A number-theoretic approach to numerical multiple integration

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Halve, W. J. M. (1981). A number-theoretic approach to numerical multiple integration. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8106). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1981

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics

Memorandum 1981-06 May 1981

A NUMBER-THEORETIC APPROACH TO NUMERICAL MOLTIPLE INTEGRATION

by

W.J.M. Halve

University of Technology Department of Mathematics P.O. Box 513, Eindhoven TheNe therlands

by

W.J.M. Halve

Abstract

A method of Hua and Wang for constructing cubature formulas is reviewed as well as an implementation of it by Moon.

1. Introduction

2. A special class of periodic functions

3. The formulas of Hua and Wang

3.1. Introduction

3.2. Number-theoretic preliminaries

3.3. The algorithm and its implications

3.3.1. The algorithm

3.3.2. Practical aspects of the algorithm

3 . 4 . Examp1es

3.5. An additional formula

3.6 • Some esdmates

4. Moon's implementation of Hua and Wang's method

3 7 7 8 9 9 1 1 12 14 15 17 4.1. 4.2. 4.3.

A description of Moon's algorithm

Periodizing techniques

Results

17

18

20

5. Comments and amplifications 22

Contents

5.2. Some consequences of Moon's choice

5.3 The numbers ~ . n,Q,,j 5.3.1. Introduction 5.3.2. A recursive relation 5.3.3. Applications for Q,

=

2,3,4 5.4. The parameter t 5.4.1. Introduction

5.4.2. Hua and Wang's parameter t

5.4.3 Moon's parameter t 24 27 27 28 29 33 33 34 34 5.5. 5.5.1. 5.5.2. 5.5.3. 5.5.4. 5.6. 5.6.1. The units Introduction The condition V1 The number g

The relation between the PQ, and the w

j

Some final remarks

Some remarks by Moon

36 36 37 38 40 42 42

Contents

5.6.2. Other real algebraic fields

5.6.3. Summary and conclusion

6. References

43

45

1. Introduction

In general, by multiple integration we mean the process of determining the value of I

R(f) :

IR(f)

=

f

f(!) dR R

for some s-dimensional region R c JR s and some function f: R -+ JR . By numerical multiple integration we mean the process of obtaining the value of an approximation of ~(f), say QR(f):

The problem is then, given R and a class of functions A , to find suitable

s

cubature formulas QR' In other words, we look for nodes

x(j ) (j = 1, . . . , N)

and weights

w. (j = 1, ••• , N)

J

such that the error E

Q_R(f)

can be estimated within reasonable bounds when f is an element of A .

s

Here we take R to be the uni t cube in JRs :

R:; C

s

I

0 S;x._J s; 1, j = 1, ...,s} •

cubature formulas for a certain special class Qf functions can be restated as a number-theoretical problem. Some years later Hua and Wang [8J proposed a method for constructing solutions of this problem. In 1974 Moon [16J im-plemented their method, which is based on the theory of real algebraic num-ber fields.

We will first state thenumber-theoretical problem in section 2 and review the algorithm of Hua and Wang in section 3. Next we describe Moon's imple-mentation in section 4 and, finally, in section 5, we make some remarks concerning several details of both the algorithm and its implementation.

2. A special class of periodic functions

In order to obtain a generalization of the well-known trapezoidal rule, Korobov [12J studied the class E of functions f : JRs -+ JR which are peri-:

s

odic with period I in each variable and which are continuous on

e .

s

Such functions f have Fourier coefficients

(1)

c(~,f)

:=

J

f(~) e-2rri(~,~)dx

e

_s

where (~,~) denotes the ordinary scalar product

s (~,~) :=

I

j=1

m.x . •

J J

In particular, by taking m = 0 in (I) we have

(2) _c(Q,f)

₌

J

f(~)

dx

e

_s

=

Ie

(f)

s

A sufficient condition for the Fourier series of f

c(~,f) e2rri(m,x)- - (?E: E JRs )

to be absolutely convergent is that

(3) (a > 1), where

III mill

:= s .IT I max {1m. I , I } J= J

and the constant associated with the O-symbol does not depend on m. Thus, (3) is equivalent with

(4) _{3. '!/ [IC<!!!,f) I} ~ _{K(f)/ III!!! IIF] ,}

K(£»O mEll s

and for the subclass Ea of functions fEE for which (4) holds, we have

s s

(5)

Korobov shows how the parameter a is related to the smoothness of the func-tions f. He proved the following lemma([13, Lemma 7J).

(*)

Lemma 1: Let a > 1 and let n

1, .•. ,ns be nonnegative integers that sum up . to lasJ. Furthermore,let f € E

s and let each of its partial derivatives

, where v₁' ••• ,v

s is any permutation of n], .•. ,ns' be

con-VI \Is

dX] •••dX

S

tinuous on

e .

Then one has f € ELasJ/s

s s

N

Now, by taking any weight vector ~ such that

I

w. = ] _{and any set of} j=l J

{!(j) s·

nodes X = € 1R

I

_{j =} 1, ••• ,N}

,

we obtain, in view of (2) and (5)

o

Q

_e

(f) :=

s

Ie

(f) +

I

c(m,f) S(!!!;x,~)

s ~d \{£}

Here S(1E;X,~) denotes the trigonometric sum

N

~

j=l

. . . 1 . d

The next observat~on ~s that ~f we choose ~ := N l an

X :=

{~(j)

:=

~ ~

I

j

=

1, ... ,N} , where

i

is the vector all components of which are 1 and ~ is taken from 7l \s {Q} , then one has

(6)

_{= {}

o ,

otherwise . Hence, for the subclass

(7)

we obtain the error in

Q

C ' viz. (*) s E

Q

(f) C s

=

which by use of (4) and (7) has the upper bound

In order to-minimize this uppe~ bound we arrive at the following number-theoretical problem. ~.e.

(8) find N E

m

and a E 7f \ fQ} such that

(*)By 0Nl

(~-,~2

is meant the right-hand si.de of (6). In general, for any

propo-{

I , if P holds si tion P we define 0p :=

( 8 )

~

-

I

111

~

III-a

met \{O},(m,a) _ 0 mod N

_-

_-

--is

minimized ..

Korobov further shows that N and a can be chosen in such a way that the expression in (8) is of order logas(N)/Na.

Several attempts have been made to solve problem (8). In this respect we refer to the work of Hlawka [7], Zaremba [zoJ,Maisonneuve [15] and Salty-kov [19J. Good survey papers on the subject have been written by Haber [5J, Moon :~16J and' Niederreiter [18J. In particular, Hua "and Wang succeeded

in obtaining constructions which will be given in· the next section. Also the paper of Haber [16J should be mentioned.

3. The· formulas of Hua arid Wang 3.1. Introduction

Before outlining the algorithm of Hua and Wang in subsection 3.3 we will mention some of the notions of the theory of algebraic numberfields in 3.2. Although these notions form-~artof the knowledge necessary to understand

the separate steps, perhaps they do not suffice to clarify the main idea of the method.

Therefore we start with a brief sketch of the ~atter. Certain algebraic number fields:IF consist of real numbers only, but have an s-dimensional

s

structure as well. This structure is represented by an additive basis W

=

{41.

