Y 3. A17

21 1)

TechnicalMemorandum

SCTM 145-53(51)

>te on the numerical evaluation of integral

-e form C' f (x)>(x) dx, with particular ref,

)04-BK (3-58)

n hI

1:1

THIS IS A SANDIA CORPORATION WORKING PAPER ANDIS INTENDED PRIMARILY FOR INTERNAL DISTRIBUTION;THE IDEAS EXPRESSED HEREIN DO NOT NECESSARILYREFLECT THE OPINION OF THE CORPORATION.

II

TID-4500 (15th Ed.)PHYSICS AND MATHEMATICS

SCTM 145-53(51)

Note on the numerical evaluation of integrals of

the form f f(x)4(x) dx, with particular referenceto the determination of the expectation of a func-

tion of a normally distributed random variable.

Systems Evaluation Department

Sandia Corporation, Albuquerque, New Mexico

ABSTRACT

A method is given for the rapid and accurate numerical integration ofintegrals of the form f! f(x)4(x) dx, where f(x) is "smooth." Tablesare given for the special case (x) = 1 e -x 2 /2c2.

1'2 n a

Case No. 417.0 September 3, 1953

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3-4

- 5-

1. Introduction. This note is concerned with the numerical integration

of integrals of the form 4'f (x) cp (x) dx, where the function f (x)

is "smooth". It is often convenient to carry out such a numerical inte-

gration using values of the function f(x) at equally spaced points. A

particularly appropriate method would appear to be one which involves the

exact integration of an integral obtained by replacing f(x) by a smooth

function interpolating the values of f(x) at the equally spaced points.

Such a function is given by the cardinal series associated with these values

(for references to the literature on cardinal series, cf. J. M. Whittaker,

Interpolation Theory).

The integral

I = f f(x) 0(x) dx

is to be approximated in terms of the values, jf(k)}, (k = 0, + 1, + 2, ... )

of f(x) at points with integral abscissas. The cardinal series, f*(x),

associated with the values, ff(k), of f(x) is given by:

f*(x) =k f(k) sin II x-k),

The integral I is approximated by

I* = _ f*(x) Q1 (x) dx,

I*= - f (k-) isin2-n(x-k) (P(x) dx,k=-c -fkx-k))

or I* = akf(k) (1)

where ak = sin x P(x) dx. (2)

Formula (1) has two important advantages, which are, of course, shared by

other methods of numerical integration:

(A) The integral I* is expressed as a linear combination of

-6-

the given values of f, and is thus particularly well-suited to calculation

with a desk calculator.

(B) The coefficients, ak, do not depend on the particular

function, f.

2. The expectation of function of a normally distributed random variable.

Suppose it is desired to compute the expected value of a function, f(x), of

a normally distributed random variable g, and that coordinates have been

chosen so that the mean of x is 0. The expectation of f(x) is given by

I = f f(x) P(x) dx,

where (x)= e x2/24

and where o is the standard deviation of x.

Let the function f(x) be evaluated at equally spaced points, and let

the distance between an adjacent pair of these points be chosen as unit. The

integral I is then approximated by

I* = r=. ak f(k), (1)

where the coefficients ak are tabulated for various d in Table 1.

For a ' 1 an approximate formula for the coefficients ak is

1 e - k2/2d2 (2)

The error for c"= 1 is 0.0006 when k = 0 and decreases as k increases.

For c = 2 the error in (2) is of the order of 10-10 and is smaller for

larger o. Thus we have the rather surprising result that for c 1 an

integration scheme which is nearly exact for the cardinal series is obtained

simply by multiplying each value of the function f(x) by the ordinate of

the probability density function at the same point and adding the results.

Whether or not this scheme is accurate for the integration of the function

f(x) itself depends, of course, on how close to the function f(x) is its

cardinal series. A rough check on the accuracy of interpolation by the

cardinal series may be obtained by plotting the points (k, f(k)), joining

them by a smooth curve, and comparing the resulting graph with the graph

of f(x).

In terms of arbitrary linear units, one concludes from the above obser-

vations that if the graph of f(x) is sufficiently smooth, then accurate

results are obtained by evaluating f(x) at points at distance d units

apart and multiplying each such value by o and by the corresponding

ordinate of the probability density function:

cx~ 1 - (kd)2/2d2f(x) 4(x) dx" o f(ko) e-,

k2 /or .00f(x) 0(x) dx M f(ko) 1 e - k2/2

If the graph of the function f(x) is not smooth enough that the cardinal

series using values at points o units apart represents the function accu-

rately, then more closely spaced points may be used. If the function is

evaluated at rioints at distance o/n units apart, the ap roximate formula

I f(x) 4(x) dx 1 f(-) e - k2/2n2-2w n

may be used.

