QUADRATURE FORMULAE WHEN ORDINATES ARE NOT
EQUIDISTANT
B Y PROFESSOR JAMES W.
GLOVER,
University of Michigan, Ann Arbor, Michigan,
U.S.A.
The formula (2) in this paper, and several others of similar type were first
derived by Hardy* and later appeared in King's Text-bookf, where numerous
applications to actuarial calculations are made. It is found in practice to yield
surprisingly close approximations, even though a small number of ordinates
are used, and on this account some further consideration of its theoretical basis
and a method of generalization may prove acceptable. The general idea is to
establish a formula of the type
uxdx = a(u_n + un) + 2 ai (u-n.+un.)
(D
= aun+
üaiün
where
th<n2< . ..
<ns<n,
to express the value of the area bounded by the curve, y = ux, the axis of x, and
the two limiting ordinates U-n and un. Then taking the origin at —n the limits
become 0, 2n (or Yin), and the formula is applied to successive sections of the
area, each standing over a base of length 2n (or Yin), to obtain a general expression for the area in terms of the extreme and certain intermediate ordinates
which are not equidistant. In some of these formulae a constant times the ordinate
at the origin is employed. In the case s = 2 the formula found by Hardy and
given in the Text-book is :
("17n
(2)
uxdx==n[.60(uQ+ul7n)+4:.79(uQn+unn)+3.11(u2n+ul5n)].
Jo
We proceed to derive this formula.
Assuming the expansion of the function by Maclaurin's theorem
.
w
(u , n2
••U0+ -rj- 4
1}
(9) ,
+ T^ 4
2)
. ns
(Q) ,
+ • • • + Tg U0 + .
it follows that
ûn = u-n+un
= 2uo+2j~ui2)
ir
+ 2 ^ - 4 4 ) + • - .+2~u^+
if
. ..
i_
and similarly for üni and üm.
*G. F. Hardy, Formidas for Approximate Summation, Journal Institute of Actuaries,
Volume 24, page 95.
fGeorge King, Institute of Actuaries Text-book, Part II, page 483 (Second edition).
JAMES W. GLOVER
832
Also the area
'n
3
6
9
71
,/»).
2^utf>+.
A = uxdx = 2nu0+2 —— u^ + 2 -r— u^ + .+
-n
\o
|5
and comparing coefficients of like
equations is obtained :
a+ d\
a+aai
(3)
a+a2ai
a + azai
a + a*ai
derivatives the following set of homogeneous
+ a2
+ßa2
+ß2a2
+ß3a2
+ß*a2
where a = ni2/n2 and ß — n22/n2.
(0, 0, 0, 0), we must have
D1-
1
1
1
1
1
a
a2
a3
1
ß
ß2
ß*
1
1/3
1/5
1/7
—n = 0
—n/3 = 0
-n/5 = 0
-n/7 = 0
-n/9 = 0
For solutions other than the identical one
=0
p2=
1
1
1
1
a
a2
a3
ß
ß2
ß3
4
4
a
/3
1/3
1/5
1/7
1/9
=0
from which it follows that
( « - l ) ( 0 - l ) ( a HLl.3
O r - ^ 3.5
- ^ + 5.7J
-l-O
a/3(a-l) 0 3 - 1 ) ( a -|S) [ - ^ - ^
+ i - l = 0.
L3.5
5.7
7.9J
All values of a and ß which satisfy both determinants are trivial except those
common to the two hyperbolas
^ _ ^ + _ !
1.3
3.5
5.7
=
0 a n d ^ - ^ + -!=0.
3.5
5.7
7.9
For example, when a = l , the ordinate uni coincides with un; when a = 0, uni
and un2 fall together.
Elimination of y leads to the quadratic
2 L t 2 - 1 4 x + l = 0,
(4)
the solution of which gives the two symmetrically situated (with respect to the
line y — x) points of intersection
[(7±2VT)/21,.(7=F2V7)/21]
of the hyperbolas or approximately (.5853096, .0813571) and (.0813571,
.5853096). Hence m = .2S522n and n2 = .76506w.
Taking (a, ß) at an intersection of the hyperbolas the determinants vanish, but
1
1
1
a
= ( a - l ) 08-1) ( a - / 3 ) * 0
ON SOME QUADRATURE FORMULAE
833
and accordingly, we may take » at pleasure, and
15a/3-5 ( a + 0 ) + 3
1
A„e„
v
J
a - -—
. » = — » = .06667»,
15 ( a - 1 ) ((3-1)
15
0l=
2(5/3
~1)
. « = — [7+§V7~]» = .55486»,
J
15 ( a - 1 ) (o-j8)
15
o2=
2(5a-l)
. n = — [ 7 - \ V T i n = .37487».
15 0 8 - 1 ) (j8-a)
15
The area obtained is
(5)
A = ujx = »[.06667 (u-n+Un) + .55486 ( « _ „ , + O + .37487 («_„,+««,)],
or taking the origin at the ordinate w_ M
2n
(6)i4
= w^x = «[.06667 («o+u 2n ) + .55486 (w„- m +«»+»,) + .37487 (« n _ Wa +w B+n ,).
o
The area is thus expressed in terms of the two bounding ordinates and two
additional pairs of ordinates symmetrically placed between them. The bounding
ordinates determine the value of n and this determines the position of the remaining four. U ux is given in functional form these six ordinates may be
computed and then the area calculated.