I

j

=

1, ... ,s} which is employed by Hua and Wang. In short, they

J

prescribe W to contain the number 1 and then would like to use U.J as the

vector ~/N. However, the numbers 41. ~ 1 are non-rational numbers, hence

J

cannot be applied directly. Instead, Hua and Wang provide sufficient con-ditions for the existence of an infinite sequence of vectors ~(t) and num-bers Nt that can be used and show how~to construct them. The parameters Nt and ~(t) satisfy

(9)

h.(t) ...;::,J ~.

Nt J (j

=

2, •••,s;

Finally, they apply the following theorem ([10,·Theorem 4 ).

s

Theorem 1: Let h(t) E: 7l \{O}, N E: IN and let an additive basis

- - t

W= {'COl

=

1, 41

2" " 41s} satisfy (9). Furthermore, let the cubature formula

Q be defined by

Nt,!!(t)

(10) _-1 _f\.l.-

f'

_h(t))

\

N N

Then E satisfies for any ~ > 1 the.inequality

Q_N ,h(t) t

-( 11 )

_{Vc->o 3 (}

_{) o[}

_sup

_IE

_(f)1

"" _{c 2 :IFs} ,~,E: > f€E~(K)_s Q_Nt,!l(t)

3.2. Number-theoretic. preliminaries

In this subsection we list some of the basic concepts of the theory of algebraic number fields. More details can be found in [3J.

1. Let p(x) be an element of ~[xJ, that is, let p(x) be a polynomial with rational coefficients. Then the zeroes of p(x) are called algebraic numbers.

2.:IF ~s an algebraic number field of degree s (over ~) i f i t ~s a

vector-s

space of dimension s over ~, containing algebraic numbers only. 3. The minimal polynomial of a number r; €:IF is the monic element of

s

~[x] of lowest degree, which has .; as a zero; s is a multiple of its degree •

. 4. The conjugates of a number r; '_'T" f . (1)

'nom~d-LO ,;; notat~on:,;

=

€:IF are the zeroes of the minimal

poly-s

(2) (s)

,;,,;

,

...

,';

.

_c5. An algebraic integer is a zero of an element of ?l[x].

6. An integlral basis W (of :IF ) is a basis of the additive group of

al-s ,

gebraic integers within:IF •

s

We remark that the set W in Theorem 1 of section 3.1 ~s an integral basis.

7. A unit is an algebraic integer whose inverse is again an algebraic integer. Hence, if u is

a

unit in:IF , we have

s (12) s II j=l u(j)

=

±1 •

8. Abasis of the multiplicative group of rank s - I of units within F

s

is called a set of fundamental units.

9. Any set of s - 1 units' hI'" . ,E

s-1} of Fs such that the matrix L of

logarithms, defined by

(13) (L) .. := loge

I

£

~i+l)

I)

lJ J (i,j

=

1, ••. ,s-l) ,

is regular, is called a (complete) set of independent units. In particu-lar, a set of fundamental units is also a set of independent units.

10. Conjugates of sums and products of algebraic numbers are calculated as sums and products of conjugates respectively.

3.3. The algorithm and its implications 3.3.1. The algorithm

Theoretically, Hua and Wang's algorithm consists of the following eleven steps.

1. Take a totally real algebraic number field F of degree s over ~ (that

s

is ,:IF c JR).

s

2. Determine an integral basis W

=

{.w

J

=

I ,wZ, .•• ,tOs} of :IFs . 3. Determine a set 'of independent' uni ts

t

= {E1' ••• ,Es-I} of

4. Form the matrix L as defined in (13). 5. Determine the value of c

1 defined by IF. s (14) c 1 := max l~i~s-l

I

(L).

.I}.

lJ

7, Construct the special unit n t defined by s-1 II j=] Lx.(t)J ] 8, , J T (]) (s)

D

t := (nt , •••,nt ) •

n~j)

; further, let cr:= sign(n t).

8, Form the matrix Q defined by

( '"') ••.~ ,'= ",(.i)w (';... , J' -- 1, • • • ,s) •

~J J

9. Determine the vector a(t) := QT_{n ,} where

- - t

s 10. Define n

t := a1(t) and note that nt =

L

j=1

11. Now let Nt := IntI and h(t) be the parameters of a cubature formula of the form (10). Here, h(t) is defined as hit):= 1, hjt) := cr a

j (t)

(j = 2, ••• ,s) •

The following two results (cf.aOJ) explain the use of n

t in the algorithm and provide sufficient conditions for the existence of Nt and h(t) as given in (9).

Theorem 2: Assume W= {WI""

tW }

to be .an integral basis of IF c. JR.. I f

s s

there exists a unit u EO IF such that

lu!

> 1 and if there is a constant

s c (IF ) > 0 such that

8

(16) ~ c (IF )

I

u ]-If(8-1 )

s (j

=

2t • • • ,s) t

s

then there exis t numbers N EO :IN,

h

EO IZ

h.

-1-...2-Iwj -

d

1

~

c' (IFs) N s -]

and c' (:IF) > 0 such that s

Theorem 3: Let

E

= {E:1 ' •.• ,E:s-l} be an independent set of units of IF sc: JR.

Then for any t E IN there exists a unit n

t whose conjugates satisfy

(17) -(2t-l)C1

e (j

=

2, .••,s) ,

(18)

!n

_t(i)

I""

_{.::. e}2CI ]. (J')_n

I

t (i,j

=

2, ••. ,s) ,

where c

1 ~s the number as defined by (14) .

3.3.2. Practical aspects of the algorithm

o

Probably the most important feature of the above algorithm is the fact that it requires only O(log(N)) elementary operations instead of(*) O(N4/3). However, before the algorithm can produce any explicit cubature formula, a number of data has to be known in advance. These data are gathered below; we shall discuss part of it in more detail in section 5 (after dealing with Moon's implementation).

I. First of all, we must have available a suitable number field IF c: JR

s

of degree s over Q, where s is prescribed and s E IN.

2. Within IF we have to know an integral basis W.

s

3. Moreover, we need a set

E

of independent units within IF •

s

4. Next, we must find out how to calculate the conjugates of the elements of both Wand

E.

(*)Korobov [13J showed that cubature formulas can be obtained using this many operations when N is the product of two distinct prime numbers. Later, Keas t [1 1] extended KorobovIs method to the case where N is the

product of J distinct prime numbers, thus reducing the number ~fnec~es­

J J

5. Additionally,we should like to have an indication of how to choose t.

6. Finally, we must be able to represent the elements of

t

in terms of the elements of W.

In the next subsection we give the two examples that Hua and Wang provide in their papers. Next, we describe an additional formula in 3.5 and we end

this section with some error estimates.

3.4. Examples

Hua and Wang propose two explicit applications in thei~ papers. The first application seems to be the most promising one (and is in fact the one which Moon implemented); we list both of them.

I. Cyclotomic fields.

number p, we can take IF to be s

.. 27fi

In

. . .

1;

=

e ~ (or any other pr~m~t~ve p-th root of

'p I f 2s +1 happens to be a prime -] :IF

.=

~(1; + 1; ), where s' p p unity). Furthermore, W:=

that is, w.

=

2cos(2~j/p) (j

= ], ...

,5).

J

As far as the elements of

t

are concerned, three possibilities are suggested. Denoting the elements of

t

by E.' (j

=

1 , ••• ,s-I), we have the following al

ter-J natives: a) sin(~ gQ,+I Jp ) sin(~

lip)

(Q,

=

1, ••• ,s-] ) where g is possible; b) E. := w. J J c) E. := w. J ~

.

J

any generator mod p of GF(p). This choice of ,units EQ, is always

(j

=

1, •.•,s-I);

where

I

w.

