User:Sintel/Generator optimization: Difference between revisions
Created page with "Some derivations on optimal generators for 2-norm (Euclidean) tunings. In what follows, <math>M</math> is an <math>n \times k</math> integer matrix that represents the temper..." |
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\end{gather} | \end{gather} | ||
$$ | $$ | ||
Now if we don't know which intervals to optimize for, a simple choice is just to optimize for each prime in the subgroup. This is equivalent to picking the identity matrix <math>V = I_{k \times k}</math>, so that the above expression simplifies to: | |||
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g = jM^{\mathsf T} (MM^{\mathsf T})^{-1} | g = jM^{\mathsf T} (MM^{\mathsf T})^{-1} | ||
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We can write the | We can write the Lagrangian as: | ||
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\begin{gather} | \begin{gather} | ||
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where we introduced a vector of Lagrange multipliers <math>\lambda</math>. | where we introduced a vector of Lagrange multipliers <math>\lambda</math>. | ||
Taking derivatives with respect to <math>g</math> and <math>\lambda</math> we get: | Taking derivatives with respect to <math>g</math> and <math>\lambda</math> we get a linear system: | ||
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\begin{bmatrix} | \begin{bmatrix} | ||