For a situation in which the integral I is to be computed for a fixed

function f and for a number of different values of d, it is desirable to

have tables of the coefficients ak for various d. Such coefficients are

given in Table I for o = 1/2, 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20. The

distance between adjacent points at which f is evaluated is chosen as the

linear unit in terms of which the standard deviation, d, is measured.

T A 3L I

A

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5

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+ 7

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13

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24

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126

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+30

+3,

+32

+33

X34

+35

+36

+37

+36

+39

+40

+41

+44"

+43

+44

+45

+46

+47

+48

+49

+50

+51

+52

+53

+ .706

+.1c0

-.018

+.008

-.005

+.00

-.00:.

+.00.

-. 001

+.001

-. 1

+.QT1

-.001

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0=1

.398

243

.054

.004

.000

.200

.176

.121

.065

.027

.009

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Q=3

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.106

-081

.055

.033

.018

.009

.004

.001

.001

.100

.097

.088

.075

.060

.046

.032

.022

.014

.008

. 004

.0u2

.001

.080

.078

.074

. 067

.058

.048

.039

.030

.022

.016

.011

.004

.003

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.G61

=6

.066

.066

.063

.059

.053

.047

.040

.034

.027

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.017

.012

.009

.006

.004

.003

.002

.001

.001

.001

d=8

.050

.049

.048

.046

.044

.041

.038

.034

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.026

.023

.019

.016

.013

.011

.009

.007

.005

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.002

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0=10

.040

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010G

.008

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.004

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.00:1

.001

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c=12

.034

.033

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.032

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.029

.028

.027

.025

024

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.017

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. v J

.008

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.006

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.001

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. 023

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G=20

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- k

- Al-

APPENDIX

Calculation of coefficients

Al. General. The coefficients ak are given by the formula

= j si x-k) 4 (x) dx. (Al)

It may also be convenient to use the expression derived below. We have

a. =) P$(x) dx * fcos (x-k) t dt,x J (xdx " A o

n0

ak = j cos kt dt . (O(x) cos xt dx

(A2)

+ L / sin kt dt . c(x) sin xt dx.iT o -o

If Q (x) is even, the second term on the right of equation (A2) vanishes,

and the coefficient ak is seen to be, except for a multiplicative constant,

the k-th Fourier cosine coefficient of the cosine transform of 40(x):

Lk jo cos kt dt I (P(x) cos xt dx. (A3)

iarl;, if ((x) is an odd function of x, we have

ak = sinkt dt L ((x) sinxtdx.

If q (x) is even, then under conditions sufficient for the convergence of t-.:

:iurier werits of the function / 4(x) cos xt dx to the proper value, we

have

I P(x) cos xt dx = ak cos kt

for It1 < 1. In this connection it is of interest to observe that on setting

-A2 -

t = 0 we have

k= o ak = 7CO. (x) dx.

Hence if f(x) is identically constant, then the integrals I and I*

coincide, where

I = 47 f(x) cP(x) dx,

1* = J f*(x) cp(x) dx,

and f*(x) f(k) sin k)

for * = ak ff(k) = C JcQ(x) dx

and I= C 4. (x) dx

if f(x) is the constant C. Thus the formula

I _ ak f(k) (A4)

is exact when f(x) is identically constant. Indeed, it can be shown that

the cardinal series associated with values of a constant function is itself

that constant; and since the formula is exact for every cardinal series, the

above observation follows.

A2. The expectation of a function of a normally distributed random variable.

Let x be a normally distributed random variable with mean 0 and standard

deviation d. Its probability density function is

((x) =-1 e - x/262

Let f(x) be a function of the random variable g. Its expectation is the

integral

- A3 -

1222I = fCOf(x)PQ(x) dx = 1Cf(x) e x/202

Since gD(x) is even, formula (A3) for the coefficient, ak,

appropriate. We have

122/01 74(x) cos xt dx = 0e4 ex/20 cos xt

- 2 2/2

dx.

is

dx

Formula (A3) then gives

ak 2 e 2/2 cos kt dt. (A5)

The remainder of this section is devoted to the evaluation of this integral.

It is clear first of all that, for large a,

ako e 02 2/2cosktdt,

ak 1de 2 2/2cos kt dt) ,

and a 1 e - k2/202. (A6)

The error involved in this approximation is not greater than

1 f -e0 2 t 2 /2 dt = ( e _ 2t2/2 dt)

[1 - d(o)], (A7)

where m(x) is the normal cumulative distribution function,

g(x) = 1 /X et 2 /2dt.

