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Line 79:
Line 79: − + − + + + − + − u_\beta^* u_\alpha' I_{\alpha\beta} I_{\alpha'\beta'}+ − + − \rho_{\gamma\delta} \rho_{\gamma'\delta'}+ − \sum_{\alpha,\beta,\alpha',\beta'}^n{+ − + − \left(u_\beta^* J^{\gamma\delta}_{\alpha\beta}\right) − − } − + − \rho_{\gamma\delta} \rho_{\gamma'\delta'}+ − \sum_{\alpha,\alpha'}^n{ − \left(\sigma_{u_\alpha}\sigma^*_{u_{\alpha'}}\right) − G^{\gamma\delta*}_\alpha G_{\alpha'}^{\gamma\delta} − } − The product of σ terms in the summation is represented by the error matrix derived from the fit. ''G'' was defined as+ − +
Error propagation in Amplitude Analysis (view source)
Revision as of 19:14, 22 November 2011
, 19:14, 22 November 2011no edit summary
</math>
</math>
:<math>
:<math>
= \sum_{\alpha\beta}^{2n}{a_\alpha b_\beta J_{\alpha\beta}}
= \sum_{\alpha\beta}^{2n}{a_\alpha a_\beta J_{\alpha\beta}}
</math>
</math>
where the sum over ''n'' complex parameters is expanded to a sum over 2''n'' real ones, and the matrix ''J'' represents the above ''2n''x''2n'' matrix of the elements of ''I''.
where the ''a'' coefficients stand for both the ''a'' and ''b'' terms
introduced above, so that the sum over ''n'' complex parameters is expanded to a sum over 2''n'' real ones. The matrix ''J'' represents the above ''2n''x''2n'' matrix of the elements of ''I''. The variance on the intensity sum is expressed in terms of the covariance matrix
''C''<sub>αβ</sub> among the fit parameters ''a''<sub>α</sub> as
<math>
<math>
\sigma_{fit}^2=
\sigma_{fit}^2=
\sum_{\alpha,\beta,\alpha',\beta'}^n{\left(
\sum_{\alpha,\beta,\alpha',\beta'}^{2n}{\left(
a_\beta a_{\alpha'} J_{\alpha\beta} J_{\alpha'\beta'}
\right) C_{\alpha\beta'}
\right) C_{\alpha\beta'}
}
}
</math>
</math>
::<math>
::<math>
= \sum_{\gamma,\delta,\gamma',\delta'}{
= \sum_{\alpha,\beta'}^{2n}{
\left(\sum_{\beta}^{2n}{a_{\beta}J_{\alpha\beta}}\right)
\left(\sum_{\alpha'}^{2n}{a_{\alpha'}J_{\alpha'\beta'}}\right)
\left(\sigma_{u_\alpha}\sigma^*_{u_{\alpha'}}\right)
C_{\alpha\beta'}
\left(u_{\beta'} J^{\gamma'\delta'*}_{\alpha'\beta'} \right)
}
}
</math>
</math>
::<math>
::<math>
= \sum_{\gamma,\delta,\gamma',\delta'}{
= \sum_{\alpha,\alpha'}^{2n}{
G_\alpha G_{\alpha'} C_{\alpha\alpha'}
}
}
</math>
</math>
where ''G'' is defined as
<math>
<math>
G_\alpha^{\gamma\delta}=\sum_\beta{ u_\beta J_{\alpha\beta}^{\gamma\delta}}
G_\alpha = \sum_\beta^{2n}{ a_\beta J_{\alpha\beta}}
</math>
</math>