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MathML

\begin{aligned} Gauss-Newton: Δ & = {(- E [H (β)])}^{-} g (β) = (X^{'} X)^{-} X^{'} r \\ Marquardt: Δ & = {(X^{'} X + λ diag (X^{'} X))}^{-} X^{'} r \\ Newton: Δ & = - H (β)^{-} g (β) = H (β)^{-} X^{'} r \\ Steepest descent: Δ & = - \frac{1}{2} \frac{\partial S S E (β)}{\partial β} = X^{'} r \end{aligned}

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\begin{align*} \mbox{Gauss-Newton: } \bDelta & = \left(-\mr{E}[\mb{H}(\bbeta )]\right)^{-}\mb{g}(\bbeta ) = (\mb{X}'\mb{X})^{-}\mb{X}'\mb{r} \\[0.15 in] \mbox{Marquardt: } \bDelta & = \left(\mb{X}'\mb{X} + \lambda \mbox{diag}(\mb{X}'\mb{X})\right)^{-}\mb{X}'\mb{r} \\[0.15 in] \mbox{Newton: } \bDelta & = - \mb{H}(\bbeta )^{-}\mb{g}(\bbeta ) = \mb{H}(\bbeta )^- \mb{X}'\mb{r} \\[0.15in] \mbox{Steepest descent: } \bDelta & = -\frac12 \frac{\partial \mr{SSE}(\bbeta )}{\partial \bbeta } = \mb{X}'\mb{r}\\ \end{align*}