
~Sn+m-l
Sn+ 1
Sn+m-l
(9)
~Sn+l
~Sn+m
~Sn
~Sn+l
~Sn+m
~Sn+m
~Sn+2m-l
~Sn+m-l
~Sn+m
~Sn+2m
Sn+ 1
Sn+m
s;
I
247
A.3. Sylvester's Expansion Theorem
On D; we use the second identity to find
Dn
=
ASn + 1
Asn + m + 1
ASn + m
ASn + 2m
ASn AS n + m 1
1
ASn + 1
ASn + m
ASn + m
AS n + 2 m -
(10) 1
Elementary determinant manipulations show that the first factor above is ( - l)k times the first factor in the denominator of Qn. Thus Qn = Q~. The proof for k = 2m + 1 is similar. A.3. Sylvester's Expansion Theorem Let A be an n x n determinant, n 2 3, with elements aij and denote the minor of element aij by Mij. Let
D=
(1)
Then (2)
[see Muir (1960, p. 132)].
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Index
A 2-process,
Aitken 5 104, 149-152 applied to power series, 151 generalized, 105, 154, 167, 184 B
Birkhoff-Poincare scales, 15-23
c Continued fractions, 156-165 Convergence equivalence to, 33 hyperlinear, 154 linear, 6 logarithmic, 6
Exponential polynomials, 203 Extrapolation, deltoids obtained by, 73-76
F Fixed points of differentiable functions, 146-148 Fourier coefficients, computation of, 205-207 Fourier series, summation of, 48-53 G G-transform, 200-205
H Hankel determinants, 14, 157 Heat conduction, equation for, 100 Hilbertian subspace, 9~94
D
Deltoid,S, 71-80 Difference equations, analytic theory, 16
E s-algorithm, 138-148 generalization of, 144-146 stability of, 141-142 Equivalence, asymptotic, I Euler's constant, 75
Implicit summation, 171-174 Interpolation, Neville-Aitken formula for, 73 Iteration functions abstract spaces, 118-119 construction of, 112-118 L
Laguerre quadrature, 91 Lebesgue constants, 48-53 255
256
Index
Lozenge algorithms, 3-5 linear, 67-76 nonlinear, 101-106 M
Means, see Transformation Method, see Transformation Modulus of numerical stability, see Numerical stability N
Numerical analysis, rational formulas for, 142-144 Numerical stability, modulus of, 29
s Saturation, 51-53 Scale, asymptotic, 1-2 Sequences complex, properties of, 5-12 Laplace moment, 84-90 iteration, 106-108 linearly convergent, 6 logarithmically convergent, 6 Taylor, 96 totally monotone, 12-14 totally oscillatory, 12-14 Stieltjes integrals, quadrature formulas for, see Quadrature Summation methods, see Transformation Sums, lattice, 232-242
o Order symbols, 1-2
T
p Pade approximants, see Rational approximations Path,3 Poisson summation formula, 238 Pollaczek polynomials, 59-63 Polynomials, orthogonal, 40-44, 80-83 Products, partial, growth of, 8
Q Quadrature, numerical, 69-71 based on BH protocol, 200-209 based on cardinal interpolation, 77-80 based on G-transform, 200-205 based on Romberg integration, 67-71 based on tanh rule, 207-209 Quotient-difference algorithm, 156-159 R
Rational approximations, 53-59 gamma function, 58 Gaussian hypergeometric function, 56-57 Pade, 54-57, 128-136 for Stieltjes integrals, 132-136 Rhomboid,S, 80-83 Richardson extrapolatin, 67-71 Romberg integration, see Quadrature
T-matrix, Abel, 66 Taylor formula, generalized, 146-148 Transformation accelerative, 3 Brezinski-Havie, 175-209 quadrature by, 200-209 e-algorithm. 120-148 multiparameter, 166-167 'TJ-algorithm for, 160 GWB, 106-108 homogeneous,S implicit summation, 171-174 Levin t and u, 189-198 linear,S Lubkin, 152-153 multiple sequences, 227-231 nonlinear,S Overholt, 108-110 probabilistic, 210-226 p-algorithm for, 168-169 regular, 3 Schmidt, 120-147 geometric interpretation of, 136-137 topological, 182-185 a-algorithm for, 169-171 Toeplitz, 24-26 applied to series of variable terms, 48-53 band,28 based on power series, 94-100
Index
characteristic polynomials for, 28 Chebyshev weights, 43--44 Euler (E, q)method, 99 Euler means, 34 (f,-'Yk) means, 51 Hausdorff, 34 Higgins weights, 45--46 Lotockil, 44 measure of, 28 nonregular, 38--40 optimal, 90-94 orthogonal, 40-43, 80-83 positive, 27
257
Richardson procedure, 67-71 generalized, 181 Romberg weights, 44--45 generalized, 181 rational approximations obtained with, 54 Riesz means, 65 Salzer means, 35-38 weighted means, 33 translative,S W, 152-153 Trench algorithm, 198-199 Triangle, 27
