# NAME

Math::Symbolic::Custom::ErrorPropagation - Calculate Gaussian Error Propagation

# SYNOPSIS

use Math::Symbolic qw/parse_from_string/;
use Math::Symbolic::Custom::ErrorPropagation;
# Force is mass times acceleration.
my $force = parse_from_string('m*a');
# The measurements of the acceleration and the mass are prone to
# statistical errors. (Hence have variances themselves.)
# Thus, the variance in the force is:
my $variance = $force->apply_error_propagation('a', 'm');
print $variance;
# prints:
# (
# ((sigma_a ^ 2) * ((partial_derivative(m * a, a)) ^ 2)) +
# ((sigma_m ^ 2) * ((partial_derivative(m * a, m)) ^ 2))
# ) ^ 0.5

# DESCRIPTION

This module extends the functionality of Math::Symbolic by offering facilities to calculate the propagated variance of a function of variables with variances themselves.

The module adds a method to all Math::Symbolic objects.

## $ms_tree->apply_error_propagation( [list of variable names] )

This method does not modify the Math::Symbolic tree itself, but instead calculates and returns its variance based on its variable dependencies which are expected to be passed as arguments to this method in form of a list of variable names.

The variance is returned as a Math::Symbolic tree itself. It is calculated using the Gaussian error propagation formula for uncorrelated variances:

variance( f(x_1, x_2, ..., x_n ) ) =
sqrt(
sum_over_i=1_to_n(
variance(x_i)^2 * (df/dx_i)^2
)
)

In the above formula, the derivatives are partial derivatives and the component variances `variance(x_i)`

are represented as "sigma_x_i" in the resulting formula. (The "x_i" is replaced by the variable name, though.)

Please refer to the SYNOPSIS for an example.

# AUTHOR

Please send feedback, bug reports, and support requests to one of the contributors or the Math::Symbolic mailing list.

List of contributors:

Steffen M�ller, symbolic-module at steffen-mueller dot net

# SEE ALSO

New versions of this module can be found on http://steffen-mueller.net or CPAN.

Math::Symbolic

Math::Symbolic::Custom, Math::Symbolic::Custom::Base,