User:Sintel/CTE tuning: Difference between revisions

Line 13:

</math>

where ''g'' is the (unknown) generator list, W the diagonal Tenney-Euclidean weight matrix, ''j'' is the [[JIP]], and V is a matrix obtained by stacking the monzos that we want to be pure. Since this is a convex problem, it can be solved using the method of lagrange multipliers. Let's first simplify:

where ''g'' is the (unknown) generator list, W the diagonal Tenney-Euclidean weight matrix, ''j'' is the [[JIP]], and V is a matrix obtained by stacking the monzos that we want to be pure. This problem is feasible if rank (V) ≤ rank (M).

== Computation ==

Since this is a convex problem, it can be solved using the method of lagrange multipliers. Let's first simplify:

<math>

Line 54:

Line 58:

Where we introduced the vector of lagrange multipliers <math>\lambda</math>, with length equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be ignored.

~~The problem is feasible if rank (V) ≤ rank (M).~~

~~== Computation ==~~

As a standard optimization problem, numerous algorithms exist to solve for this tuning, such as [[Wikipedia: Sequential quadratic programming|sequential quadratic programming]], to name one.

@@ Line 13: / Line 13: @@
 </math>
-where ''g'' is the (unknown) generator list, W the diagonal Tenney-Euclidean weight matrix, ''j'' is the [[JIP]], and V is a matrix obtained by stacking the monzos that we want to be pure. Since this is a convex problem, it can be solved using the method of lagrange multipliers. Let's first simplify:
+where ''g'' is the (unknown) generator list, W the diagonal Tenney-Euclidean weight matrix, ''j'' is the [[JIP]], and V is a matrix obtained by stacking the monzos that we want to be pure. This problem is feasible if rank (V) ≤ rank (M).
+== Computation ==
+Since this is a convex problem, it can be solved using the method of lagrange multipliers. Let's first simplify:
 <math>
@@ Line 54: / Line 58: @@
 Where we introduced the vector of lagrange multipliers <math>\lambda</math>, with length equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be ignored.
-The problem is feasible if rank (V) ≤ rank (M).
-== Computation ==
 As a standard optimization problem, numerous algorithms exist to solve for this tuning, such as [[Wikipedia: Sequential quadratic programming|sequential quadratic programming]], to name one.