Delta-rational chord: Difference between revisions
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=== Direct least-squares error === | === Direct 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 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 '' | 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 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 | ||
<math> | <math> | ||