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<a name="Nonlinear-Least_002dSquares-Weighted-Overview"></a>
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<p>
Next: <a href="Nonlinear-Least_002dSquares-Tunable-Parameters.html#Nonlinear-Least_002dSquares-Tunable-Parameters" accesskey="n" rel="next">Nonlinear Least-Squares Tunable Parameters</a>, Previous: <a href="Nonlinear-Least_002dSquares-TRS-Overview.html#Nonlinear-Least_002dSquares-TRS-Overview" accesskey="p" rel="previous">Nonlinear Least-Squares TRS Overview</a>, Up: <a href="Nonlinear-Least_002dSquares-Fitting.html#Nonlinear-Least_002dSquares-Fitting" accesskey="u" rel="up">Nonlinear Least-Squares Fitting</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Weighted-Nonlinear-Least_002dSquares"></a>
<h3 class="section">39.3 Weighted Nonlinear Least-Squares</h3>

<p>Weighted nonlinear least-squares fitting minimizes the function
</p>
<div class="example">
<pre class="example">\Phi(x) = (1/2) || f(x) ||_W^2
        = (1/2) \sum_{i=1}^{n} f_i(x_1, ..., x_p)^2 
</pre></div>

<p>where <em>W = diag(w_1,w_2,...,w_n)</em> is the weighting matrix,
and <em>||f||_W^2 = f^T W f</em>.
The weights <em>w_i</em> are commonly defined as <em>w_i = 1/\sigma_i^2</em>,
where <em>\sigma_i</em> is the error in the <em>i</em>th measurement.
A simple change of variables <em>\tilde{f} = W^{1 \over 2} f</em> yields
<em>\Phi(x) = {1 \over 2} ||\tilde{f}||^2</em>, which is in the
same form as the unweighted case. The user can either perform this
transform directly on their function residuals and Jacobian, or use
the <code>gsl_multifit_nlinear_winit</code> interface which automatically
performs the correct scaling. To manually perform this transformation,
the residuals and Jacobian should be modified according to
</p>
<div class="example">
<pre class="example">f~_i = f_i / \sigma_i
J~_ij = 1 / \sigma_i df_i/dx_j
</pre></div>


<p>For large systems, the user must perform their own weighting.
</p>



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