Delta-rational chord: Difference between revisions

Line 72:

which can be plugged back into the error formula to obtain the error. (We multiply the 1:''r''1:''r''2:...:''r''''n'' chord by α in order to compare it to the target DR chord on the same isodifferential series.)

~~<!--~~

==== Partially DR ====

When the DR signature has one or more free ("+?") terms, the optimization problem becomes a multivariate one: we have one variable <math>\alpha_i, i \ge 2</math> for each free term, as well as the variable <math>\alpha_1</math> for the root. However, solving it is not much more difficult than the univariate case.

Line 88:

\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>

~~We treat the DR chord as a union of fully DR chords~~

~~<math> \beta_i : \beta_i + \delta_{i,1} : \cdots : \beta_i + \sum_{l_i=1}^{n_i} \delta_{i,l_i}, </math>~~

~~each chord being approximated by~~

~~<math> \beta_i : \beta_ir_{i,1} : \cdots : \beta_i r_{i,n_i}. </math>~~

~~Here we require that each <math>\beta_i</math> be above the previous chord.~~

~~This turns the objective function into a sum of univariate objective functions~~

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

~~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 72: / Line 72: @@
 which can be plugged back into the error formula to obtain the error. (We multiply the 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> chord by α in order to compare it to the target DR chord on the same isodifferential series.)
-<!--
 ==== Partially DR ====
 When the DR signature has one or more free ("+?") terms, the optimization problem becomes a multivariate one: we have one variable <math>\alpha_i, i \ge 2</math> for each free term, as well as the variable <math>\alpha_1</math> for the root. However, solving it is not much more difficult than the univariate case.
@@ Line 88: / Line 88: @@
 \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>
-We treat the DR chord as a union of fully DR chords
-<math> \beta_i : \beta_i + \delta_{i,1} : \cdots : \beta_i + \sum_{l_i=1}^{n_i} \delta_{i,l_i}, </math>
-each chord being approximated by
-<math> \beta_i : \beta_ir_{i,1} : \cdots : \beta_i r_{i,n_i}. </math>
-Here we require that each <math>\beta_i</math> be above the previous chord.
-This turns the objective function into a sum of univariate objective functions
-<math> \displaystyle{ \sum_{k_i=1}^{n_i} \Bigg( \beta_i r_{i,k_i} - \beta_i - \sum_{{l_i}=1}^i \delta_{i,l_i} \Bigg)^2. } </math>
-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 ==