Lyness - Introduction to Lattice Roles and their Generator Matrices 1989

IMA Journal of Numerical Analysis (1989) 9, 405-419

An Introduction to Lattice Roles and their Generator Matrices

J. N. LYNESS

Mathematics and Computer Science Division, Argonne National Laboratory,9700 South Cass Avenue, Argonne, IL 60439-4844, USA

[Received 31 May 1988 and in revised form 30 October 1988]

For the one-dimensional quadrature of a naturally periodic function over itsperiod, the trapezoidal rule is an excellent choice, its efficiency being predictedtheoretically and confirmed in practice. However, for 5-dimensional quadratureover a hypercube, the s-dimensional product trapezoidal rule is not generally costeffective even for naturally periodic functions. The search for more effective ruleshas led first to number theoretic rules and then more recently to lattice rules. Thissurvey outlines the motivation for and present results of this theory. It isparticularly designed to introduce the reader to lattice rules.

0. Introduction

LATTICE rules are generalizations of the number theoretic quadrature rules ofKorobov (1959) for integration over an s-dimensional hypercube. They were firstintroduced by Sloan & Kachoyan (1984, 1987). The rules use an abscissa set ofpoints that lie in the intersection of the hypercube and an integration lattice. Thetheory presented previously (some of which is described in more detail in Section7) was based firmly on the geometric character of the lattice. Up to now, little usehas been made of the algebraic properties of the lattice generator matrices.

In this paper we develop a new approach, based on the definition of a latticethrough its generator matrix. This has led to two new characterizations of anintegration lattice. One (Definition 3.3) demands that it be a sublattice of a scaledversion of the unit lattice. The other is simply that its reciprocal lattice shouldhave a generator matrix composed of integers (Theorem 5.2). These definitionshave led in turn to a self-contained and relatively uncomplicated theory; thisinvolves a particularly simple proof of the Poisson summation formula for latticerules (Corollary 6.1).

In Section 1 we describe the one-dimensional lattice rule (the trapezoidal rule)and so cover completely the one-dimensional specialization of all the theory inthis paper. In Section 2 we introduce the problems that arise when one attemptsto generalize the one-dimensional theory. Sections 3 to 5 then deal with the basictheory of lattice rules; we treat lattices, integration lattices, lattice rules, thereciprocal lattice, and integer lattices in that order. This material is mainly aquestion of interrelated definition. Then in Section 6 we return to the analysis ofthe quadrature rule error and show that many terms in the Poisson summationformula drop out naturally when it is applied to a lattice rule.

© Oxford Univtnity Prea 1989

by guest on January 27, 2011im

ajna.oxfordjournals.orgD

ownloaded from

http://imajna.oxfordjournals.org/

4 0 6 J. N. LYNESS

Section 7 is devoted to a brief outline of previous and current work on latticerules.

1. The one-dimenskmal background

The statement that the trapezoidal rule is good for periodic functions is oftenmade. In this section we discuss this statement. This provides a suitablebackground for the subsequent introduction to lattice rules.

We shall treat an integrand function f(x) that is ( ^ [ 0 , 1 ] with p s= 0. Let / / beits exact integral over the integration interval [0,1], and let/(x) be the periodiccontinuation of f(x) defined as the sum of the Fourier series of f(x) with respectto this interval. We shall denote the Fourier coefficients by

M for all integer r. (1.1)Jo

Let

Qf=i^f(xj) with i>y=i (i.2)y-i y-i

be a general quadrature rule, and

"'"' -; M>+%t (£)+H - ip(£be the m-panel trapezoidal rule. These are both approximations to / / = ao(f), theexact integral. By substituting the Fourier series of f(x) into (1.2) we find ageneral form of the Poisson summation formula, namely,

Q? = If+ £ «,a)G(e2-te) (1.4)r*0

where the multiplier of the rth Fourier coefficient is

dr(Q) = Q(e2"i") = £ w, exp Qnirx,), (1.5)7 - 1

i.e. the result of applying the rule Q to the indicated exponential function.The Poisson summation formula (1.4) expressed the quadrature error Qf'-If

as a series. We seek to construct Q so that the most significant terms in this seriesdrop out. These are usually the earliest terms. To do this, we seek Q that for afixed value of v annihilates as many as is feasible of the coefficients dr(Q)- Thisturns out to be a well-defined problem. The solution is a one-parameter system ofrules that comprise /?(m) together with all its offset (displaced) versions, withm = v or v — 1. We find

dr(Rim)) = l when r/m = integer, and 0 otherwise. (1.6)

