Delta-rational chord: Difference between revisions
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=== Naive least-squares error === | === Naive least-squares error === | ||
Rewriting a bit, if 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> has delta signature +ε<sub>1</sub> +ε<sub>2</sub> ... +ε<sub>''n''</sub> (where the chord is 1:1+ε<sub>1</sub>:...), let <math>D_i = \sum_{k=1}^i \delta_i</math> and <math>E_i = \sum_{k=1}^i \epsilon_i | Rewriting a bit, if 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> has delta signature +ε<sub>1</sub> +ε<sub>2</sub> ... +ε<sub>''n''</sub> (where the chord is 1:1+ε<sub>1</sub>:...), let <math>D_i = \sum_{k=1}^i \delta_i</math> (the ''target'' delta signature) and <math>E_i = \sum_{k=1}^i \epsilon_i</math> (the ''approximant'' delta signature). Then the resulting linear least-squares optimization problem is | ||
<math> | <math> | ||
\displaystyle{ \min_x \sqrt{\sum_{i=1}^n \Bigg( | \displaystyle{ \min_x \sqrt{\sum_{i=1}^n \Bigg( D_ix - E_i \Bigg)^2 } } | ||
</math> | </math> | ||
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<math> | <math> | ||
x = \displaystyle{\frac{\sum_{i=1}^n D_i E_i}{\sum_{i=1}^n | x = \displaystyle{\frac{\sum_{i=1}^n D_i E_i}{\sum_{i=1}^n D_i^2},} | ||
</math> | </math> | ||