For o = 1 this upper bound on the error is 0.00064. For o = 2 the

error in formula (A6) is not greater than 6.78 (1011). For larger o,

of course, the error is smaller.

For k = 0 and for arbitrary o we have, from (A5),

a = fe - c2t 2/2 dt = 1 f2( 0s.-I e _ 02t2/2 dt),o 19 o o 2R

= [5( wc) - 1/2], (AS)

where again G(x) is the normal cumulative distribution function.

It remains to develop methods for computing ak from (A5) for o < 1

and k > 1. On expanding the exponential in a power series, equation (A5)

becomes

ak = 1 (0)j2 2j cos kt dt,

ak? _oJ 22 u cos wku du,

or a = (-l) k (1)j20 Ck J, (A9)

1 2where Ck,j = (-1t2j cos vkt dt. (A10)

The constants (- 1 )k Ckj are tabulated in Lowan and Laderman, Journal of

Mathematics and Physics, 1943. For a situation in which these tables are

not sufficiently extensive, the recursion formula derived below may be used.

We have, from (Ala), on integrating by parts,

1 1C = (-1)k [t2j sin kt] _ f 2j-1 sin irkt dt ,k~j sk o ik o

k+1 k1CkJ = (-1) k(2 )Jf t2~ 1 sin 'kt dt, for j Zl, k> 1.

A further integration by parts yields

- A5 -

C =(l1k+1(2 j) [-cos ,ikt (t2j-1 1 2j-2 cos Jkt dtkj = rk Ak o + k o

Ckj = 2 [1-(2j-1) C ] (All)kj (urk)2 k, j-1

for j ?_l, kilo .Clearly

Ck,0 = 0 if k t_1,

Ck,0 "IC00= 1.

The following alternative series expansion of ak converges rapidly for

small 0 and large k. We have from (A5)

2

g=k 4 0, -e u /2 cos audu,K =O dku

ak n e(u) cos du,

where (u)e= -u/2

Hence 1"ak4(u) cos du

[r sin $(u)] - QT'O*(u) sin-du

=-j[- -kcos 4'(u)] + ko "(u)cos du

= (-1)k 2 2 In "(u) cos - du

kA k o (3

= (-1)k ( ) (-1)k 4 (Is) ek k

= (- 1 )k f (-1)r (2) *(2r+1) (d),k2 ro k

where $(2r+1) (n'r d) is the (2r + 1)-st derivative of $(u) = e -u /22cw

-A6-

evaluated at u = 1 d. Hence

ak ()k d j0 o(l)r (k)2r *2r+l)( 0 ). (Al2)

The Hermite polynomials, H,(x) are defined by

(9 ) 44(i)(x) = (-1) H (x) $(x),

so that equation (A12) may be rewritten

(-)klf= g(id) (- )2rH (d ). Alkk2k+o H2r+1

We have H0(x) = 1, H(x) = x, H2(x) = x2 - 1, H3(x) = x3 - 3x,

H4 (x) =x 4-6x2 + 3, H5(x) = x5 - 10x3 + 15x, H6 6x) 6-15x4 +45x2 - 150

Further Hn(x) may be determined from the recurrence relation,

H(x) = x H (x) - H (x).

For small k neither of the above series developments converges rapidly. On

expanding the cosine in (A5) in a power series, we have

= j ()jk2j 1 4 2j e - d2t2/2 at.ak jNo o(2je)dt.

Now l" t2je - d2 t2 /2dt = 1 J u2 j je-u2/2 du.W 0 0d2j+l o

Let A(x) =x u2je-u/2 du. (A14)

Then 1k -1 k (- L(k)2J A (d). (A15)

We have A (x2) = 1 -uj-1 - u2/2] + 2- I uj-2 e - u2/2 du,

A.(x) = - x2j-1 4(x) + (2j-l) A- 1 (x) (j > 1), (A16)

where

We have

where

Further Ai(x) can

- A7 -

e x/2*(x) = e

AO(x) = Q(x) - 1/2,

x 2

g~x =t---2edt,

A1 (x) = Aox - x (x),

A2(x) = 3 Ao(x) - (x) [x3 + 3x],

A3(x) = 15 AO(x) - *(x) [x5 + 5x3 + 15x],

A4 (x) = 105 A(x) - *(x) [x' + 7x5 + 35x3 + 105x].

be determined from the recurrence relation (A16).

SYSTEMS EVALUATION DEPARTMENT - 5120

mn