Consequently, by applying (1.4) to the trapezoidal rule, we have found that

R™f = If+ £ aml(f). (1.7)(— — oo



ownloaded from


LATTICE RULES AND THEIR GENERATOR MATRICES 407

This rule then is clearly suitable for functions fix) whose Fourier coefficientsar(f) decay rapidly with r. It is relatively well known that the decay rate isconnected with the degree of continuity of j(x). An immediate demonstration ofthis statement follows by applying integration by parts to the right-hand side of(1.1) to obtain the Fourier coefficient asymptotic expansion

valid when / e C*p)[0,1]. It is, of course, also valid when / e C^^O, 1) in whichcase/^*"1^) and/ ( f l -1 )(l) should be replaced by the appropriate right-hand andleft-hand derivatives respectively. Since J(x) is periodic we have

(1.9)

which, of course, is zero if the (q — l)th derivative of fix) is continuous.

THEOREM 1.1 (See e.g. Copson, 1965; page 21.) When fix) and its first p-\derivatives are continuous,

aril) = (f(p\0+) - / ^ (0 - ) ) / (2 j r i ry + 1 + o(r^p+1)). (1.10)

When the derivatives of f(x) of all orders are continuous, this implies that ar(J)decays more rapidly than any inverse power of r. One can show that when fix) isanalytic in the strip |Imx| < L, there exists M such that

\ar(f)\<Me~Mr. (1.11)

These results about the rate of decay of the Fourier coefficients imply that, whenf{x) is not C , this rate depends on the degree of continuity of fix), and whenfix) is analytic, this rate depends on a measure of the magnitude of the region ofanalyticity.

The statement that the trapezoidal rule is good for naturally periodic functionsis an abbreviation for the following set of statements.

(a) The trapezoidal rule is better for functions fix) whose Fourier coefficientsdecay more rapidly.

(b) These are functions whose periodic continuations fix) have higher degreesof continuity.

(c) The appearance of fix) is more naturally periodic according as /(x) hashigher degree of continuity.

This one-dimensional theory contains three principal features which we shallwant to generalize to s dimensions:

(i) The Poisson summation formula (1.4) which expressed the error Qf — Ifas a sum of terms ar(J)driQ)- This is easy to generalize.

(ii) An implied ordering of the Fourier coefficients. That is, we made thereadily justifiable assumption that ar(J) is likely to be more significantthan a,(J) when \r\ < \t\. There is no self-evident generalization in sdimensions. In Section 2 we simply report three somewhat arbitrary butreasonable possibilities, one of which has been widely studied.



ownloaded from


4 0 8 J. N. LYNESS

(iii) The derivation of the rule Q which 'zeroed out' the coefficients dr(Q)associated with the more significant Fourier coefficients ar(J). In onedimension, the result that /?(m) is the solution has the status of a theorem.In s dimensions, the corresponding nonlinear equations are more compli-cated, and no general solution is known. The reason they are morecomplicated stems from the more involved ordering in (ii) above. Sections2-6 are devoted to describing the nature of ad hoc attempts to findapparently reasonable rules. These include number theoretic rules andlattice rules.

2. Problems arising in the s-dhnenskHial generalization

The J-dimensional generalization of features (i) and (ii) at the end of Section 1has been treated extensively in the theory of number theoretic rules, and wehave little to add. In the context of lattice rules, this theory is covered in Lyness(1988) using the same notation as is used here. Briefly, the one-dimensionalformulas (1.1), (12), (1.4), and (1.5) have obvious ^-dimensional generalizations.We treat the unit hypercube H; [0,1)J; the Fourier coefficients are

JH~"du, du2 • • • du, (2.1)

and the Poisson summation formula takes the form

Qf = lf+2atf)dr(Q) (2.2)re/V)

where as expected

dr(Q) = Q(e2nir-X) = S w, exp (2nir • x,). (2.3)

In (2.2) we have used the terminology re A^to indicate that all components of rare integers. (Later we shall introduce the unit lattice A, to denote precisely thisset of points.) The details about the convergence of the Poisson summationformula are complicated but play no significant role in this theory, /(x) is the sumof the ^-dimensional Fourier series of/(x). This is a periodic continuation of /(x)and may be defined by

/(*)=i 2 2 • • • 2 /(*./,. y^-.y.j) (2.4)C 7i-0 ;j-0 j,-0

where[{**} xk * integer,

^*J 1/ xk = integer.

