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