Delta-rational chord: Difference between revisions
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ArrowHead294 (talk | contribs) m →Least-squares error: Oops, «nowrap» isn't supposed to go inside LaTeX math |
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We can replace the 1 with x, vary x and ask, "By at least how much do the deltas have to be off for any x?" | We can replace the 1 with x, vary x and ask, "By at least how much do the deltas have to be off for any x?" | ||
Rewriting a bit, if 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> has delta signature {{nowrap|+ε<sub>1</sub> +ε<sub>2</sub> ... +ε<sub>''n''</sub>}} (where the chord is written to start on 1, i.e. 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 ''approximating'' delta signature). Then the resulting linear least-squares optimization problem is | Rewriting a bit, if 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> has delta signature {{nowrap|+ε<sub>1</sub> +ε<sub>2</sub> ... +ε<sub>''n''</sub>}} (where the chord is written to start on 1, i.e. 1:{{nowrap|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 ''approximating'' delta signature). Then the resulting linear least-squares optimization problem is | ||
<math> | <math> | ||
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=== Partially DR === | === Partially DR === | ||
For approximating a target delta signature of the form <math>+\delta_1 +? +\delta_3</math> with the chord <math>1:( | For approximating a target delta signature of the form <math>+\delta_1 +? +\delta_3</math> with the chord <math>1:(1 + E_1):(1 + E_2):(1 + E_3)</math> (where the +? is free to vary), the least-squares problem is | ||
<math> | <math> | ||