Here we have used braces to denote the fractional part. This somewhatcumbersome definition of/(x) disguises an intrinsically simple concept, and thedifference between /(x) and /(x) also plays no significant role in this theory.



ownloaded from


LATTICE RULES AND THEIR GENERATOR MATRICES 4 0 9

We now turn to step (ii) of the list at the end of Section 1, that of decidingwhich of the s-dimensional Fourier coefficients ar(J) are the more significant. Weneed this to establish a rule construction criterion, in this case a criterion fordeciding which coefficients dr(Q) should be 'zeroed out'. This problem isdiscussed in some detail in Lyness (1988). Let us define three measures.

P(r) = fxr2 • • • f, where f, = max (1, \r,\)

S(r) = |r,| + \r2\ + • • • + \rs\ (2.5)

M(r) = max (|r,|).

(P, 5, and M are the product, the sum, and the maximum respectively of thenon-zero components of r.) The traditional criterion in this area (Korobov, 1959;Zaremba, 1972) incorporates the view that

ar(f) is more significant than a,(f) when P(r) < P(t).

For functions that have low-order derivative discontinuities, this statement isrealistic. For naturally periodic functions, e.g. /(x) analytic in all components,examination of Fourier coefficients suggests strongly that one should use S inplace of P. The use of M, which leads to the product trapezoidal rule, has little torecommend it. Further information is available in Lyness (1988). Correspondingto each measure, there are indices p and 5 to indicate to what extent a rule Q has'zeroed out' dr(Q)- Thus,

Q has Zaremba index p when dr{Q) = 0 for 1 s£ P(r) < p except r = 0

Q has overall degree 6 when dr(Q) = 0 for 0 < S(r) =e 6 (2.6)

Q has product degree 6 when dr{Q) = 0 for 0 < M{r) =s 6.

Note that in all cases d9(Q) = 1.In one dimension all these three measures become the same, reducing to the

one used in Section 1.Step (iii) of the list at the end of Section 1 involves determining cost effective

rules that satisfy the construction criterion we have chosen.If we choose the measure M(r) above, there is an exact s-parameter solution;

this is the .s-dimensional product of s displaced trapezoidal rules, each having mpanels. The parameters are the displacements which may all be different. Anysuch rule which uses m' function values of / is of product degree m — 1.

In contrast, the measures P(r) and S{r) are difficult to handle. An analyticsolution requires solving large, somewhat ad hoc sets of nonlinear equations. Asis usual in this sort of situation, progress continues only on a trial and error basis.One makes an educated guess about the form of what one thinks might be asuitable rule. Then one tries to choose the incidental parameters to maximize thedegree 6, or the Zaremba index p. And, of course, progress in this search will bequicker, at least in the early stages, if one chooses the rule form so that 6 or p isrelatively easy to measure - or to bound.



ownloaded from


4 1 0 J. N. LYNESS

In general, at first sight, for every candidate rule Q, one has to calculate

= £ w^i (2.7)

for various values of r = (rlv.., r,), searching for a rule for which many of these arezero. The first 'simplification' we can make is to consider only rules whoseone-dimensional projections are /V-panel trapezoidal rules. Then at least some ofthe dr(Q) are zero, i.e. those having 5 — 1 zero components and one nonzerocomponent less than N. Such a rule would use some, but not necessarily all, ofthe abscissas on the lattice (l/N)Ao, i.e. abscissas for which all components areinteger multiples of 1//V. We start by considering rules using TV points which havethis property.