I

>

I

w.

1

> ••• >

I

w.

I.

~1 ~2 ~s

These two choices are only possible (that ~s, yield independent sets of units) if the following condition VI is satisfied:

{

.)

VI 1

ii)

2 is a generator mod p of GF(p) or

oP

=

7 mod 8 and 2 has order s mod p •

Corresponding to the cases a),b) and c) we observe that the computation of the conjugates can be performed as follows.

a) W(i) := (i,j

=

1, ... ,s); gj W_gi+j b) w.(i) := w.. _(i;j

₌

_{1, ... ,s);} J ~J c) w.(i) := _{w(i+j)mods(i,j}

₌

1, ..•,5). J 2. Dirichlet fields

If s happens to be a power 0f 2, say s

=

2h

,

t en we may ta eh k IF

:=

s

~(;p;,

...

,~), where the numbers PI, ... ,Ph are distinct primes. Now the set

E

is derived as follows.

F b f {I } 4 rt 1 h . . . • bl (*)

or any su set I 0 , ••• ,h, I r ~, so ve t e m~nlm~zat~on pro em

with d

_r

:= IT

i n

p .•

~

Then 8

_r

:= ~(u + ~v), where

u

+ ~v is the solution of (19). Thus we have s - 1 independent units which are rearranged according to

(*)The equations

Hx.

+

vcr:

v.)

(j

=

1 •••• ,.)

J J J

x. + Id.v. (j = .+1"",s-1) •

J J J

Here a number j S , corresponds to a s~bset I

j of {l, ... ,h} for which

u

and

v

are both odd, while a number j > , corresponds to a subset I.

]

for which

u

and

v

are both even.

Next,an integral basis W is defined by

r

e: • (j

₌

1, ••• ,,)

t

] wI := w_{j +1} := I([""" _(j

₌

,+1 , ••• ,5-1 )

.

]

Finally, the (j+l)~st conjugate of ~ is calculated by

(j+1)

Ii\

3.5. An additional formula I{k}nI. , (-1) J I

p

k (j

=

1, .•. , s-l; k= 1 , ..• ,h) •

Apart from the cubature formulas Q

h N as g~ven in (10), Hua and Wang present

-'

another type of formulas that can be calculated with less effort. We note

h.

that in (9) rational numbers NJ are used to approximate w.,the w. being

] J

the elements of an integral basis of JF • This is done because the vector

s

w cannot be used directly in an equal weight formula (w

j

=

N)'

Taking

W= {WI=1,w

2' •••,ws} as an integral basis of JFs ' we have the following cubature formula for Ea

5-1 (20) 1 n~

Q:

n.c

(f):= -

I

V n • f(jw*) ,"", s-1 N. n n,,,,,,] -J=-n"" (*) where n E: :IN, are determined by

Jl, :=

r

a

1 ,

N •- (2n + 1)9, and the coefficients 1-1 n •

n,,,,,]

smallest integer greater than or equal to x.

(21) ( =:

nt

\ ' 11 zj

L ' " Jl,' j=-nt n, ,J

Here the vector _w

*

_{is def~ned}

.

_{by ~\}

*

_{:= (w} T

2, .••,ws) • The great advantage of formulas (20) is that neither independent units nor conjugates are re-quired.

3.6. Some estimates

Evidently, any approximation and a cubature formula in particular, is not of much use unless we have some indication of the size of the error it produces. Hua and Wang derive the following estimates (c;. section 3.1), which hold for any ~ > 0 and any a > 1. One has

(11 ) EO (f)1 'N ,h(t) t -(22) _{[sup f} a feE s_1(K) E

*

(f) I Qn,t;C 1

s-In terms of N

=

(2n + 1)

fa1

the estimate in (22) is of order E

*

QN'C

-a/fal

'

s-1

O(N ) and thus (11) is always of smaller order. As mentioned in tion

2

there are other methods which yield better estimates, namely

=

sec-. -as

a

o

(log (N) IN ) .However such methods require considerahly more effort than the O(log(Nt»)operations necessary to express n

t in terms of w, •J

We conclude this section with some estimates that occur

in

Hua and Wang's papers. These estimates were used by Hua and Wang to deri~e (II') and (22).

(j

=

1, ••• ,s) • (23) s

I

j=l

le.1

J (24) a. (t)

I

11. - .-J::..-_ J n t (j

=

1, •.• ,5) • The number c

1 in these formulas is again the number defined by (14). The

numbers e. in (23) and (24) are the components of the vector ~ E

7! ,

deter-J

mined By (~,~) = 1 corresponding to the general case of

w.

Ultimately, Hua

T

and Wang consider the case 1 E W, which implies e

=

(1,0, ••• 0) •

Then the estimates (23) and (24) reduce to

2c I 1 2c l . . 1 s [w(i)\'!n

I

- 5-1 - s-1

In

_{t - ntl} :s; _e

I

_{= e (S-O!l1} t i=2 1 t 2e I 0 5 a. (t)

_{I

I

w

~i)

I

lj(S-l)l\

1-

5 - 1

I

w. _ J :s; _{e +} N

J

l1_t = J n_{t i=2 J t} 5 _ _8_ \·Iw~i)\·r

1

5 - 1 ~L J ' "\ ~=2 (j=2, ..., s ) .

4. Moon's implementation ofHua and Wang's method

Moon implemented Hua and Wang's method in the case of cyclotomic fields and compared his results to related known results which were obtained by applying Korobov's and ZarembaIs methods. Furthermore, he investigated how

the smoothnessparameter a and the periodizing techniques influence the error terms. After dealing with the exact algorithm, which we present in subsection 4.1, we describe in 4.2 some of the periodizing techniques Moon has investigated. A short summary of Moon's results is given at the end of

this section.

4.1. A description of Moon's algorithm

1. Check if 2s + I is a prime number (If not, terminate.)

2. Take w. := 2cos(27fj/p) (j

₌

I , ...,s) . (Thus IF := ~(1; + 1;-1).)

J s p P

3. Take E: • := w. such that

lSI

I

>

Is

2

1

>

...

>

I

E:

!

and use the largest

J 1. S

]

of

E.

s - 1 elements E

l , ••• ,E:s-1 as elements

4. Form the matrix L of logarithms as defined in (13).

5. Determine c₁ as given in (14).

6. Choose a number t ~ 0 and solve the linear system (15).

7. I f in performing step 6 it turns out that det(L)

=

0, then stop. 8. Round the components of !(t) to the nearest integers, that is, set(*)

YJ' (t) := [x.(t)].

J s

9. Use the fact that

I

j=1 w.

=

-I to determine a representation of J s-1 10. In order to transform II j=1 s-1 II j=1 y. (t) E:. J J in terms of w••J y. (t) s. J J into s

L

k.w., set up a multiplication j=1 J J (*) Cn +

!I

is defined to be n +1 in case n € 7l.

table for e. using J

w.w.

=

w • • + w • •

1 J l+J l-J (i,j

=

I, ...,s) .

II. Detenune

.

<*)

~(t):·=S"2 S"2~T = (pI - 2J)k and compute Nt by taking Nt :=

s

I

I

a. (t)

I.

Then form h( t) by taking

J

-j=1

h. I (t) :-= !a.(t) I

J+ J (j

=

1, •••,s-1).