Hardy had chiefly in view the case where the ordinates are known for
certain integral values of the arguments, and in this connection notice that
when n = S.5n, the values of ni = 2A2n and «2 = 6.50», nearly, and this led to
taking them arbitrarily as »i = 2.5 and n2 = 6.6. When these values are substituted
back in the first three equations of (3) new values of a, ax, a2 are found and the
area expressed in (6) reduces to Hardy's formula (2). In other words, it arises
by a slight displacement of the values of a, ax, a2, »i, n2. It thus appears that
(6) gives the area A based on and true to the first nine terms of the expansion
of ux by Maclaurin's theorem and Hardy's formula (2) gives a close approximation to this value.
This method of approach leads to an easy generalization which will be
pointed out briefly for the case s = 3, that is, when three pairs of symmetrically
placed ordinates conforming to certain conditions are given between the extreme
ordinates bounding the area. The latter would then take the form:
p
(7).
=
3
uxdx = a(u-.n+un)+Tiai(u-n.+un.),
J —n
*=1
and expanding by Maclaurin's theorem to thirteen terms, and comparing coefficients up to and including the twelfth derivative, we have the following set
of homogeneous linear equations:
JAMES W. GLOVER
834
a+ ax + a2 + a 3
a+aai
+ß a2 +y a 3
a+a2ai +ß2a2 +y2as
a+a3ai +ßza2 +yzaz
a-fa 4 ai +ßAa2 + 74&3
a+a^ai +ßha2 +y5a^
a + a 6 a i +/36a2 +y«a*
(8)
—n
=0,
—n/Z = 0 ,
—n/ò = 0 ,
-n/7 = 0 ,
-n/9 = 0 ,
-n/ll = 0,
- w / 1 3 = 0,
where
a — ni2/n2,
ß = n$2/n2,
and y — n^/n2.
The determinants whose vanishing is a necessary and sufficient condition for
solutions other than the identical one are taken from the matrix:
1
1
l a
1
a2
1
a3
1
a4
1
a5
1
a6
1
ß
ß2
ß*
04
ß*
/3 6
\
y
y%
y%
y^
y§
y$
1
1/3 1/5 1/7 1/9 1/11 1/13
=0
and may be expanded as follows:
Dl = k[-f
3
ßy
aß+ßy+ya
3.5
i+ß+y
5.7
+ -i- 1-0,
ßy_
5
aß+ßy+ya
5.7
a+ß+y
7.9
+
|3+<37+7<x
7. 9
a.+ß+y
9.11
+—
L 5.7
7.9
J
2
> /3V = 0,
11.13J
where, apart from a constant factor,
fe = ( a - l ) ( 0 - 1 ) ( T - 1 ) (a-jS) (ß-y)
(7-a).
All the solutions common to A , D2, and D 3 , are trivial (involving the coincidence
of two or more of the ordinates), except the points of intersection of the three
cubic surfaces
xyz
xy+yz+zx
, x+y+z
1
5.7
3.5
7.9
(9)
xyz
3~5
xyz
5~7
xy+yz+zx
5~7
x+y+z
7.9
xy+yz+zx
x+y+z
7.9
9.11
1
9.11'
1
11.13
These points have abscissae satisfying the cubic equation
x3+px2+qx+r = Q
where p, q, r, are determined by the equations
ON SOME QUADRATURE FORMULAE
JL + iL +
1-3 + 3.5
5.7
(10)
JL + Ì- +
Ù + 5.7
S
5^7 + 7.9
7.9
,
P
9.11
+
1
7.9
= 0,
1
9.11
= 0,
835
1
= 0.
11.13
The actual cubic in this case is
429x3-495x2+135x-5 = 0
(11)
and its roots are all real, distinct, and positive, and accordingly lead directly to
the numerical coefficients in the expression for the area A. The roots are
a = .0438062, ß= .3501091, 7 = .7599309, and they furnish the following values
of the quantities »1, n2, » 3 , entering in the quadrature formula:
m = wi'n=.20930» f
n2 = n2n — .59170»,
» 3 = » 3 '» = .87174»,
where primed letters are used to denote the corresponding numerical coefficients
of ». It will be convenient to replace the arbitrary » by \Mn\ the formula then
becomes
r*M»
[Ma',
,
. L Mai' ,
,
,1
(12)
A=
M * = » ~—(U-n'n + Un>n)+ i — - (W-n> + « » > )
}-\Mn
L 2
»=1 I
J
or, taking the origin at the extreme left hand ordinate,
mMn
(13)
VMa'
A = Uxdx^ni —~(u0+uMn)+
Jo
L 2
3
Ma-'
2 — t (uHM-Mn>.)n+uHM+Mn'.)n)
<=i
2
1
.
J
This expression for the area agrees with that obtained from the series
expansion to thirteen terms. If » is any integer and M is an integer so chosen
as to make the Mn'\ integral (nearly) a quadrature formula is obtained in terms
of ordinates at integral abscissae by a slight displacement of these values, that is,
of the positions of the paired ordinates. This displacement in turn modifies
slightly the values of a / , a2, and a 3 ', as determined from the first five
equations in (8). The closeness of the approximation to the area depends
upon the displacement of the paired ordinates determined by the cubic (11)
and this displacement is in turn affected by the choice of the multiplier M.
A great variety of formulae may thus be obtained for each value of s. When
« = 1 and M is small, say not more than 10, the formula may be applied to
successive sections of the area and the result combined in a single quadrature
formula, as in the case of Simpson's formula. On the other hand n may be
taken sufficiently great to cover a long range Mn on the abscissa axis, and the
area obtained with the use of only eight ordinates, or in general, 2 ( s + l ) ordinates.