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## The Error Matrix.

The Minuit processors [MIGrad]MIGRAD and [HESse]HESSE normally produce an error matrix. This matrix is the inverse of the matrix of second derivatives of FCN, transformed if necessary into external coordinate space , and multiplied by the square root of [SET ERRordef]ERRORDEF. Therefore, errors based on the Minuit error matrix take account of all the parameter correlations, but not the non-linearities. That is, from the error matrix alone, two-standard-deviation errors are always exactly twice as big as one-standard-deviation errors.

When the error matrix has been calculated (for example by the successful execution of a command MIGrad or HESse) then the parameter errors printed by Minuit are the square roots of the diagonal elements of this matrix. The commands SHOw COVariance and SHOw CORrelations allow the user to see the off-diagonal elements as well. The command SHOw EIGenvalues causes Minuit to calculate and print out the eigenvalues of the error matrix, which should all be positive if the matrix is positive-definite (see below on Migrad and positive-definiteness).

The effect of correlations on the individual parameter errors can be seen as follows. When parameter `N` is FIXed, Minuit inverts the error matrix, removes the row and column corresponding to parameter `N`, and re-inverts the result. The effect on the errors of the other parameters will in general be to make them smaller, since the component due to the uncertainty in parameter `N` has now been removed. (In the limit that a given parameter is uncorrelated with parameter `N`, its error will not change when parameter `N` is fixed.) However the procedure is not reversible, since Minuit forgets the original error matrix, so if parameter `N` is then RELeased, the error matrix is considered as unknown and has to be recalculated with appropriate commands.

Janne Saarela
Mon Apr 3 15:36:46 METDST 1995