Random quadratures of improved accuracy

RANDOM QUADRATURE!3 OF IMPROVED ACCURACY* S.M. ERMAKOV (MOSCOW) (Received

29

June

1963)

Since a Monte-Carlo method is essentially statistical, the apparatus of mathematical statistics can be widely employed when such a method is used. Gn the other hand such a method possesses a number of specific features, due to the fact that we construct artificially a random magnitude, and the construction is not as a rule unique. These features bring a Monte Carlo method close to classical computational methods, and in particular, to the theory of mechanical quadratures. Several quadrature formulae with random base-points, which enable the error of a Monte Carlo method to be reduced in certain cases, are discussed in [l] L [3]. Interpolation-quadrature formulae with random basepoints are obtained in [4] under extremely general assumptions. The present article discusses quadrature formulae with random basepoints of improved accuracy, which are in a sense an analogue of Gaussian quadratures. The size of the error of the quadrature formulae obtained is estimated by evaluating the dispersion of the corresponding quadrature sum. 1. As in [4], we shall consider coefficient oj .of the function F: aj

=

the

problem

of

vych.

mat..

4, NO. 3.

(1)

Vj (Q) f (Q) dQ*

We shall assume that D is a region of k dimensions, summable in D and that the functions g+,, (m = 0. 1. normalized in D.

Zh.

the Fourier

s

D

l

evaluating

550-554. 213

1964.

that f is square . . . , N) are ortho-

S.M.

214

To evaluate

the

integral

(l),

Ernakov

we form the’ quadrature

Kn,jIfl= $ Ai (Qo~QI,...,Q,)f (QJ,

sum Kn, j [fl :

Qi~D.

(2)

i=o

We assume that

is accurate Qi are

the

for

situated

quadrature

the

formula

functions

randomly

q+,(Q)

in D with

(w = 0,

. . . . N),

1.

probability

density

that

the

Points

G(Qe,

Ql,.

. . ,Qnlr

and that M. o. K,,j

We represent

f(Q)

orthonormalized

by a Fourier

functions

[f] = aj.

series

including

qo,

(4)

in the ql,

complete

system

of

. . . . ‘pN:

f(Q) = i cr,cp,(QL m---o and we evaluate follows

from

where pl

111is

Al,,,

tained

in

of

coefficient

K,,j [cP,L

of

[cp,]

=

formulae

with <

of

the

It

random magnitudes

1~~1, u,,, = ~/ul;,,,j 1 held

for

111= sole

(-

i)j

smaller

1, are

Q,) can be uniquely imposed on the distribution

and the

~,,j[fI.

the

IT,,,]. The equations

quadrature

formulae

ob-

[4].

n < N and DK,, j [q,,,]

K,,j

random magnitude

correlation

a1 = I/o&j

AntO,,, . ..a tions

the

that

= 0 and DK,,j

The quadrature which

dispersion

and (5)

the

Kw,j 1~11 ’ and n = N,

the

(4)

(5)

dispersions,

obviously

*.

.T

‘Pj_1

(let lj TU (Qi)s imposed

i.e.

those

interest.

for

When n = N the

determined with relatively weak restricof the points Qi. In this case

det II f (Qi)v TO (Qi).

condition

of

on the

(Oil. 1

‘Ft.

‘~jsl

(Qi)t . .

’

1

CF,~(Qi)II’:

(Qi) !I:

coordinates

of

the

Q,

is

Random

When n < N, extra

are

imposed

on the such

Obviously. quadratures. tain

quadrature of

is

the

formulae

(“1

of

(p = 0,

1,

. . . , “1).

that

is

independent

D,

. . . . n2)

1 = n

obtain

E

(4).

/I2

of

quadrature

of

improved

accuracy

quadrature

formula.

Let D

fJ1 and D2,

(q

1,

= 0,

and let

Q,,,;

Q,,,

formulae

in D,,

of

is (6)

of

and ~“,,(I?

exact are

for

us define

= (XP, It

. . , np).

Y, and D2 independent

which

and numerator

ob-

interpolation-

the

XP E assumed here

is

X. Let 9’hc.Y) tp = 0.

1.

~‘~(.Y)~“U(Y) equal

n,

(h = 0.

