Delta-rational chord: Difference between revisions
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<math> x : x + \delta_1 : \cdots : x + \sum_{l=1}^n \delta_l.</math> | <math> x : x + \delta_1 : \cdots : x + \sum_{l=1}^n \delta_l.</math> | ||
We can vary x and ask, "By at least how much (in the linear domain) does the approximating chord have to be off for any x?" When a specific x achieves this minimum, the resulting chord with delta signature {{nowrap|+δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub> | We can vary ''x'' and ask, "By at least how much (in the linear domain) does the approximating chord have to be off for any ''x'' > 0?" When a specific ''x'' > 0 achieves this minimum, the resulting chord with delta signature {{nowrap|+δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub> | ||
}} is taken to be the DR chord that is being approximated. | }} is taken to be the DR chord that is being approximated. | ||
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<math> | <math> | ||
\displaystyle{ \underset{x}{\text{minimize}} \sqrt{\sum_{i=1}^n \Bigg( 1 + \frac{D_i}{x} - f_i \Bigg)^2 } } | \displaystyle{ \underset{x}{\text{minimize}} \sqrt{\sum_{i=1}^n \Bigg( \frac{x + D_i}{x} - f_i \Bigg)^2 } = \underset{x}{\text{minimize}} \sqrt{\sum_{i=1}^n \Bigg( 1 + \frac{D_i}{x} - f_i \Bigg)^2 } } | ||
</math> | </math> | ||