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 $u\left(y\right)$ in $y$, where $y$ is a function of variables $x_i$, is

where $u\left(x_i\right)$ is the error in $x_i$, and the partials are evaluated at the central values of the $x_i$.

When correlations are included, the formula also involves the covariances $u\left(x_i\,x_j\right)$ between the quantities-with-error.

The covariance $u\left(x_i\,x_j\right)$ can be expressed in terms of the correlation $r\left(x_i\,x_j\right)$ and errors $u\left(x_i\right)$, $u\left(x_j\right)$ as:

where $u\left(x_i\right)$ and $u\left(x_j\right)$ are the errors in $x_i$ and $x_j$.

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.

$\mathrm{with}\left(\mathrm{ScientificErrorAnalysis}\right)\:$

$a≔\mathrm{Quantity}\left(10.\,1.\right)\:$

$b≔\mathrm{Quantity}\left(20.\,1.\right)\:$

$\mathrm{combine}\left(ab\,\mathrm{errors}\right)$

$\mathrm{Quantity}\left(200.\,22.36067977\right)$

$\mathrm{combine}\left(ab\,\mathrm{errors}\,\mathrm{rule}=\mathrm{round}\left[2\right]\right)$

$\mathrm{Quantity}\left(200.\,22.\right)$

$\mathrm{combine}\left(\frac{b}{a}\,\mathrm{errors}\right)$

$\mathrm{Quantity}\left(2.000000000\,0.2236067977\right)$

$\mathrm{SetCorrelation}\left(a\,b\,0.1\right)$

$\mathrm{combine}\left(ab\,\mathrm{errors}\right)$

$\mathrm{Quantity}\left(200.\,23.23790008\right)$

$\mathrm{combine}\left(\frac{b}{a}\,\mathrm{errors}\right)$

$\mathrm{Quantity}\left(2.000000000\,0.2144761059\right)$

$\mathrm{combine}\left(\frac{b}{a}\,\mathrm{errors}\,\mathrm{correlations}=\mathrm{false}\right)$

$\mathrm{Quantity}\left(2.000000000\,0.2236067977\right)$

$\mathrm{with}\left(\mathrm{ScientificConstants}\right)\:$

$\mathrm{e5}≔\mathrm{Constant}\left(h\right)\mathrm{Constant}\left(c\right)a$

$\mathrm{e5}≔\mathrm{Constant}\left(h\right)\mathrm{Constant}\left(c\right)\mathrm{Quantity}\left(10.\,1.\right)$

$\mathrm{combine}\left(\mathrm{e5}\,\mathrm{errors}\right)$

$\mathrm{Quantity}\left(1.986445824\times {10}^{\mathrm{-24}}\,1.986445824\times {10}^{\mathrm{-25}}\right)$

$\mathrm{e6}≔\frac{\mathrm{Constant}\left(m\left[e\right]\right)}{\mathrm{Constant}\left(m\left[p\right]\right)}$

$\mathrm{e6}≔\frac{\mathrm{Constant}\left(m_e\right)}{\mathrm{Constant}\left(m_p\right)}$

$\mathrm{combine}\left(\mathrm{e6}\,\mathrm{errors}\right)$

$\mathrm{Quantity}\left(0.0005446170214\,5.176301769\times {10}^{\mathrm{-14}}\right)$