From abb4acaa1c32c0b0067dc9d972134c8afe6b5395 Mon Sep 17 00:00:00 2001 From: Claudio Maggioni Date: Fri, 14 May 2021 15:33:26 +0200 Subject: [PATCH] midterm: small correction --- .../Claudio_Maggioni_midterm.md | 16 +++++++++++++--- .../Claudio_Maggioni_midterm.pdf | Bin 631173 -> 632528 bytes 2 files changed, 13 insertions(+), 3 deletions(-) diff --git a/Claudio_Maggioni_midterm/Claudio_Maggioni_midterm.md b/Claudio_Maggioni_midterm/Claudio_Maggioni_midterm.md index b6c830d..513eaeb 100644 --- a/Claudio_Maggioni_midterm/Claudio_Maggioni_midterm.md +++ b/Claudio_Maggioni_midterm/Claudio_Maggioni_midterm.md @@ -366,11 +366,21 @@ respectively in the files `trust_region.m`, `dogleg.m`, and `cauchy.m`. ### Taylor expansion +We first compute the gradient and the hessian of the energy function: + +$$\nabla f\left(\begin{bmatrix}x_1\\x_2\end{bmatrix}\right) = \begin{bmatrix} +\frac{d f(x)}{d x_1} \\ \frac{d f(x)}{d x_2} \end{bmatrix} = +\begin{bmatrix}48x_1^3 - 16x_1x_2 + 2x_1 - 2\\2x_2 - 8x_1^2\end{bmatrix}$$ + +$$ \nabla^2 f\left(\begin{bmatrix}x_1\\x_2\end{bmatrix}\right) = \begin{bmatrix} +\frac{d^2 f(x)}{d x_1^2} & \frac{d^2 f(x)}{dx_2 x_1} \\ \frac{d^2 f(x)}{dx_1 +x_2} & \frac{d^2 f(x)}{d x_2^2}\end{bmatrix} = \begin{bmatrix}144x_1^2 -16x_2 + +2 - 16 & -16 \\ -16 & 2 \end{bmatrix}$$ + The Taylor expansion up the second order of the function is the following: -$$f(x_0, w) = f(x_0) + \langle\begin{bmatrix}48x^3 - 16xy + 2x - 2\\2y - 8x^2 -\end{bmatrix}, w\rangle + \frac12 \langle\begin{bmatrix}144x^2 -16y + 2 - 16 & --16 \\ -16 & 2 \end{bmatrix}w, w\rangle$$ +$$f(x_0, w) = f(x_0) + \langle \nabla f(x_0), w\rangle + \frac12 \langle +\nabla^2 f(x_0) w, w\rangle$$ ### Minimization diff --git a/Claudio_Maggioni_midterm/Claudio_Maggioni_midterm.pdf b/Claudio_Maggioni_midterm/Claudio_Maggioni_midterm.pdf index b77d057bf7567d69c338b50ae8583358f4e19f07..2d1b093b6a96d3d7690dae9ba57e25ea6acd6b35 100644 GIT binary patch delta 108842 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