An obvious candidate is a number theoretic rule

Qf = {VN)'2KjzlN), (2.8)i-i

where z = (zx,..., z,) has integer components. Let us assume for this illustrationthat TV is a prime number, and 0 < zt < N for all i. It is easy to verify that allone-dimensional projections are TV-panel trapezoidal rules. And with thisstructure we find many zero coefficients. Specifically,

If we consider only rules of the form (2.8) defined by parameters TV and z, we candetermine which dr(Q) are zero simply by seeing whether r • z/N is an integer, aprocess that is much faster than carrying out the sum in (2.7).

There is a vast literature devoted to number theoretic rules. Major contributorsinclude Korobov (1959, 1960), Hawka (1962), Zaremba (1966, 1972), Niederrei-ter (1978), Hua & Wang (1981), and Haber (1983). A reference list with overseventy entries may be found in Niederreiter (1988). We now return to thecomparatively simple general theory of lattice rules.

3. Lattices and integration lattices

A lattice rule is a quadrature rule that uses as abscissas a finite selection ofpoints of a particular kind of lattice. In this section we introduce lattices.

An 5-dimensional lattice is an infinite array of points in j-dimensional spacehaving certain regularity and periodicity properties. The simplest example is theunit lattice, denoted by AQ which comprises all points having integer components.A scaled unit lattice denoted by 6A^, comprises all points all of whosecomponents are integer multiples of «5. It is convenient to define the unit vectorse) (J = 1,..., s). The ;th component of et is 1 and the other components are all zero.Thus the scaled unit lattice 6AQ comprises all points p of the form p = E/_iwith ft = (/J,,..,/i,)6 4



ownloaded from



A widely used definition of a lattice follows.

DEFINITION 3.1 A lattice A comprises a set of points such that

p,qeA => p + qe A andp-qe A (3.1)

and there exists no limit point, i.e. there exists positive e(A) such that

\p-q\** e(A) unless p = q. (3.2)

Corresponding to every s-dimensional lattice A is a non-singular 5 X 5 generatormatrix A . The rows of A , namely, ar = (ar>1, ar2,-,ar^), are elements of Achosen in such a way that

peA O p = 2^r (3.3)

for some A = (A^..., A,) e AQ. The rows of a generator matrix are sometimestermed a 'set of generators of A'. The right-hand relation of (3.3) may be written

(3.4)

LEMMA 3.1. Any nonsingular s xs matrix A generates an s-dimensional lattice A.

Proof. Qearly the set of points (3.3) satisfy (3.1). When A is nonsingular wehave

\iA\2^o\M2^o VA#0,

where a is the smallest singular value of A. This establishes (3.2).

These results about generator matrices are introduced and discussed in greaterdetail in most texts on the geometry of numbers (see, e.g. Cassells, 1959:Chap. 1).

There are many different generator matrices corresponding to the same latticeA. To see this we note that the set of points defined by (3.4) is unaffected whenone replaces

fl'by"a' (3.5)ar by ar + \ia, (r*t, /z = integer).

So one may apply elementary row operations (using integer coefficients) to agenerator matrix without altering the lattice A. It can readily be shown that anygenerator matrix of A may be obtained from any other generator matrix of Ausing elementary row operations with integer coefficients. It follows that:

(i) if all elements of one generator matrix of A are integers, then all elementsof any generator matrix of A are integers;

(ii) the transformations do not alter |detA| which is a constant. Thus,N = N(A) = |det A\~l depends on the lattice.

The interpretation of this constant follows from treating the matrix A as a spacetransformation p T = ATAT from A-space, in which is embedded the AQ lattice, top-space, in which is embedded the A lattice. These lattices are in one-to-onecorrespondence with each other. Thus |det A\ is the ratio of the point densities in



ownloaded from


4 1 2 J. N. LYNESS

these two lattices. Since the point density of A) is 1, |det Ap 1 is the point densityof A. We shall state a particular case of this fundamental result in Theorem 3.1.In some of the proofs below, we denote a scaled up or down version of a lattice Aby kA where A: is a nonzero scalar. (The corresponding generator matrices are Aand kA.)

We shall now define a special class of lattice.

DEFINITION 3.2 An integration lattice A is a lattice that contains the unitlattice AQ.

The rest of this paper, and the theory of lattice rules, is exclusively concernedwith integration lattices.

We shall see below that, when A 3 AQ, the structure of the lattice is periodicwith period 1, and IdetAp1 is precisely the number of points N(A) in the unithypercube. We restate this critical result as a theorem.