4.2. Periodizing techniques

So far the functions f considered were periodic. Cubature formulas which are designed only for periodic functions may be applied to non-periodic functions as well, by using periodizing techniques. Moon discusses a number of transformations that map certain non-periodic functions into periodic ones. Mor~ precisely, he regards (among others) the fol-lowing two operators P I and

P~

(£ E: IN \ {I}) :

P 1(f) := _lXE:C f(l -

1

2x

-ll)

- s

P~(f)

:= s IXE:C f<:!Q,(~.P TI Ti(x_j )

,

j=1 s where O=I, .••s) and x TQ,(x) := (1£-1)

e(~:~))J

z£-l

(i

-z)£-I dz =

°

B (£,t)

=

(1£ - 1) --=-x-,--~ B(£,£) (x E: [O,IJ). (*)Here I l'S the 'd_{1 entlty matrix with}.

matrix, ire. J =

j jT

entries 8 .. while J is the all-one lJ

In general, Moon considers periodizing transformations P that satisfy the following two conditions, i.e.

a) Q,

Q,

f E H - P(F) E E

s s

where f E H iff all of its partial derivatives

s

(0 ~ n. ~ Q,

J j = 1, ••. ,s)

are continuous on

e

(and hence are bounded ~n absolute value by

s

some constant K(f) > 0) ;

b)

₌

_{Ie (P(f»}

s

Korobov proves a lemma concerning condition a) (cf.C13, Lemma 12]).

which runs as follows.

Lemma 2: Let Q, E IN\{l} and let f E

H~(K)

be the subclass of HQ, with

s s

K(f) ~ K. If for each n = 0,1, .•••1-2 and each j = 1••.•• s

=

then f E EQ,(K).

s

We note that PI produces functions

~n

E;, while

P~

produces functions in EQ,

s

4.3. Results

1. By means of the algorithm in 4.1 Moon derives cubature formulas for dimensions s

=

3,5,6 and 9.

2. Using seven kinds of tastfunctions and applying P₁ and

P~ (~

=

2,3,4), Moon compares Hua and Wang's formulas with Korobov's and Zaremba's. From the experiments it appears that

i) Zaremba's formulas in general have a better performance than Korobov's in most cases, but not always. Roughly speaking, they give errors of the same order of magnitude. This agrees with Moon's claim that Korobov's "optimal coefficients" and Zaremba's "good lattice points" are basically the same.

ii) Hua and Wang's parameters give errors about one order larger than Korobov's and Zaremba's. (That is, they differ by a factor ten. )

iii) If s

=

3,4 or 5,

P~

and

pi

perform much better than both

P~

and Pl' However,

P~

often yields smaller errors than

pi

(in particular this is true when s

=

6 or s

=

9). Apparently, there is no use in taking ~ (and thus ~) too large, because of the in-crease in K(f).

iv) When s is rather large Moon advises to use

P~

as periodizing transformation for non-smooth functions and Pl for smooth integrands.

v) Whenever higher precision ~s required, Moon suggests to take an even more complicated (and thus more expensive) periodizing technique.

that Nt increases rapidly with t,and further observes the occurrence of very high values of NO in'higher dimensions. For instance, if s

=

14 one has NO

=

385,806.

4. Another important remark is made by Moon with respect to the error bounds. From his numerical experiments he concludes that none of the

theoretical upper bounds used seems to be very realistic, as these estimates turn out to be several orders of magnitude higher than the actual errors observed in most cases. Moreover, in practical cases it may be difficult to derive the value of K(f) := max {Ic(m,f)

I

1

5. Comments and amplifications 5.1. Introduction

As mentioned earlier, Moon does not implement Hua and Wang's ideas to full extent, but restricts himself to cyclotomic fields of a special nature. Part of our investigations consisted iri finding out for which dimensions s S 40 either Hua and Wang's method or Moon's algorithm may yield cubature formulas. The results are listed in Table 1. Its second column contains descriptions of the type of formulas which are feasible. By C(p) we mean the formula constructed by use of the cyclotomic field of degree

Hp - 1)

=

s. Furthermore, D(h) denotes a formula constructed by means of a Dirichlet field of degree s

=

2h• Formulas of type (20) are denoted by C*(p) and D*(h) respectively. The third column contains either the symbol Y if a feasible integration formula of type C(p) may be derived by Moon's algorithm, or the symbol N if this is not the case.

s type feasibLli ty

at

s type feasibility of

<Moon's algorithm Moon's algorithm

* * C (5),D (1) 21 C(43) N 2 C(5),D(1) Y 22 C* (47) 3 C(7),D*(2) Y 23 C(47) Y 4 C*(1),D (2) 24 5 C(11) , C* (3) y ₂₅ C*(53) 6 C(13) Y 26 C(53) Y 7 C*(17),D* (3) 27 * _C*(59} 8 C(17),C (19),D(3) N 28 9 CO 9) y _{29 C(59) , C* (61)} y 10 C*(23) 30 C(6l) Y 11 C(23) Y 31 D* (5) 12 32 C* (67) ,D(5) 13 C*(29) 33 C(67) y 14 C(29) ,C*(31) Y 34 C* (71) 15 C(31 ),D* (4) N 35 C(71) ,C* (73) y 16 D(4) 36 C(73) N 17 C* (37) 37 18 C(37) y ₃₈ C* (79) 19 C*(41) 39 C(79) Y 20 C(41),C*(43) N 40 C*(83) Table 1

The rest of this section deals with some consequences of Moon's choice (section 5.2), the numbers P t . (section 5.3), the parameter t

(see-n, ,J

tion 5.4), the units P

5.2. Some consequences of Moon's choice

Although Moon's choice is rather restrictive (there are only 15 symbols Y in Table 1 and 5 symbols N), it has the clear advantage that the units

e. depend in a simple way on the w.. From the program text in Moon's

J J

paper it appears that he does not make full use of the information his restriction provides. In particular, we suggest the following two simpli-fications.

1. The determination of the nwribers e .. Moon uses a procedure Ifsortl f to

J

sort the numbers Iw.1 (j = 1, ... ,s) according to their absolute J

values. However, from the definition w. J (cf.Figure 1) that . 2 '

=

2cos(-!l) it is evident p 1> !Iwsl > !lw11 > !lws- 1

1

> ... > !Iw

I.

rtl

Hence 2 ~ Iwsl > \w_2sl > ••. > Iw 2₁ s as ws (2j+1)

=

ws_j and ws (2j)

=

w. ._J

2. The determination of the nwriber (Jj' As we now have established e. = w •

J SJ

(i)

and furthermore know the conjugates w. = w.. , we can derive a simpler

J lJ

expression for the number c₁ as given In (14). Moon evaluates c 1 by

S (t+l'lr

calculating

I

]log(le~ '1) I for each i

=

1, ••. ,s-] separately and

j=l J

then determines the largest s-1 values by means of a sorting process. Instead, we proceed as follows.

{

s-1 }

max

I

Ilog (

I

w..

I) I

=

w s m(z) ~ Figure I. 21Ti/p 1:; =e p

s

{

Ilog(loo. 21)1 }

=

~

. Ilog(loo. \)

I -

min =

j=1 J 2:S;i:S;s 1.S

= c - min

{

\log (100 • 21 ) I

}

= c - min { Ilog (100j

I) 1}

=

0 _2:s;i:s;s 0

l:S;j:S;s 1.S

j#r~l

= c - min _{{llog(IS j}

I)' } .

0

l:s;j:S;s-1

This last expression can be reduced still further. To that purpose we define integers j and j such that

+

-\00.

