BOP tuning: Difference between revisions

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First, we note that for any tuning map, call it <math>T</math>, there is an associated vector called a '''signed error map'''. This is given by <math>E = T-J</math>, where <math>J</math> is the JIP. If we assume that we are in a prime-limit, then the signed error map will contain the signed error on each prime. We assume that all errors are unweighted so far.

For any such error map, we can then divide the unweighted error by the desired weighting on each prime to get the weighted error map. This can be viewed as a change of coordinates to a "weighted" coordinate system. Assuming our tuning maps are row vectors, this can be expressed as a matrix right-multiplication by some diagonal weighting matrix <math>W</math>, where the diagonal entries are the desired weights on each prime. We will then denote the weighted error by <math>F = E\cdot W</math>.

For any such error map, we can then divide the unweighted error by the desired weighting on each prime to get the weighted error map. This can be viewed as a change of coordinates to a ''weighted coordinate system''. Assuming our tuning maps are row vectors, this can be expressed as a matrix right-multiplication by some diagonal weighting matrix <math>W</math>, where the diagonal entries are the desired weights on each prime. We will then denote the weighted error by <math>F = E\cdot W</math>.

For any such error map, we can then evaluate the maximum ''unsigned'' weighted error on the primes. This quantity is equal to the maximum of the absolute values of the coefficients of the signed weighted error map, or equivalently the [https://en.wikipedia.org/wiki/Norm_(mathematics)#Maximum_norm_(special_case_of:_infinity_norm,_uniform_norm,_or_supremum_norm) L-inf] norm of the signed error map in the weighted coordinate system.

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 First, we note that for any tuning map, call it <math>T</math>, there is an associated vector called a '''signed error map'''. This is given by <math>E = T-J</math>, where <math>J</math> is the JIP. If we assume that we are in a prime-limit, then the signed error map will contain the signed error on each prime. We assume that all errors are unweighted so far.
-For any such error map, we can then divide the unweighted error by the desired weighting on each prime to get the weighted error map. This can be viewed as a change of coordinates to a "weighted" coordinate system. Assuming our tuning maps are row vectors, this can be expressed as a matrix right-multiplication by some diagonal weighting matrix <math>W</math>, where the diagonal entries are the desired weights on each prime. We will then denote the weighted error by <math>F = E\cdot W</math>.
+For any such error map, we can then divide the unweighted error by the desired weighting on each prime to get the weighted error map. This can be viewed as a change of coordinates to a ''weighted coordinate system''. Assuming our tuning maps are row vectors, this can be expressed as a matrix right-multiplication by some diagonal weighting matrix <math>W</math>, where the diagonal entries are the desired weights on each prime. We will then denote the weighted error by <math>F = E\cdot W</math>.
 For any such error map, we can then evaluate the maximum ''unsigned'' weighted error on the primes. This quantity is equal to the maximum of the absolute values of the coefficients of the signed weighted error map, or equivalently the [https://en.wikipedia.org/wiki/Norm_(mathematics)#Maximum_norm_(special_case_of:_infinity_norm,_uniform_norm,_or_supremum_norm) L-inf] norm of the signed error map in the weighted coordinate system.