combine quantities-with-error in an expression
combine( expr, errors, opts )
expression containing quantities-with-error
(optional) equation(s) of the form option=value, where option is one of 'rule' or 'correlations'; determine behavior
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 u⁡y in y, where y is a function of variables xi, is
where u⁡xi is the error in xi, and the partials are evaluated at the central values of the xi.
When correlations are included, the formula also involves the covariances u⁡xi,xj between the quantities-with-error.
The covariance u⁡xi,xj can be expressed in terms of the correlation r⁡xi,xj and errors u⁡xi, u⁡xj as:
where u⁡xi and u⁡xj are the errors in xi and xj.
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.
a ≔ Quantity⁡10.,1.:
b ≔ Quantity⁡20.,1.:
e5 ≔ Constant⁡h⁢Constant⁡c⁢a
e6 ≔ Constant⁡meConstant⁡mp
Overview of ScientificErrorAnalysis
ScientificErrorAnalysis and ScientificConstants
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