1

=

. J+ _{I:s;j :S;s}min

, 100.

J_

I

= _1:s;j:o;s

max

{loo·l<l}J

Then, obviously, min {!log(loo./)I} = log(min{\oo.

1,100.

I-I}).

l:s;j:s;s J J+ J_

I t is cle.ar that there are only two values of <p in the range (211" ,

~

]

p p

. 11" 211"

for which \cos(~)

I

=

!;

in fact,lp =

3

and ~

=

:r .

Hence, the numbers j+ and j_ can be found among rtl , ltJ ' r~l and

lfJ .

For the two cases(k) modulo 3 that can be distinguished for the value of s, namely a) s

=

3a and b) s

=

3a + 2, we find respectively: ad a) j+

=

a,

L ::

20 and ad b) .j+ = 2(a + I) , j_ = 0 + 1.

Now we use the relations

cos (p) - cos(q)

₌

2 sin (p+q) sin(q-P)

2 2

sin(p) - sin(q) = 2 cos (p+q) sin(P-q)

2 2

o

:0; x<- .. - x_{- 6}11" 3 :0; sin(x) :0; x

11"

(*) The case s = 3a + I does not occur, since then p = 2s + 1

=

6a + 3 which is either not a prime number or does not satisfy p ~ 5.

Tr-to obtain (with CL := 3p )

~ ~ ~ ~)

ad a) € := cos(3) - cos (3 + (p-2)CL) > cos(3) - cos (3 + 2CL

=

(~ ) . ( ) • (~ CL) • eCL) = 2 sin

3"

+ CL Sl.n CL > 2 nn

3" -

2'

Sln

2'

=

= cos(i - CL) - cos(;) =: 0,

whence

I

Wj

+.1

<

I

Wj -' - I

as 0 < 0 < € implies Iwj_'-lwj+1 = (1-2€)(1+2o) < 1

ad b) .\ := cos(.! - 2et) -

cos(~)

= 2sin(.! - CL) sin (et) :::;

~etsin(.!

- et)

3 3 3 - ~ 3

J.1 := cos (.!) - cos (.! +~) = 2 sin (2:. + ~) sin (~) < a sin(1!. +~) ,

3 3 2 3 2 2 - 3 2 and sin(.!) - sin(.! - a)

=

3 3 • ('IT a) . (~) S1n

3

+

2 -

s~n3

=

whence A < 'IT • l ' as < a - 33 l.mp l.es 'IT a 2cos (- - -) 3 2 'If a 2 cos (- + -) 3 4 sin(~) sin(~) et

::>2'

> 1+4a sin (; -a) ~ 1+4 sin (1-a) sin (et)

=

1 + 2.\ ,

from which we deduce Iwj+llw

j-'

=

(2.\ + 1)(1 - 2J.1) > 1 .

The restriction CL

::>

'IT/33 excludes the case p=5 (5=2). Moreover, 5=2 implies

is not one of the

fjl

=

f%l ,

whence (25) W

r

₁

1

flog

(Iw

I)

I

I

r

₁

1

€ ••

J Nevertheless, the result

holds for the case s =2 also, as then'

Iw

5.3. The numbers ~. n .

n,N,J 5.3.1. Introduction

These numbers are defined by (21) in section 3.5; we recall that the num-bers E. become superfluous when usi.ng the ~ n .• Moreover, as is shown

J n,N,J

by the results in Table 1, it frequently happens (13 times out of 36)

that the only way to obtain a cubature formula of Hua and Wang of a prescribed dimension is by means of the ~ .•

n, t,J

It seems a straightforward but tiresQrne computation to determine the num-bers ~ n . explicitly', especially for large values of t. Fortunately,

n,N,J

from Moon's experiments (cf. 4.3 2iii1) it follows that probably only values of t ~ 4 are interesting. Here we will first derive a recursive relation for the numbers ~ n • and then apply. the relation for t = 2,3

. n,N,J .

and 4.

5.3.2. A recursive relation

For t E IN \{I} we may wri te

(26) nt \ 11 zJ .-L I"' t ' .-j=-nt n, ,] n

I

k= -n k z

₌

=: n(Q.-I)

L

i= -n(Q.-1)

n

i+k

k=~-n

fln ,Q.-I,i z,

Comparing coefficients of equal powers we find

(27) fl • n,t,J

=

net-I) n

L

I

fln,n-I,i O. k . i=-n(,Q,-I) k=-n N 1.+,J

(j

= -nQ., ••• , nQ.) .

Furthermore,we note the symmetry in ,the definition, that is

(28)

(I

_j

I

_{~ nQ.) .}

(29)

nO,-l )

I

p ~ 1 . ~. k .

i

=

-n~ -1 ) n,N - ,~ ~ + ,J

Changing the order of summation in (27) and taking into account (29) we have (30) whence ]l

.=

n,t,J min{n,j+n(2-1)} \' (J' = 0,1, .•• ,n2) , L l1n ,2-1,j-k k=max{-n,j-(Q,-l)n} (31 ) _{11 . •} n,2,J min{n,

jj

l+n(2-1)}

=

I

.

]l I 1 k=max{-n,lj!-n(1-1)} n,2-1, j -k (j = -n2, ... ,n2) . 5.3.3. Applications for Q, = 2,3,4

From definition (26) we tmmediately have

(32) _]l

₌

n,1 ,j (j

=

-n, .•• , n) •

Applying (30) with Q,

=

2 and using (32) we get

11_{n, . ,J}2 ' =

min{n,j+n} n

I .

1

=

2

k=max{-n, j-n} k=j-n

= 2n

+

1 - j (j = 0, ... ,2n) •

From the symmetry relation (28) it follows that

(33) _{1.1 2 ' = 2n +1 - Ij I}

n, ,J (j

=

-2n, ... ,2n) .

Hua and Wang mention these numbers(*1n one of their papers ([ 10,p •48S]).We apply- _

(* )

(30)again,-nowwith !L = 3; of course, use is also made of (33). Consequently (34) _{11 3 .} n, ,J min{n,j+2n} = kImax{-n,j-2n}

~n,2,j-k

= n

L

.

(2n +1 -

jj -

k

1)

k= -n +max{O, j-n} (j = 0, 1 , ••.,3n) •

A slight complication arises: we have to distinguish between two different cases a) and b).

a) 0 :<;; j :<;; n. In this case it is easily seen that the right-hand side of

(34) can be expressed as (35) _{~ 3 .} n, ,J n =

L

(2n +1 - Jj - k I ) k= -n b) n :<;; J :<;; 3n yields (36) _{~ 3 .} n, ,J

=

n

I

(2n +1 -

I

k - j

I)

= k=j-2n n

L

(2n +1 ....j +k)

=

k=j-2n

=

!(3n - j + 1)(3n - j + 2).

We note that if j

=

n formulas (35) and (36) yield the same result, namely

2

~ = 2n + 3n + 1, while ~ 3 3 = 1. Combining (35) and (36) in a

n, 3, n n, , n

single formula we have

Taking Q,

=

4 we apply (30) once more, in combination with (37). The gene-. ral type of formula that results has the form

(38) _{11 4 .}

n, ,J (j = 0,1, ... , 4n) .