. . . , n2)

are orthonormalized of base-points of

respectively

net

Yq).

in

= tt?\,,(Q).

the

on the net

when

to

the

corresponding

Arguments sults: 1)

to

formula

or net

+ 1) points

As = det, I/f (X,, i.e.

is

in D,. The products ~‘h(.Y)p”,, (Y) shown that, given the distribution

formula

denominator

of

of

it

a quadrature

be orthonormalized

orthonormalized n. It is easily quadrature

of

+ l)(np Yq

formulae.

some examples

regions

product

system

quadrature

than

formula,

Cartesian

form

in a sense an analogue of Gaussian it is much more difficult to case,

quadrature

as the

215

accuracv

random base-points.

example

iterative

the

us consider

with

be the

1,

are

improved

the

region

Let

accuracy

of

classical

an arbitrary

The simplest

of

base-points

formulae.

improved 2.

constraints

As in the

them for

of

qundratures

if

x,,

similar

. . . . X”

Y,), ‘pi W,),

quadrature

to

1

those

are

sum

employed

distributed

. . ., V,,, t-l’,) II:‘.

z,,,,,

in

If1

[41

is

lead

in D, with

to

the

the

followina

prohahility

re-

density

216

S. M. Eraakov

1

[det(lv;(X,L * * *?vpc (x,)\pl’,

(nl + I)! and Yo,

Y1,

. . . ,

in

Y,,

D, with

the

probability

density

1

(nn + I)! Bidet(1‘PO”(Y,& . - *, ‘P*, (Y,) (pl? then

cp; (Xl

jlK,,,2ljl = \

M.O.

‘P;(Y)

f(X, Y), dXdY:

i>

5 dY[S

If1 = %$ (Q) dQ - i

ml,,&

2)

A=1 Da

-

dXf (Xv y)

(X)]a -

‘p;

01

5 ( dX [ \ dYf (X, Y) ‘P;0j2- [5P);(X) cp;; 0’) f (X,Y) dx fl]‘. p=1 DI

DI

LJ

If we had not assumed that the base-points of the which is exact for n,,(Q) lie on the net, we should virtue of Theorem 2 of [4], the folllowing expression of

the

corresponding

quadrature

sum K,,~,“?

It

is

easily

)i=oy=o

shown that.

whatever

:

1

5 [ 1 f (Q) cp; (S) cp; 0’) dQ

*KnI,n, If1 = 1 P (Q) dQ - 2 D

[fl

quadrature formula have obtained, by for the dispersion

* .

LJ

f may be

DK*,, ‘1? [i] >, DK n,, ?I?I/l. Naturally, quadrature other to

the

hand,

arrange the

net

as such

in the

3) The case formula,

of

are of the

a formula

arbitrary special

is

interest

there

of

and not

is are

it

achieved

exact

for

summable is

that

imposed kn such

to

it

form On the

is

better

them arbitrarily.

because

any functions

of

the of

fact

the

form

11 = I. . . . ,)I?,

functions.

when the

on the

pays

functions.

choose

, )II, and gl (A’) v;;(1). square

(7)

mean that

products

has been obtained,

is here

I r= 1, .

Suppose

not

for

on the net, formula

type

does

exact

dispersion

quadrature

where g and gl

ture

are

base-points

6 (k’) & (W.

constraints

obtained

which

as soon its

The reduction that

result

formulae

maximum number of

base-points

constraints,

i.e.

of the

the

quadra-

coordinates

Random

of

n base-points

base-point in the Qo,

be definite,

vector

arguments

density

is

also

Qi. in Qo,

the

of

the

symmetry

of

the

coordinates

of

a given

of

. . . , Q,).

which

. . . . 0,.

Hence

it

follows

if

the

in D.

that

the

quadrature

co-

formula

equation

111'(Qo,Q1,. ..,Qn)= \ f (Q) ‘Pj (Q) dQ. h

\ ~Qo ‘,,j

If1

Kn , j

217

accuracy

(7) are symmetrical assume that the proin fact depends only on

distributed

D

In view

in terms

Q1,

ordinates of Q0 are uniformly is accurate for constants. Thus we must satisfy

improved

say QO). The relations Thus it is natural to

function-G(Qo,

symmetric

M.O.

of

can be expressed

(to

bability

quadraturcs

relations

(7)

and the

function

G(Q,,,

. . ,Q,j,

we obtain

(--

I? (n

f

1)

s

det Ilq, (Q,,,), . . . , ‘~j-1 (Qm)* Pj+l (0,)

dQo f(Qo)

...l~‘n (Qrn)lI:

=

det Iho (Q,,,), ...,(P,, (Q,,,Il::

D

= \ ‘Pj (Q) f (Q) dQ.

b

we have

i.e.

almost

everywhere

G (Qor . . ., Q,) = The condition

for

(-I)jvj

symmetry

det

II(pO(0,).