THEOREM 3.1 When A is a generator matrix of an integration lattice A, i.e.A 3 AQ, then the number of points of A in the hypercube [ 0 , 1 / is precisely

Definition 3.2 is a little cumbersome in some cases because, to verify that a set ofpoints forms a lattice, one has to verify the criterion (3.2) about the absence oflimit points. It is convenient to replace this criterion as follows.

We define

djx = min {\pf\ : Pj # 0, p e A}.

i.e. djx is the smallest nonzero xt component of any lattice point. For a generallattice, dj need not exist. But since A => AQ, it follows that dj exists and is aninteger. We may take d = lcmy (dt) in the following definition.

DEFINITION 3.3 An integration lattice A comprises a set of points such that

p,q G A => p + q e A and p — q e A

and there exists an integer d such that

d-'Ao^A^Ao. (3.6)

This means that every point of A lies on the scaled unit lattice, and that everypoint of AQ is also a member of A. The advantage of this definition is that it isself-contained; there is no reference to limit points and it does not require theprior definition of a lattice (except the scaled unit lattice).

Yet another definition of the same concept follows.

DEFINITION 3.4 An 5-dimensional integration lattice A may be denned by any set

{n,,zi :i-l,...,t},

where t and n( are positive integers and zt e AQ. A comprises all points of the form

P = 2 W«i + X WJ (3-7)



ownloaded from



where A, and /iy may be any integers. Note that the same point p may be definedusing different selections of A, and \Xj. Nevertheless, it is trivial to see that, if wetake d = n^n2 • • • n,, all the postulates of Definition 3.3 are satisfied.

In Definition 3.4, we employed t + s parameters A, and \i-t to describe anj-dimensional system. Naturally, when the generator matrix definition (3.4) isused, only s parameters are required. The question arises as to what condition ona generator matrix A is required to ensure that A is an integration matrix. Theanswer is that A~l should have only integer elements. This is one of the mostcrucial facts of the theory and is proved in Section 5.

We have mentioned that the integration lattice A having a generator matrix Ahas precisely N = |det A\~l points within the unit hypercube H = [0,1)1. Let usdenote these points by

A n / / = {x, ,...,*„}. (3.8)

Since the lattice contains the unit lattice AQ, it follows that every point in thelattice has the form

/>=** + 2 V ; with A: e [1 , N]. (3.9)y-i

The lattice is, of course, infinite in extent. But it is periodic with period 1 in eachdirection. If one imagines all space to be subdivided into unit hypercubes, one ofwhich is Ily-i [A;, A, + 1) for integer Ay, the pattern of points of A within eachhypercube is identical to the pattern in the unit hypercube [ 0 , 1 / . It is this basicset of points that is used in the lattice rule.

4. The lattice rale Q(A)

We are now in a position to define a lattice rule Q(A). Given an integrationlattice A, we define the abscissa set A(Q) as the set in (3.8), i.e.

nH. (4.1)

Then the lattice rule is

l ^ 2 f(p) (4-2)

with/(x) being the periodic continuation of f(x) defined in (2.4). Here v(Q) = Nis the order of the set A(Q). This is the number of function values of/required toevaluate Qf. (The number of function values of / required exceeds \{Q) by atleast T — 1 since all lattice rules require /(0) which is the average of f(x) takenover all ¥ vertices of H.)

A completely self-contained definition of the abscissa set for a lattice rule is thefollowing. We define the 'fractional part' of a vector in a standard way, i.e.y = {x} when y e H and y — x e AQ. The abscissa set comprises N points xh with

xteH and x, E /V"'AI

having the property that

*,-,*, 6i4(G) => {or, + xt) eA(Q) and {x, -x,} e A(Q). (4.3)



ownloaded from


414 J. N. LYNESS

The abscissa set A(Q) forms a group of order N under the operation 'plus' wherex 'plus' y = {x + y}. This aspect of the abscissa set is treated in considerable detailin Sloan and Lyness (1989).