Again we have to consider two separate cases·, viz. c) and d). c)

a ::;;

j ::;; 2n. Then an elementary computation shows that

n (39)

L

=

L

(3n2 + 3n + 1 - j2 - k2) = (2n+l)(3n2 + 3n + 1 _j2)+ 1 k=-n 1

- r

(n + 1) (2n + 1), (40)

_L

2 J-n =

L

k= -n (n2 + n + j2 2kj + k2 - (2n+1)(j

-k»=~(j-l)j(j

+1) d) 2n ::;; j ::;; 4n. Then (41 )

_L

1

=

n

I

(3n2 + 3n + 1 - j 2 + 2kj - k 2)

=

k=j-3n and

=

(4n + 1 - j)(3n2 + 3n + 1 _ j2) + + j(j - 2n)(4n + 1 -j) -i(4n + 1 - j) {2n2+2(j-3n)2+ 2n (j-3n)+4n-j}

(42) n

L ::

I

2 k::j-3n + n + j2 _ (2n + ]}(j - k) - 2kj + k2) :: (4n+] -j)(n2 +n+j2_(2n+l)j) + f(2n+I-2j)(j -2n)(4n+l-j) + + i(4n +1 - j){2n2 + 2(j - 3n)2 + 2n(j - 3n) + 4n - j} •

Substituting (39) and (40) into (3'8.) we obtain

(43) )l 4 . :: 5.;.31 n3 + 8n2 + 4..;;32 n + 1 +

U

3 n, ,J

(j

=

0, 1 , ••• ,2n) •

Likewise, substitution of formulas (41) and (42) in (38) gives

(j :: 2n, .•. ,4n) •

It ~s easy to verify that (43) and (44) both yield

p :: 1-3] n 3 + 4n2 + };.32 n + ] n,4,2n while Pn,4,4n

=

1 .

(45) 1.1

=

(5 3 J _{n3 + 8n}2 ₊ 4-3 2_{n+ J _ 2nj 2 +}

_~

_I

_{j [3 _ j 2 -}

_~

_Ij

_I)

₊ n,4,j (j = -4n, ••. ,4n) .

From formulas (32), (33),(37) and (45) it is clear that the recursive for-mula (31) produces expressions for 1.1 n • which in general are polynomials

n,N,J

of degree t - 1 in

rj

I.

This is another reason (cf. section 5.3.1) why ~ should not be taken too large, although larger values of t do not in-crease the number of cases that have to be considered. We omit details

. . . (*)

w~th respect to th~s last assert~on.

5.4. The parameter t 5.4.1. Introduction

We recall that the parameter t appears ~n step 6 of Hua and Wang's algorithm (cf. subsection 3.3.J), and, as a consequence, shows up in Theorem 3. In his algori thm (cf. sec tion 4.J) Moon takes t to be nonne-gative, while he further mentions (cf. subsection 4.3.3) that the number of nodes Nt increases rapidly with t. Thus, it may be important to

ob-tain a lower bound to for t. In the sequel we shall see how Hua and Wang on the one side, and Moon on th~ other side, treat the parameter t and the lower bound to respectively.

, as n

I

f. . . However, one then has to

• Jl+ ..·J_t

Jt=-n of f(j~*).

shows that one can avoid computing the ~ n •

n,N,J n

L

j =-n 1 values f. J (*)Niederreiter [l7] nt

L

~

.

f.

=

j=-nt n,t,J J s tore the 2nt +1

5.4.2. HuaandWang's parametert

Hua and Wang do not mention any lower bound for t explicitly in their papers. This is not surprising as they only use the parameter t to prove that special units l1

t with Inti -+ co as t -+ 00 can be produced; this is a direct consequence of (lnand (J2). From (17) one may derive the lower bound to

=

~ in order to have 111t

I

> 1, which is part of the con-dition of Theorem 2 (cf.subsection 3.3.1), Although Hua and Wang compute several cubature formulas explicitly ([9, pp.975-977I), they do not point out what are the ~alues of t that correspond to the cubature formulas exhibi ted. Apparently, they do not solve the system (15) directly but. first solve the related system

, . "

(46) La

=

a

...

where L consis ts of th.e last first row of

L.

s - Z rows of L - J'tT

--1'

!i.

being the

Obviously,

L

consists of s - 2 linear independent rows, and thus the solution of

Afterwards,

(46) is completely determined up to a multiplicative constant.

~

.al as_Z)T

the ~ector ----' ..•'-a--- is uniquely determined by (46).

s-1 s-1·

a

s-1 is chosen suitably large.

We

~bserve that one may always

start with solving (15) for t

=

to and then obtain solutions of (15) for (2c]t +1 )

other values of t by taking !.(t)

=

\2c

1

t

O

+ 1 ~(tO)' where c₁ is defined by (]4).

5.4.3. Hoon's parameter t

However, at first sight this value seems too small as from (17) it

fol-lows that to ~. !. In order to prove (17), Hua and Wang ([10,p.488]) pro-ceed as follows. One has

y.(t)

10g(I~~i)

I)

~

j J

This result can be improved by rounding x.(t) to the nearest integer

J

y.(t):= [x.(t)] instead of taking the truncated value Lx.(t)J • Thus,

J J J

Moon's algorithm implies !y.(t) - x.(t)! ~ ! instead of Iy.(t) - x.(t)1 < 1,

J J J J and hence (47) s-l

I

j=1

C)

y.(t) log(I~.~ \) J J s-1 ~I j=l (') x.(t) log(I~·~

I)

+ J J s-1 +!

I

Ilog(

I

~J~i)

I) I

~

-(2c 1t + 1) + !c1 • j=l

The final member of (47) has to be negative

~n

order to warrant !

n~i)

I

<1 •

. 1 1

In other words, using Moon's algorithm one may take t > - - - - •

o

4 2c

1 It is not clear, however, that c

1 < 2, which is necessary in order to enable to = O. Moreover, it now follows that it may be advisable to

choose c

1 as small as possible. As follows from (25) the smallest c1 is

ntiJ:t; obtained by taking the~. to be th.e largest s: - 1 elements: of the

J

set

{I

w.

I I

j

=

1,..., s }, unl ess s·

=

2.

J

note that from (23) we have Nt

=

In.tl + o(\ntl-S-1), while (47) together wi th (12) yields IT) I > t (S-1){2clt+l-~Cl } e

Thus Nt increases exponentially with t.

5.5. The units Pi 5.5.1. Introduction

We have defined the numbers Pi in 3.4.] a) as a feasible choice for the independent units E. when dealing with cyclotomic fields. Moreover, from

J

Table 1 in section 5.] we know that in several cases (e.g.

p e {17,3I,41,43,61,73}) the simpler choices 3.4.1 b) and 3.4.1 c) do not work. We note that before we can actually use the numbers. Pi to com-pute a special: unit T)t' we must express PQ. ~n terms of w

j ' Hua and Wang do not supply this information, but refer (cf.[9]) to a book by Fricke(*)

([4, p.225]) for a proof of the fact that tIie numbers P

t (t

=

1, ••. ,s-l)

form an independent set of units.

In what follows we shall first discuss the condition VI as a means of avoiding the use of the P

t, next we shall deal with the number g that occurs in the definition of the P9, and,finally,we shall point out how the P

t and the wj are related.

(*)

Fricke also mentions that P

5.5.2. The condition VI

Hua and Wang prove that condition V] as stated in subsection 3.4.1 is necessary and sufficient for the w. (j = ] •••••s-]) to form an independent

J

set of units(*) ([9. Theorem 5.lJ). Moon does not use condition VI in his algorithm but determines the matrix L and then verifies whether L

is regular or not. We may reformulate condition VI as follows. (i) 2 is a generator of GF(p)

Vi

or

(ii) -2 1S a generator of GF(p) .

