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fix: Improve formula presentation
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docs/src/dev/how_it_works.md

Lines changed: 10 additions & 6 deletions
Original file line numberDiff line numberDiff line change
@@ -102,24 +102,28 @@ two calls to ``f`` to evaluate the gradient:
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x_3 \\
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x_4
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\end{bmatrix}
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\vec{x}_{\epsilon} = \begin{bmatrix}
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```
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1st call:
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```math
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\to \vec{x}_{\epsilon} = \begin{bmatrix}
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x_1 + \epsilon_1 \\
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x_2 + \epsilon_2 \\
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x_3 \\
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x_4
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\end{bmatrix}
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\to
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f(\vec{x}_{\epsilon}) = f(\vec{x}) + \frac{\delta f(\vec{x})}{\delta x_1} \epsilon_1 + \frac{\delta f(\vec{x})}{\delta x_2} \epsilon_2
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\vec{x}_{\epsilon} = \begin{bmatrix}
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f(\vec{x}_{\epsilon}) = f(\vec{x}) + \frac{\delta f(\vec{x})}{\delta x_1} \epsilon_1 + \frac{\delta f(\vec{x})}{\delta x_2} \epsilon_2,
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```
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2nd call:
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```math
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\to \vec{x}_{\epsilon} = \begin{bmatrix}
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x_1 \\
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x_2 \\
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x_3 + \epsilon_1 \\
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x_4 + \epsilon_2
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\end{bmatrix}
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\to
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f(\vec{x}_{\epsilon}) = f(\vec{x}) + \frac{\delta f(\vec{x})}{\delta x_3} \epsilon_1 + \frac{\delta f(\vec{x})}{\delta x_4} \epsilon_2
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f(\vec{x}_{\epsilon}) = f(\vec{x}) + \frac{\delta f(\vec{x})}{\delta x_3} \epsilon_1 + \frac{\delta f(\vec{x})}{\delta x_4} \epsilon_2.
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```
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This seeding process is similar for Jacobians, so we won't rehash it here.

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