88
where ¢ is a distribution whose amplitude is of the
order of the magnitude of error in the measurement of
g(x). In other words, small errors in g(x’) may be
amplified to such an extent that the exact solution of
Eq. (1) [deleting ¢(2’)] becomes physically meaningless.
iq. (1) is usually approximated bya linear system of
equations, or equivalently, a matrix equation. The in-
terval (a,b) issubdivided into n part a = a1 < te--* <
x, = 6b, and Eq. (1) is approximated by
solution of Eq. (2) which minimizes
m
m
nm
2
a Die; = » Pi e ajifi — us) :
j=l
j=l
i=l
where a;; = Aj; and p; are appropriate weight
If there is no a priori evidence that some of the
more accurate than others, then one mayset |
for all 7. The minimization may be subject to t
straint which arises when the total activity C is
(see Section A below) and which has the discre
1
gal, m,
2, Kiwi: =gjte
(2)
where w; are weights that depend on the quadrature
formula chosen, Ki = K(x5,0/), fi = f(t), 93 =
g(x3), €; = e(z;) and 21,22 +--+ tm the points at which
g is measured. In order that the system of linear equations become meaningful, it is necessary to have m => n.
Introducing the notation A = (w.K;:),f = (fi. =
(gi), and e = (e;), Eq. (2) can be written in the matrix
form
Af=g+e.
» wifi = C,
t=1
and to other constraints such as that of being suff
smooth.
A typical smoothness constraint is to minin
sum of the squares of the second differences. Tl
lem then is to minimize the function
ACh, see fa hy)
(3)
The method used for the solution consists in finding the
(figs — 27+ fia)’ + y(sus
I
+ Y
FWHM=8.5 cm
“+ Si2fo)?
Tt
3,
La
+ ¥°(8njn—1fn—a + Sandu) «
FWHM = 15.0 em
g(x’) DATA SMOOTHED
The last two terms in the above equation con
h=icm
conditions at the end points of the interval (a.
the factors \ and y° are Lagrange multiplie:
4
smoothness of the solution depends on y’, the
FWHM=15.0
10
given to minimize the second differences in th:
h=2cm
;
Y
i
FWHM =15.0cm
g(x') DATA NOT SMOOTHED
case. Four curves are plotted in Figure 70 (see
comments in Section C), giving y as a functiol
error
h=tem
2
L
oO
2
j=l
1071
10°
my
e= > Pye;
50
|
100
150
200
250
e7/\94
Fic. 70—Representation of y's as a function of
l
300
|
350
for different cases.
Normally, in a given experiment a lower and a
bound forthe value of e? of Eq. (4) can be estima
the y to be chosen ought to be the one which ¢
acceptable value of e?.
VARIATION OF THE EXPERIMENTAL PARAMETER:
A, The Knowledge of the Total Amount C of
activity
The computer program vields both quantitie-
f(x) from the experimentally measured values :
“ Sha 4