It turns out that in Some q.ses condi tions(i) and(ii) do not hold. To that end

(**)

-we note that

(48)

{

-1 is a square mod p ~ p _ 1 mod 4

2 is a square mod p ~ p :: ±l mod 8 •

As each square mod p is certainly not a generator of GF(p). we have to distinguish between four different cases.

a) s _ 0 mod 4 (p _ mod 8) - , ( i ) /I ,(ii)

b) s :: mod 4 (p _ 3 mod 8) _ I (ii)

c) s - 2 mod 4 (p _ -3 mod 8) - either (i) /I (ii) or I (i) II I (ii) •

d) s - -1 mod 4 (p :: -1 mod 8) _ I(i) .

(*)Because the w. (j = } ••••,s) are each others conjugates. it easily J

follows that any s -} of w. (j = 1 •••••s) form an independent set of

J

units.

(**)The second asserti.on in (48) 15 a consequence of p2-1

2~

=

(-l)~mod

p ,

Thus we may conclude

s -

a

mod 4 • VI does not hold.

5.5.3. The number g

A number g € {l, ••. ,p-l} is said to be a generator of GF(p) if gJ :: 1

mod p implies j ::

a

mod (p-l). Consequently, the number g which appears in the definition of the PfL is more than just "an integer" as Moon des-cribes it ([16, p.32]).

In order to find a generator of GF(p) one may proceed as follows. First of all one may consult the literature. For example, Cohn ([2, Table II, p.256]) lists the orders mod p of the numbers j = 2, ••~,p-l for all prime numbers p < 100. If for some reason (for instance, because s is too large) we are not successful this way, then what remains is to compute a generator g. Figure 2 contains an outline of such a computation on which a computer program may be based. Beforehand it is appropriate to make the following remarks.

i) The number of generators of GF(p) is given by ~(p-1), where ~ de-notes Euler's function defined by

(49) ~(n) :

=

#{m € :IN

I

mS n , gcd (m, n)

=

I} •

We note that for any n'E: .:IN ,one has ~(n) > O.

ii) Let v(j) and r(j) be the exponent and residue of J, i.e. let

{

v(j) ::== min{m E: :IN

I

jm _

r(j) jV(j) mod p .

±1 mod p}

Then we have, taking r(j) = ±I ,

a) g is a Kenera.:tox ... v(g) = s and reg) = -I

b) v(j) = v(p-j) and r(j) = -1V(j) r(p-j) (j = Z, ••• ,p-Z),

J

v(j) i f r(j) = 1 c) _{order of J} = fzv(j) (j = Z, ••• , p-Z)

.

if r(j) = -I

iii) If j is not a generator, then nei ther any power of J is as V(jm)

I

V(j)

I

FOR J := 3, ••• , S DO KAN[JJ := 1;1,

_'

_, ." II ,

.

Y: N

's

:: 0 MOD 41

>

t

"

'f

I

KAN[ZJ := -I;G := 3;

,

~

..

" I KAN[ZJ _{:- I}

.

_{G :-}

2:1

1.1 I.... WHILE KAN[GJ 1: I DO G := G + I;

I

KAN[GJ := O;v := 1 •

,

R := Gd l-f"' V := v + 1;R := (R

*

G)MODP;I

"

11\ Y

_P

11\ ... IRI 1? "-. '1 : = KANe IR 11-1;

t-t

= " . KANCI RI.1 S1

...

-v

=

_"

_N , ,IIy.

_"".

<

R

=

-1 v S ODD?> N

"

,

; SET ALL KANCJJ= 0 TO KAN[JJ = I ; t

-I

G := -R

*

G; [

The following remarks concerning Figure 2 are in order.

I. Information about the candidates J

=

±2, •.. ,±s for being a generator of GF(p) is given by the array KAN; one has

KAN[J] < 0 • ±J is not a generator

_,

KAN[J]

=

0 • ±J is being checked,

KANeJ]

=

• ±J has not been checked yet.

2. The operation X MOD Y delivers a value in the range [-~Y,+~Y].

3. Except for the last statement G := -R

*

G in Figure 2 the numbers G, v and R satisfy the relation GV

=

R'MDD P.

4. The number G that is ultimately produced is the smallest generator in absolute value.

5.5.4. The relation between the P₂ and.the w j

In this subsection we derive the desired representation of the P

2 in terms of the w.

=

2cos(2~j).

Use is made of the notation

~

:= erri/p and

~(k)

:=

~k;

J P

furthermore we let g denote a generator of GF(p) and we take h E: {g,p-g}

to be even.

In view of the the definition of P we then have the following equations:

t . (50) = ( 2+1) ( t + l ) ~ g - ~ -g 2 2 d g ) - d-g ) = dh2+1) _ ~(_h2+1)

dh

t) -

d-h

t) =

=

+

If we define the exponents ind(n) E {O,l, •.• ,s-l} by

ind(n) := min{m E :IN u {O} Igrn _ ±n mod p} ,

then it follows that also h ind{il)

=

±n mod p. Hence we can rewrite the last member of (50) as

(51) _PQ, = (-1) g

~h

g (h (2j - 1»0 +9- I;;.(Q,(.h 2J - 1))-l..} = j-l

= (-I )g fh {1;;(hQ,+ind(2j-I» + l;;(hQ,+ind(2j-l»-1 } = j=1

= (-I )g fh w

ind(~h)+ind(2j-l)+Q,-1

.

j=1 g

It now becomes apparent

instead of

conjugates were chosen as

w(~):=

gJ why in 3.4. 1 a)

(k)

= Wij : one then has pQ, ="P t+k (i) w. J w . . instead of 1+J g (k) 0 P

t - PQ,+ind(k)"' which we would havehad otherwise. However, expression (51) is still not easy to work with. But we can draw another conclusion from it by rewriting (51) as

(52) .£

=

(-I)g A W

here A 1S a matrix whose elements consist of zeroes and ones with the

iTA = !h iT. This equation, together with (52) and taking into account that i T

~

= -1, yields

(53) _J.. T

_{.e = (-}

I )g+I

.!

h •

Now we easily deduce from (53) that the units p~ (t=l, ••• ,s) form an in-tegral basis if and only if h = 2 or equivalently if{p9- [9-=1, ••• , s} =

{We Ij=l, ••• ,s}, which in turn is the same as g= ±2, i.e. condition

J

VI holds.

5.6. Some final remarks 5.6.1. Some remarks by Moon

Moon ([16, p. 36J) mentions two inequalities from Hua and Wang's work (cf.[IO, pp. 487-488J), namely (54) (55) In(j)_t

I

~ c (IF )_s

I

ntl- s-11 In (j)

I

_{~ e}-(2t-l)c(lF ). s t (2 ~ j ~ s) , (2 ~ j ~ s) •

He remarks that in practice the right-hand side of (55) is often greater than the bound in (54). As an example he takes t = 1, s = 3, (whence N = 1692) and finds .247 and .034 as the approximate values of the bounds in (55) and (54) respectively. He further remarks that (54) corresponds to (16) in Theorem 2 and (55) to (17) in Theorem 3, while Theorem 3 was meant to prove condition (16) of Theorem 2. We like to comment on these

remarks of Moon.

