Delta-rational chord: Difference between revisions
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=== Measuring the error of an approximation === | === Measuring the error of an approximation === | ||
==== Linear error ==== | ==== Linear error ==== | ||
Say we want the linear error of a chord 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> as an approximation to a fully delta-rational chord with signature +δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub>. Solving the linear least-squares optimization problem | Say we want the linear error of a chord 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> (in increasing order) as an approximation to a fully delta-rational chord with signature +δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub>. Solving the linear least-squares optimization problem | ||
<math> \displaystyle{ \min_\alpha \sum_{i=1}^n \Bigg( \alpha r_i - \alpha - \sum_{l=1}^i \delta_l \Bigg)^2 } </math> | <math> \displaystyle{ \min_\alpha \sum_{i=1}^n \Bigg( \alpha r_i - \alpha - \sum_{l=1}^i \delta_l \Bigg)^2 } </math> | ||