A graphic description of the construction of a lattice rule is the following. Onechooses an integer mesh ratio d and chooses t points Xi,...,x, on the latticed - 1 /V Then one constructs the lattice A by including 0 and all points «,,j = l,...,s, and all points obtainable from any existing pair using ordinary vectoraddition and subtraction. The abscissa set A(Q) is the set of N points of A lyingin H = [0 , iy, i.e. those whose components all lie in [0,1). And the rule is givenby (4.2) above. Incidentally N is a factor of or a multiple of d. In this process, ifwe had chosen / = 1 we would have found a number theoretic rule (2.8) above.However, with O l w e might have found a rule not expressible in that form.

If one returns to Definition 3.4 of an integration lattice, one might define

*k=zklnk (k = l,...,t),

and generating the lattice then leads to the following form for a lattice rule.

THEOREM 4.1 (Sloan & Lyness, 1989) The form {t-cycle)

Qf=

where z, e AQ and nt and t are positive integers, is a lattice rule, and all lattice rulescan be written in this form. This is called a t-cycle form if n^,..., n, are each greaterthan 1.

Note that, since f(x) =/({*}), the effect of the generators ejt j = l,...,s, isautomatically taken into account. It is readily verified that all the points in A n Hare included in this finite sum. The only nontrivial problem concerns thecircumstance that points may be counted more than once. However, one mayshow that if this set includes 0 more than once, say k times, then it includes everyother point precisely k times, and vice versa. In this case we term form (4.4) a^-repetitive form of Q. While it would be ridiculous to program directly the^-repetitive form, it can be used with confidence to derive theoretical results. Weshall make use of this form in Section 6.

To obtain a form (4.4) directly from the /4-matrix, one sets

Zj/n, = aj, (4.5)

where Z/ e A>- This is possible since we recall each element of A is a rationalnumber. Using these values of zt and nt in (4.4) gives an s-cycle form that is likelyto be repetitive. This is easy to prove, but not completely self evident. One has toconvince oneself that the integers ns provided by (4.5) are suitable as upper limitson the summations in (4.4).

5. The reciprocal lattice AL

Corresponding to every lattice A is its reciprocal (polar, dual) lattice,sometimes denoted by Ax . It may be defined as follows.



ownloaded from



DEFINITION 5.1 r e Ax if and only if

r 'p = integer for all p e A. (5.1)

One has to show that, given A, points r defined above exist, and then that they doindeed form a lattice. This is essentially established in Theorem 5.1. After this, itis straightforward to derive a host of relations of which the following are only aselection.

(A^ = A; (dAoy = d-1Ao( ' }

Naturally, the generator matrices of a lattice and its reciprocal lattice are closelyrelated.

THEOREM 5.1 When A has generator matrix A, then Ax has generator matrixB = (ATy\Proof. Denote by Ax the reciprocal lattice of A and by A(B) the lattice whosegenerator matrix is B. We have to show Ax = A(B).

(a) When r e A(B) and p e A we have

p • r = kABTfiv = kfiy = integer

which, according to (5.1), is the condition that r e Ax.(b) When r e Ax, since aj e A we have aj • r = /^ = integer and so

giving

This, according to (3.3), is the condition that r e A(B). O

We now treat the case when A is an integration lattice; then B has a specialproperty described below.

THEOREM 5.2 A nonsingular s xs matrix A is the generator matrix of anintegration lattice A if and only if B = (AT) -1 has only integer elements.

Proof. Nearly all the proof follows from applying the second and third membersof (5.2) to A and AQ giving

A 2 A o O A-'-cAo. (5.3)

(a) If A(A) is an integration lattice, it satisfies the left-hand relation. Ax whosegenerator matrix is B then satisfies the right relation. Thus all elements ofAx have integer components. Since the rows of B are elements of A±, itfollows that the elements of B are all integers.

(b) Since B is nonsingular, Lemma 3.1 assures us that A(B) exists and since ithas integer elements, Ao3 A(B). Since A = (B7)'1, A(A) is the reciprocallattice of A(B). Thus A(A) 3 AQ and so is an integration lattice. D

The reciprocal lattice of an integration lattice, like the integration lattice, isperiodic in structure. We note that, when N = |det A\~l is the number of points of



ownloaded from


4 1 6 J. N. LYNESS

A lying in [0,1)J, we find that AL has period N. That is, Ax has Ns~x points rylying in [0fNY and this pattern of points is repeated in every hypercubeII/=i [kjN , (A; + 1)N) such that XEAQ. Much of the burden of constructing latticerules lies in examining this reciprocal lattice to find points near the origin. Thelarge number N*~* of points that may be involved in a cursory search adds to thecomputational difficulty of this task.