First of all we recall that Hua and "t.Jang use the notation c(IF) for any

s

constant depending only on IF . This kind of notation is probably the

s

explanation why Moon takes both constants c(IF ) in (54) and (55) to

s

be equal to c

1 as defined by (14). Apparently this is wrong as the correct 2c

1

values in (54) and (55) are c(IF

s)

=

e and c(IFs)

=

cl respectively.

Next we observe that (55) does not imply (54), but (18) does. Indeed,

Intl-l s I

n~j)

I

( min

{In~j)

I})S-l = IT ;:: j=2 2:>;j:>;s 'by (I2), while

In~i)

I 2c₁

{I

n(j)

I}

:>; e m1.n

.

2:>;j:>;s t

Substituting the correct value for c(IF ) in (54) we obtain approximately

s

.4 instead of .034. Furthermore,we find c₁ ~ l.4 and hence to ~ -.1.

Finally, Moon concludes from his experiments that his bounds (that is -(2t-l )c

1

and e ) get closer to e'ach' other when t

1.n-creases. As both riumbers tend towards. zero as t -+ 00,' this 1.S not

surprising.

5.6.2. Other real algebraic fields

Besides the examples in section 3.4 (cyclotomic fields and Dirichlet fields), other real algebraic fields are mentioned in [11. Essentially Hua and Wang consider three types of fields different from the ones in 3.4; we review them briefly here.

i) General cyclotomic fields. W

=

~(~

+

~-1),

where

~q

is a

primi-s q q

i

tive q-th root of unity and q ~s a prime power, q

=

p say. Then 27T

W

=

Q(2 cos(--)) has degree s

=

~~(q), where ~ is defined by (49);

s q

i-I

s_ ::c.;

!

(p-I)p . We add here that one may take

W:= {w

O:= I} U {Wj := 2 cos (21tjjq) Ij = I, ••• ,~~(q) - I}

as an integral basis of W (see [4, pp. 201-202]) and

s

_ ~2j+1

q

_I

I < 2j + 1 < q - I , gcd(2j + 1,q) = 1}

as a set of independent units (cf.[13, pp. 84-85J), ~p being a primi-tive p-th root of unity.

ii) Higher order root fields. For example,

~(~)

is a real algebraic number field of degree s

=

4.

iii) Generalized Fibonacci "fields". Actually, these are not fields; what is meant is the set of generalized Fibonacci numbers F. defined by

J F := I , 5-1

••..••• ' :=

F 5-2 := 0, F s+m := FS+m-1 + •••• + Fm+1 + Fm (m ~ 0) •

These numbers are used to obtain N and a as follows:

N := F

5.6.3. Summary and conclusion

However complicated the theory used by Hua and Wang may be (their error estimates are based on the theory of discrepancies which we did not even mention here), their method can be applied without much difficulty. Although one cannot prescribe the number of nodes N in advance, the order of magni-tude of Nt is roughly determined by the choice of t. In this section 5 we have made several remarks which can be seen as minor improvements of the

theory of Hua and Wang and Moon's algorithm. However, these improvements are not such as to enable the construction of cubature formulas with Nt nodes in less than O(log(N

t)) elementary operations. Hua and Wang [1 OJ show that cubature formulas can indeed be constructed using O(log(N_t)) operations. This number of operations is crucial, as the application of a cubature formula with N nodes requires N function evaluations, hence OeN) operations. Other number-theoreti.c methods to solve (8) so far did not achieve better results; more than O(N) (cf. the footnote in subsection 3.3.2) operations are needed instead of O(log(N

t)). Hence, the cost of finding a cubature formula by any other number-theoretic method known so far is more expensive than its application. We end with two important remarks of Hua and Wang, taken from [1J. The first one is that they ob-served that the "simple" cyclotomic fields (as given in subsection 3.4.1) performed better than all other applications of their method. The second one is that the "simple" cyclotomic fi~ld method is, of course, restricted to dimensions s

=

p; 1 , but seems good enough to be applied to

dimen-. * * , ( 1 f d'

s~ons s <s, s be~ng close to s. One then computes a formu a or

~men-*

sion s and deletes the last s - s coordinates in order to obtain a cuba-ture formula for dimension s .

*

In this respect a lemma of Korobov (cf.[1t, Lemma 8J) which states EU C Ea is of importance.)

*

s

6. Refe~ences

[I] Cheung, W.C., L.K. Hua and Y. Wang, Number-theoretic methods in the approximation of multidimensional integrals, Acta Math.Appl. Sinica

l

(1978), 106-114 (in Chinese).

[2] Cohn, H., "A second course in number thoery". Wiley, New York -London, 1962.

[3] "Encyclopedic dictionary of mathematics". (by the Mathematical Soci-ety of Japan), MIT Press, Cambridge, Massachusetts and London, England, 1977.

[4J Fricke, R., "Lehrbuch der Algebra", III. F. Vieweg, Braunschweig, 1928. [5] Haber, S., Numerical evaluation of multiple integrals, SIAM Rev. 12

(1970), 481-526.

[6] Haber, S., Experiments on optimal coefficients, in "Applications of number theory to numerical analysis". (ed. by S.K. Zaremba), Aca-demic Press, New York, 1972, pp. 11-37.

[7J Hlawka, E., Uniform distribution modulo 1 and numerical analysis, Compositio Math. ~ (1964), 92-105.

[8J Hua, L.K. and Y. Wang, On diophantine approximations and numerical integrations (I), (II), Sci. Sinica

II

(1964), 1007-1010.

[9J Hua, L.K., and Y. Wang, On numerical integration of periodic functions of several variables, Sci. Sinica.!.i (1965), 964-978.

[10] Hua, L.K. and Y. Wang, On uniform distribution and numerical analysis (I), Sci. Sinica ~ (1973), 483-505.

[]]J Keast, P., Optimal parameters for multidimensional integration, SIAM

J. Numer.Anal.l£ (1973), 831-838.

[]2J Korobov, N.M., On approximate calculations of multiple integrals, Dokl.Akad. Nauk SSSR 124 (1959), 1207-1210 (in Russian). [13J Korobov, N.M., "Number-theoretic methods in approximate analysis".

Fitzmatig, Moskow, ]963 (in Russian).

[l4J Lang, S., "Cyclotomic fields".Springer-Verlag, Berlin, 1978.

[]5J Maisonneuve, M., Recherche et utilation des "bons treillis". Program-mation et resul tats numeriques, in "Applications of number theory to numerical analysis". (ed. by S.K. Zaremba), Academic Press, New York 1972, pp~"-121-200~

[]6J Moon, Y.S., "Some numerical experiments on number theoretic methods in the approximation of multidimensional integrals". Tech. Rep. 72, Department of Computer Science, University of Toronto, To-ronto, 1974.

[]7J Niederreiter, R., Methods for es.timating discrepancy, in "Applications of number theory to numerical analysis". (ed. by S.K. Zaremba), Academic Press, New York, ]972, pp. 203-236.

[18J Niederreiter, H., Quasi-Monte Carlo methods and pseudo-random numbers, Bull.Amer. Math.Soc. 84 (1978), 957-1041.

[]9] Saltikov, A.I., Tables for evaluating multiple integrals by the method of optimal coefficients, U.S.S.R. Comput.Math. and Math. Phys. 3

(1963), 181-]86.

[20J Zaremba, S .K. Good lattice points, discrepancy and numerical inte-gration, Ann. Mat.Pura Appl.

1l

(1966), 293-317.