The combine( expr, errors, opts ) command combines quantities-with-error in a mathematical expression, or in other words, propagates the errors through an expression.

The opts argument can contain one or more of the following equations that determine the behavior:

'rule' = name

If the optional parameter rule=name is given, the rounding rule name is applied to the result of combine. Otherwise, the default rounding rule is used ('digits', or as set by ScientificErrorAnalysis[UseRule]).

'correlations' = true or false

If 'correlations'=true, combine/errors uses correlations defined between the quantities-with-error combined. The default value of 'correlations' is true. If 'correlations'=false, combine/errors ignores any correlations defined between the quantities-with-error.

If no correlations have been directly defined between the quantities-with-error in expr (using ScientificErrorAnalysis[SetCorrelation]), 'correlations'=false does not produce a result different from the default.

'correlations'=false has no effect on further induced error analysis calculations. That is, when combine/errors requires the variance of a quantity-with-error with functional dependence, that calculation is performed using correlations.

The result of combine/errors is a quantity-with-error returned in a Quantity object.

The uncertainty is calculated using the usual formula of error analysis involving a first-order expansion with the variances of the quantities-with-error.

The error  in , where  is a function of variables , is

where  is the error in , and the partials are evaluated at the central values of the .

When correlations are included, the formula also involves the covariances  between the quantities-with-error.

The covariance  can be expressed in terms of the correlation  and errors ,  as:

where  and  are the errors in  and .

ScientificErrorAnalysis[Variance] and ScientificErrorAnalysis[Covariance] are used to calculate the variances and covariances of the quantities-with-error. Thus, any quantity-with-error combined can have functional dependence on other quantities-with-error.