the

of

G(Q,,

‘Pj (Ql+l)

(Pj-l(Qm)s(Pj+l (Qm)*~.~.~n(Qm)(I:

. . . , Q,)

1 --= n,

A ft1

‘41 Al is

(00) det ITJO(Q,),.. .,T,(Q,,,)IC

(n + 1) dct ll~~(Q,)~~..v

‘Pj ((21) -=

where

in D:

cofactor

of

. . . ,vn (Q,)

mean that the coefficients G (Qo, Q1, . . ., Q,,) E const..

the

II:.

element

also

gives

1, . . .( n -

o,(Ql)

of

us the

equations

1,

the

(8) determinant

When I = 0 and p0 = con&..

equations

(8)

of the quadrature formula are all equal and Zhen 1 # 0, the equation A, x qj (Q,) obviously

holds. The quadrature example

of

nomials of quadrature base-point) higher

than

formulae

formulae

of

this

obtained type

in

[I]

and [21 offer

which

are

exact

for

the first degree. The author has not succeeded formulae with a maximum number of constraints which would be exact for algebraic polynomials the

first.

an elementary

algebraic

poly-

in obtaining (with one free of degree

218

S.M. Eraokov

If D

Is

a unit

with one free formula which

and f is

a periodic function of period unity construction of a quadrature formula la possible, provided there is a determinate

hypercube

a trivial

in all it8 variables.

base-point Is exact for

periodic

functions.

Thus

the

quadrature

formula

f

(Xl, . . . , Xk) dXl*.-dXk

=: +

may be associated

with

the

s

f(X,,...,Xk)dX1...dXkz+;

f (‘1.i

i=i

where tl, . .. . & [O,

, . *l xk,i)

j (Xl,*,

formula

D

interval

; i=l

are random magnitudes

I].

Obviously.

+ El,

. . * t x,,<

+

E/J

uniformly dfstributed over the

the same inequality will hold for the re-

mainder of the random quadrature formula as for the remainder of the determinate formula. 4. of

Let

us

improved

f (Q)dQ =

quote

another

accuracy

for

example

k [f] = $

(f (x1,0, . . . ,x,,,)

the

unit

of

an elementary

hypercube,

quadrature

formula

namely

+ f (x1,1, . . . ,Kk,l) + f (xl,%..., x,,,)},

% where

(i = 1,2 ,..., k),

Xi,3 = vi-X&-X&

magnitude8 distributed

and

Xi,r,

Xi,0

are

XiTb+Xi~~ < 1, Xi.0

in the region

random

>, 0

With the

probability density

This

formula

is

accurate

dependent variablea. in all its variables. centre

of

gravity

of

the

six base-pclnlx which pendent variables. It

is

easily

f a8 a k-tuple evaluate

the

is

verified

a constant

and second

aerles

hypercube, accurate

also

for

the

s L,

in Lenendre ^ K [f].If

I3 “L,....,n¶, ml,..., mk

If

are we can easily

that mdo. i? [f] is

di8perBiOn

DR VI = f

fpr

It is obviously convenient Using base-point8 which

polynomials f is even in

pouers

of

the

in-

f

is an even function symmetrical about the obtain a formula with

first

j(Q) dQ.

powers

On

of

the

inde-

representing

and using (8). we can all its variables. we have

(i + 2 fJ (miL,:)!! i=l

) ,

mi>

2.

Random

This

quadrature

quadratures

formula

is

of

fairly

improved

simple

219

accuracy

and suitable

for

practical

com-

putations.

Translated

D.E.

by

Brown

REFERENCES 1.

Hammersly,

I.M.

antithetic

and Morton,

variates.

Proc.

K.W.,

A new Monte Carlo

Cambridge

PhiIos.

Sco.,

technique 52,

for

449-475,

1956.

2.

3.

Morton,

K. W. ,

for

evaluating

Bakhvalov.

A

generalization integrals.

N. S.,

WC;I/, Ser.

ApprOximate

matem.,

mekhan.,

of J.

the

Math.

antithetic Phys.,

evaluation astron.,

of fiz.,

variate

36,

No.

UIUitiplC!

khimii,

3,

technique 2R9-293.

1957.

integrah.

NO.

4.

Vesti.

3-18.

1959.

4.

Ermakov,

S.M.

Monte Carlo NO.

4.

Polynomial

and Zolotukhin,

V.G.,

method.

veroyatnostei

473-476.

1960

Teoriya

approximations i

cc

primeneniya,

and a 5.