6. The Poisson summation formula of a lattice rale

The reader cannot have failed to notice that, after Section 2, no resultspertaining to the accuracy or feasibility of lattice rules have been presented.Sections 3,4, and 5 dealt with the concepts of lattice, lattice rules, and reciprocallattice, without mentioning at all why such concepts are relevant. In this sectionwe return to the analysis.

We recall that our error analysis is based on the Poisson summation formula(2.2), namely

Of -V = 2 arif)dr(Q),

and our concern is to reduce this error by choosing Q in such a way that thecoefficients dr(Q) in the significant terms vanish. The basic result of this section isTheorem 6.1 which links these coefficients to the reciprocal lattice Ax. Surpris-ingly enough, the most straightforward proof of this theorem stems from theredundant but graphic Definition 3.4 of the lattice and the corresponding f-cycleform (4.4) of the lattice rule. We recall that dr(Q) is the result of applying Q tothe function exp(2jtir -x). Replacing/(r) by this function in (4.4) gives

</,((?) = G(exp2ra>-x)

THEOREM 6.1 Let Q be a lattice rule having lattice A. Then

whenreAx, (6.2a)whenr$Ax. (6.2b)

Comment. The search for efficient lattice rules is based on constructing rules forwhich dr(Q) = 0 for r satisfying properties outlined in Section 2. Thus the wholethrust of the construction is shifted to examining properties of the reciprocallattice. A second comment is that (6.2a) is much easier to establish than (6.2b).

Proof. Expanding (6.1) we find

dr(Q) = Qie2""-*) = II ^ 2 exp (2**7* ̂ ) . (6.3)

The kth term of this product is 1 or is 0 according as zk • r/nk is an integer or isnot an integer. Consequently, there are only two possible values for dr(Q). These



ownloaded from



arewhen zk • r/nk = integer for all k = 1,..., /,

(6-4)otherwise.

We now apply (6.4) to the reciprocal lattice Ax.(a) When r = A±, then by Definition 5.1 r«x = integer for all x e A. Since

each of zk/nk is an element of A, it follows that zk • r/nk = integer,it = 1,...,/, and in view of (6.4) dr{Q) = 1.

(b) Conversely, when dr(Q) = 1, each term in the product (6.3) is 1. Thuszk • r/nk = integer, k = 1,..., f. It follows that, for a general abscissa x of(4.4), x • r = E*=i/*z* • r/nk is also an integer. Since x • r is an integer forall x e A, it follows that reAx.

Parts (a) and (b) above establish that reAx is a necessary and sufficientcondition for dr(Q) = 1- In view of (6.4) the only other value which dr(Q) cantake is zero. This establishes (6.2). •

COROLLARY 6.1 For a lattice rule Q(A)f, the Poisson summation formula is

where Ax is the reciprocal lattice of A.

We may now reconsider the situation described in Section 2 where weintroduced the Zaremba p-index and the trigonometric degrees of a rule Q. If weare dealing with lattice rules Q(A), these degrees are simple properties of thereciprocal lattice A±. The search for rules that are cost effective using any ofthese definitions may be accomplished by searching reciprocal lattices. Each pointon such a lattice corresponds to an error term. We look for lattices whose latticepoints are not close to the origin; here the term close may be defined in terms ofP(r), S(r), or M(r) of Section 2.

7. Other work on lattice rules

The author is developing further theory based on the approach of this article,that is, exploiting various properties of the generator matrices A and B of thelattice A and its reciprocal. This work seems at present to be almost orthogonal totwo other areas in which significant progress has recently been made.

