Lens Equation Error Propagation
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with: (1) Functions of several variables. (2) Evaluation of partial derivatives, and the chain rules of differentiation. (3) Manipulation of summations in algebraic context. At this mathematical level our presentation can be briefer. We error propagation derivative can dispense with the tedious explanations and elaborations of previous chapters. 6.2 THE error propagation formula physics CHAIN RULE AND DETERMINATE ERRORS If a result R = R(x,y,z) is calculated from a number of data quantities, x, using differentials to estimate error y and z, then the relation: [6-1] ∂R ∂R ∂R dR = —— dx + —— dy + —— dz ∂x ∂y ∂z
holds. This is one of the "chain rules" of calculus. percent error calculus This equation has as many terms as there are variables. Then, if the fractional errors are small, the differentials dR, dx, dy and dz may be replaced by the absolute errors ΔR, Δx, Δy, and Δz, and written: [6-2] ∂R ∂R ∂R ΔR ≈ —— Δx + —— Δy + —— Δz ∂x ∂y ∂z Strictly this is no longer an equality, but an approximation toError Propagation Calculus Examples
DR, since the higher order terms in the Taylor expansion have been neglected. So long as the errors are of the order of a few percent or less, this will not matter. This equation is now an error propagation equation. [6-3] Finally, divide equation (6.2) by R: ΔR x ∂R Δx y ∂R Δy z ∂R Δz —— = —————+——— ——+————— R R ∂x x R ∂y y R ∂z z
The factors of the form Δx/x, Δy/y, etc are relative (fractional) errors. This equation shows how the errors in the result depend on the errors in the data. Eq. 6.2 and 6.3 are called the standard form error equations. They are also called determinate error equations, because they are strictly valid for determinate errors (not indeterminate errors). [We'll get to indeterminate errors soon.] The coefficients in Eq. 6.3 of the fractional errors are of the form [(x/R)(∂R/dx)]. These play the very important role of "weighting" factors in the various error terms. At this point numeric values of the relative errors could be substituted into this equation, along with the other measured quantities, x, y, z, to calculate ΔR. Notice the character of the standard form errorbe down. Please try the request again. Your cache administrator is webmaster. Generated Thu, 20 Oct 2016 06:39:51 GMT by s_wx1126 (squid/3.5.20)
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