ó ¡¼™\c@ sždZddlmZmZddlmZmZddlmZm Z m Z m Z m Z m Z mZddlmZmZmZd„Zd ed„ZdS( s'Tools for arithmetic error propagation.iÿÿÿÿ(tprint_functiontdivision(trepeatt combinations(tStSymboltAddtMultsimplifytPowtexp(t RandomSymboltVariancet CovariancecC s$t|jƒdkr |jdS|S(Ni(tlentargs(tvar((s5lib/python2.7/site-packages/sympy/stats/error_prop.pyttc C s›|j}t|ƒdkr‚||kr1tdƒSt|tƒrPt|ƒjƒSt|tƒrutt|ƒƒjƒStdƒSnt|ƒ}tt t |t ||ƒt ||ƒƒƒ}t|t ƒr9t |Œ}|r—gt |dƒD]1\}}dtt|ƒt|ƒƒjƒ^qì} |t | Œ7}q—n^t|tƒrgt||ƒD]\} } | | d^qX} t|dt | Œƒ}|r—gtt |dƒt |dƒƒD]E\\} } \}}dtt|ƒt|ƒƒjƒ| | ^qº} |t | Œ7}q—n|t|tƒra|d} |d|| |dd} t| ƒ}n6t|tƒr‹t|d|dƒ}n t|ƒ}|S(s¸Symbolically propagates variance (`\sigma^2`) for expressions. This is computed as as seen in [1]_. Parameters ========== expr : Expr A sympy expression to compute the variance for. consts : sequence of Symbols, optional Represents symbols that are known constants in the expr, and thus have zero variance. All symbols not in consts are assumed to be variant. include_covar : bool, optional Flag for whether or not to include covariances, default=False. Returns ======= var_expr : Expr An expression for the total variance of the expr. The variance for the original symbols (e.g. x) are represented via instance of the Variance symbol (e.g. Variance(x)). Examples ======== >>> from sympy import symbols, exp >>> from sympy.stats.error_prop import variance_prop >>> x, y = symbols('x y') >>> variance_prop(x + y) Variance(x) + Variance(y) >>> variance_prop(x * y) x**2*Variance(y) + y**2*Variance(x) >>> variance_prop(exp(2*x)) 4*exp(4*x)*Variance(x) References ========== .. [1] https://en.wikipedia.org/wiki/Propagation_of_uncertainty iii(RRRt isinstanceR R tdoitRtlisttmapt variance_propRRRR t _arg0_or_varRtzipRR R ( texprtconstst include_covarRtnargstvar_argstvar_exprtxtyttermstatvtb((s5lib/python2.7/site-packages/sympy/stats/error_prop.pyR sD.      D0[  N((t__doc__t __future__RRt itertoolsRRtsympyRRRRRR R t sympy.stats.symbolic_probabilityR R R RtFalseR(((s5lib/python2.7/site-packages/sympy/stats/error_prop.pyts 4