Delta-rational chord: Difference between revisions

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Suppose that the target delta signature is

<math>+\! \delta_{11} +\! \delta_{12} +\! \cdots +\! \delta_{~~1n_1~~} +\!? +\! \delta_{21} +\! \delta_{22} +\! \cdots +\! \delta_{~~2n_2~~} +\!? \ \cdots +\!? +\!\delta_{m1} +\! \delta_{m2} +\! \cdots +\! \delta_{~~mn_m~~}.</math>

<math>+\! \delta_{1,1} +\! \delta_{1,2} +\! \cdots +\! \delta_{1,n_1} +\!? +\! \delta_{2,1} +\! \delta_{2,2} +\! \cdots +\! \delta_{2,n_2} +\!? \ \cdots +\!? +\!\delta_{m,1} +\! \delta_{m,2} +\! \cdots +\! \delta_{m,n_m}.</math>

Writing ''a:b:c:...'' as [''a'', ''b'', ''c'', ...] for readability, the chord to be approximated is

<math>

[\alpha_1, \alpha_1 + \delta_{11}, \alpha_1 + \delta_{12}, ..., \alpha_1 + \sum_{l=1}^{n_1} \delta_{1l}, \\

[\alpha_1, \alpha_1 + \delta_{1,1}, \alpha_1 + \delta_{1,2}, ..., \alpha_1 + \sum_{l=1}^{n_1} \delta_{1,l}, \\

\alpha_1 + \alpha_2 + \sum_{l=1}^{n_i} \delta_{1l}, \alpha_1 + \alpha_2 + \sum_{l=1}^{n_1} \delta_{1l} + \delta_{21}, ..., \alpha_1 + \alpha_2 + \sum_{l=1}^{n_1} \delta_{1l} + \sum_{l=1}^{n_2} \delta_{2l}, \\

\alpha_1 + \alpha_2 + \sum_{l=1}^{n_i} \delta_{1,l}, \alpha_1 + \alpha_2 + \sum_{l=1}^{n_1} \delta_{1,l} + \delta_{2,1}, ..., \alpha_1 + \alpha_2 + \sum_{l=1}^{n_1} \delta_{1,l} + \sum_{l=1}^{n_2} \delta_{2,l}, \\

..., \\

\alpha_1 + \cdots + \alpha_m + \sum_{i=1}^{m-1} \sum_{l_i=1}^{n_i} \delta_{~~il_i~~} + \delta_{m1}, ..., \alpha_1 + \cdots + \alpha_m + \sum_{i=1}^m \sum_{l_i=1}^{n_i} \delta_{~~il_i~~}].

\alpha_1 + \cdots + \alpha_m + \sum_{i=1}^{m-1} \sum_{l_i=1}^{n_i} \delta_{i,l_i} + \delta_{m,1}, ..., \alpha_1 + \cdots + \alpha_m + \sum_{i=1}^m \sum_{l_i=1}^{n_i} \delta_{i,l_i}].

</math>

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and turning the objective function into a sum of univariate objective functions

<math> \displaystyle{ \sum_{l_i=1}^n \Bigg( \beta_i r_{l_i} - \beta_i - \sum_{{l_i}=1}^i \delta_{l_i} \Bigg)^2. } </math>

<math> \displaystyle{ \sum_{l_i=1}^n \Bigg( \beta_i r_{l_i} - \beta_i - \sum_{{l_i}=1}^i \delta_{i,l_i} \Bigg)^2. } </math>

The Hessian of the objective function is positive-definite, thus the global optimum can be found by setting all individual partial derivatives to zero.

The Hessian of the resulting objective function is positive-definite, thus the global optimum can be found by setting all individual partial derivatives to zero.

== DR and RTT ==

@@ Line 77: / Line 77: @@
 Suppose that the target delta signature is
-<math>+\! \delta_{11} +\! \delta_{12} +\! \cdots +\! \delta_{1n_1} +\!? +\! \delta_{21} +\! \delta_{22} +\! \cdots +\! \delta_{2n_2} +\!? \ \cdots +\!? +\!\delta_{m1} +\! \delta_{m2} +\! \cdots +\! \delta_{mn_m}.</math>
+<math>+\! \delta_{1,1} +\! \delta_{1,2} +\! \cdots +\! \delta_{1,n_1} +\!? +\! \delta_{2,1} +\! \delta_{2,2} +\! \cdots +\! \delta_{2,n_2} +\!? \ \cdots +\!? +\!\delta_{m,1} +\! \delta_{m,2} +\! \cdots +\! \delta_{m,n_m}.</math>
 Writing ''a:b:c:...'' as [''a'', ''b'', ''c'', ...] for readability, the chord to be approximated is
 <math>
-[\alpha_1, \alpha_1 + \delta_{11}, \alpha_1 + \delta_{12}, ..., \alpha_1 + \sum_{l=1}^{n_1} \delta_{1l}, \\
+[\alpha_1, \alpha_1 + \delta_{1,1}, \alpha_1 + \delta_{1,2}, ..., \alpha_1 + \sum_{l=1}^{n_1} \delta_{1,l}, \\
-\alpha_1 + \alpha_2 + \sum_{l=1}^{n_i} \delta_{1l}, \alpha_1 + \alpha_2 + \sum_{l=1}^{n_1} \delta_{1l} + \delta_{21}, ..., \alpha_1 + \alpha_2 + \sum_{l=1}^{n_1} \delta_{1l} + \sum_{l=1}^{n_2} \delta_{2l}, \\
+\alpha_1 + \alpha_2 + \sum_{l=1}^{n_i} \delta_{1,l}, \alpha_1 + \alpha_2 + \sum_{l=1}^{n_1} \delta_{1,l} + \delta_{2,1}, ..., \alpha_1 + \alpha_2 + \sum_{l=1}^{n_1} \delta_{1,l} + \sum_{l=1}^{n_2} \delta_{2,l}, \\
 ..., \\
-\alpha_1 + \cdots + \alpha_m +  \sum_{i=1}^{m-1} \sum_{l_i=1}^{n_i} \delta_{il_i} + \delta_{m1}, ..., \alpha_1 + \cdots + \alpha_m + \sum_{i=1}^m \sum_{l_i=1}^{n_i} \delta_{il_i}].
+\alpha_1 + \cdots + \alpha_m +  \sum_{i=1}^{m-1} \sum_{l_i=1}^{n_i} \delta_{i,l_i} + \delta_{m,1}, ..., \alpha_1 + \cdots + \alpha_m + \sum_{i=1}^m \sum_{l_i=1}^{n_i} \delta_{i,l_i}].
 </math>
@@ Line 104: / Line 104: @@
 and turning the objective function into a sum of univariate objective functions
-<math> \displaystyle{ \sum_{l_i=1}^n \Bigg( \beta_i r_{l_i} - \beta_i - \sum_{{l_i}=1}^i \delta_{l_i} \Bigg)^2. } </math>
+<math> \displaystyle{ \sum_{l_i=1}^n \Bigg( \beta_i r_{l_i} - \beta_i - \sum_{{l_i}=1}^i \delta_{i,l_i} \Bigg)^2. } </math>
-The Hessian of the objective function is positive-definite, thus the global optimum can be found by setting all individual partial derivatives to zero.
+The Hessian of the resulting objective function is positive-definite, thus the global optimum can be found by setting all individual partial derivatives to zero.
 == DR and RTT ==