The first, chronologically, is work by Sloan & Kachoyan (1984, 1987). Aconvenient review of that work is given in Sloan (1985). This generalizes to latticerules some of the deeper number theoretical results previously established only inthe context of number theoretical rules (sets of good lattice points). An exampleis a generalization of a result due to Hua & Wang (1981); in the notation of (2.5)and (2.6), this is to the effect that

Qf-If^cd(s, *)p~'r(l + lnp)tr~1

when f(x) has Fourier coefficients satisfying



ownloaded from


418 J. N. LYNESS

The second area, currently under investigation, exploits the group character(see (4.3)) of the abscissa set A(Q). This leads to a canonical form for a latticerule expressed in the form (4.4). Specifically, every rule has a set of positiveinteger invariants nx,...,ns. Each divides the previous one, and Qf, expressed inform (4.4) using these values of nh is nonrepetitive. The rule is termed to be ofrank t when n, > 1 and nt = 1 for / > t. The number theoretic rules of Korobov areall rank 1. Other rules, including the product trapezoidal rules, are of higherrank. These ideas are developed in a sequence of articles (Sloan & Lyness, 1989)and summarized in Sloan & Walsh (1988).

Acknowledgment

It is a pleasure to acknowledge the cooperation of Drs Tor S0revik and PatKeast in preparing this article. This work was supported by the AppliedMathematical Sciences subprogram of the Office of Energy Research, USDepartment of Energy, under contract W-31-109-Eng-38.

REFERENCES

CASSELS, J. W. S. 1959 Introduction to the Geometry of Numbers. Berlin: Springer-Verlag.COPSON, E. T. 1965 Asymptotic Expansion. Cambridge: Cambridge University Press.HABER, S. 1983 Parameters for integrating periodic functions of several variables. Math

Comp. 41, 115-129.HLAWKA, E. 1962 Zur angenaherten Berechnung mehrfacher Integrale. Monatsh. Math.

66, 140-151.HUA Loo KENG, & WANG YUAN 1981 Applications of Number Theory to Numerical

Analysis. Berlin: Springer-Verlag.KOROBOV, N. M. 1959 The approximate computation of multiple integrals (Russian).

Dokl. Akad. Nauk. SSSR 12A, 1207-1210.KOROBOV, N. M. 1960 Properties and calculation of optimal coefficients (Russian). Dokl

Akad. Nauk SSSR 132, 1009-1012; Soviet Math. Dokl. 1, 696-700.LYNESS, J. N. 1988 Some comments on quadrature rule construction criteria. In:

Numerical Integration III (G. Hammerlin & H. Brass, Eds), ISNM vol. 85. Basel:Birkhauser Verlag, 117-129.

NIEDERREITER, H. 1978 Quasi-Monte Carlo method and pseudo-random numbers. Bull.Amer. Math. Soc. 84, 957-1041.

NIEDERREITER, H. 1988 Quasi-Monte Carlo methods for multidimensional numericalintegration. In: Numerical Integration III (G. Hammerlin and H. Brass, Eds), ISNMvol. 85. BirkhSuser Verlag, 157-171.

SLOAN, I. H. 1985 Lattice methods for multiple integration. J. Comp. Appl. Math. 12,13,131-143.

SLOAN, I. H., & KACHOYAN, P. 1984 Lattice for multiple integration. Proc. Cent. Math.Anal. Aust. Natl. Univ. 6, 147-165.

SLOAN, I. H., & KACHOYAN, P. J. 1987 Lattice methods for multiple integration: theory,error analysis and examples. SIAM J. Numer. Anal. 24, 116-128.

SLOAN, I. H., & LYNESS, J. N. 1989 The representation of lattice quadrature rules asmultiple sums. Math. Comp. 52, 81-94.

SLOAN, I. H., & WALSH, L. 1988 Lattice rules - classification and searches. In: NumericalIntegration III (G. Hammerlin and H. Brass, Eds), ISNM vol. 85. Birkhauser Verlag,251-260.



ownloaded from



ZAREMBA, S. K. 1966 Good lattice points, discrepancy and numerical integration. Ann.Mat. Pura Appl. 73, 293-317.

ZAREMBA, S. K. 1972 La me'thode des "bons treillis" pour le calcul des int6gralesmultiples. In: Application of Number Theory to Numerical Analysis (S. K. Zaremba,Ed.) London: Academic Press, 39-119.



ownloaded from


Lyness - Introduction to Lattice Roles and their Generator Matrices 1989

Documents

Transcript of Lyness - Introduction to Lattice Roles and their Generator Matrices 1989